LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES
Series Editors
H. G. Dales Peter M. Neumann
LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES Previous volumes of the LMS Monographs were published by Academic Press, to whom all enquiries should be addressed. Volumes in the New Series will be published by Oxford University Press throughout the world.
NEW SERIES 1. Diophantine Inequalities R. C. Baker 2. The Schur Multiplier Gregory Karpilovsky 3. Existentially Closed Groups Graham Highman and Elizabeth Scott 4. The Asymptotic Solution of Linear Differential Systems M. S. P. Eastham 5. The Restricted Bumside Problem Michael Vaughan-Lee 6. Pluripotential Theory Maciej Klimek 7. Free Lie Algebras Christopher Reutenauer 8. The Restricted Burnside Problem 2nd edition Michael Vaughan-Lee 9. The Geometry of Topological Stability Andrew du Plessis and Terry Wall
10. Spectral Decompositions and Analytic Sheaves J. Eschmeier and M. Putinar 11. An Atlas of Brauer Characters C. Jansen, K. Lux, R. Parker, and R. Wilson 12. Fundamentals of Semigroup Theory John M. Howie 13. Area, Lattice Points, and Exponential Sums M. N. Huxley 14. Super-Real Fields H. G. Dales and W. H. Woodin
15. Integrability, Self-Duality, and Twistor Theory L. Mason and N. M. J. Woodhouse 16. Categories of Symmetries and Infinite-Dimensional Groups Yu. A. Neretin
Area, Lattice Points, and Exponential Sums M. N. Huxley College of Cardiff University of Wales
CLARENDON PRESS 1996
OXFORD
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Library of Congress Cataloging in Publication Data Huxley, M. N. (Martin Neil) Area, lattice points, and exponential sums/M. N. Huxley. (London Mathematical Society monographs; new ser., 13) Includes bibliographical references and index. 1. Exponential sums. 2. Lattice theory. 3. Functions, Zeta. I. Title. II. Series: London Mathematical Society monographs; new ser., no. 13. 1996 512'.73--dc20 QA246.7.H89 95-38370
ISBN 0 19 853466 3 Typeset by Technical Typesetting Ireland Printed in Great Britain by Biddies Ltd., Guildford, Surrey
To Trish, Nicholas, and Clare
Contents
Notation
xi
Introduction
1
Part I Elementary methods The rational line 1.1 Height 1.2 The Farey sequence 1.3 Lattices and the modular group 1.4 Uniform distribution 1.5 Approximating a given real number 1.6 Uniform Diophantine approximation
1.
Polygons and area 2.1 Counting squares 2.2 Jarnik's polygon 2.3 The discrepancy of a polygon 2.4 Fitting a polygon to a smooth curve 2.
The integer points close to a curve 3.1 Introduction 3.2 The reduction step 3.3 The iteration 3.4 Swinnerton-Dyer's method: the convex polygon 3.5 Swinnerton-Dyer's method: counting quadruplets 3.6 Expanding the result 3.7 Points on the curve 3.
The rational points close to a curve 4.1 Major and minor sides of the polygon 4.2 Duality 4.3 A linear form often near an integer 4.
Part II The Bombieri-Iwaniec method Analytic lemmas 5.1 Bounds for exponential integrals 5.
87 87
Contents
viii
5.2 Partial summation for exponential sums 5.3 Rounding error sums 5.4 Poisson summation 5.5 Evaluating exponential integrals 5.6 Bilinear and mean square bounds
89 93 98 104 114
Mean value results 6.1 The simple exponential sum 6.2 The lattice point discrepancy 6.3 The mean square discrepancy
126 126 128
The simple exponential sum 7.1 Motivation 7.2 Major and minor arcs 7.3 Poisson summation 7.4 The large sieve on the minor arcs 7.5 A preliminary calculation 7.6 Long major arcs
142
8.
The exponential sum for the lattice point problem 8.1 Major and minor arcs 8.2 The major arcs estimate 8.3 Poisson summation on the minor arcs 8.4 The large sieve on the minor arcs
167 167 170 172 178
Exponential sums with a difference 9.1 Major and minor arcs 9.2 The major arcs estimate 9.3 Poisson summation on the minor arcs 9.4 The large sieve on the minor arcs
185 185 187 188 193
6.
7.
9.
Exponential sums with modular form coefficients 10.1 Modular forms 10.2 The Wilton summation formula 10.3 Farey arcs 10.4 Wilton summation on the Farey arcs 10.5 A large sieve on the Farey arcs 10.6 Towards a mean square result 10.7 Jutila's third method 10.
135
142 144 149 152 158 160
197 197
200 208 210 213 220 225
Part III The First Spacing Problem: `integer' vectors The ruled surface method 11.1 Preparation: divisor functions 11.2 Families of solutions 11.3 A comparison argument 11.
235 235 241 249
Contents
ix
The Hardy-Littlewood method 12.1 Integrals that count 12.2 The minor arcs 12.3 The major arcs 12.4 Extrapolation
255
The First Spacing Problem for the double sum 13.1 Families of solutions 13.2 Good and bad families 13.3 The problem with a perturbing term
272
12
13.
255 259 262 270
272 278 283
Part IV The Second Spacing Problem: `rational' vectors The First and Second Conditions 14.1 The Coincidence Conditions 14.2 Magic matrices 14.3 The Second Condition 14.4 A family of sums
287
Consecutive minor arcs 15.1 Parametrizing rational points 15.2 Coincidence over a short interval 15.3 Linearizing the Fourth Condition 15.4 Coincidence over a long interval 15.5 Extending the Taylor series
307
The Third and Fourth Conditions 16.1 Counting coincident pairs of minor arcs 16.2 Sums with congruence conditions 16.3 Eliminating the centres of the arcs
329
14.
15.
16.
287 289 295 300
307 311 315 320 323
329 336 339
Part V Results and applications Exponential sum theorems 17.1 The simple exponential sum 17.2 Exponential sums with a parameter 17.3 A congruence family of sums 17.4 Sums with T large
349
Lattice points and area 18.1 Exponential sums 18.2 Integer points and rounding error 18.3 Lattice points inside a closed curve 18.4 A family of lattice point problems 18.5 Rounding error and integration
372
17.
18.
349 357 361 363
372 377 384 393 397
Contents
x
Further results 19.1 Exponential sums with a difference 19.2 A major arc estimate 19.3 Exponential sums with a large second derivative
406
Sums with modular form coefficients 20.1 Exponential sums 20.2 Mean value theorems 20.3 The modular form L-function
423
21. Applications to the Riemann zeta function
438
21.1 Introduction 21.2 The order of magnitude in the critical strip 21.3 The mean square 21.4 Gaps between zeros 21.5 The twelfth-power moment
438 441 443 447 451
22. An application to number theory: prime integer points 22.1 Preparation 22.2 Type I double sums 22.3 Type II double sums 22.4 Prime numbers in a smooth sequence
452
19.
20.
406 413 415
423 429 435
452 456 459 463
Part VI Related work and further ideas 23. Related work
467
23.1 Integer points close to a curve 23.2 The Hardy-Littlewood method 23.3 Other Farey arc arguments 23.4 Higher dimensions 23.5 Exponential sums with monomials
467 470 471 473 475
Further ideas 24.1 Comments on the method 24.2 Subdivision without absolute values
478
References
484
Index
491
24.
478 479
Notation GENERAL The lower case letters a to w denote integers unless stated otherwise. The letter p always denotes a prime number, and q always denotes a positive integer, usually the denominator of a rational number. The exceptions are as follows:
dy/dx etc., are derivatives. e and i are used for the usual mathematical constants when these occur. r, s, and t are used for variables where the notation is traditional: r for the
polar coordinate; s for arc length and for an exponent, and in Fourier transform formulae as in Dirichlet series; t for a real variable, usually `time', sometimes the imaginary part of a complex exponent s = o, + it; u and v are sometimes used for variables. x and y are always real variables, and z is a complex variable. Upper case letters are used for sets, matrices, and for real numbers (which may be integers). Coefficients (functions of one or more integer variables) are
usually denoted a(... ), b(... ), or c(... ). Functions of one or more real or complex variables are e(... ), f (... ), F(... ), g(... ), G(... ), h(... ), H(... ), k(... ), or K(...) and sometimes u(... ), v(... ), and w(... ). Greek letters are used for real and complex numbers and variables.
Theorems and lemmas are numbered uniquely. Theorem 18.3.2 is the second result in Chapter 18, Section 18.3. The equations in its proof are numbered (18.3.7) onwards, indicating that they occur in Section 18.3 of Chapter 18.
NUMBER-THEORETIC NOTATION The relation m I n (`m divides n (without remainder)') between m and n
means that m is positive and that n is a multiple of m. The relation a = /3(mod m) (`a is congruent to /3 modulo m') between real numbers, usually integers, means that a - /3 is an integer n and that m I n. For integers m and n, (m, n) denotes the highest common factor (defined for m and n not both zero) and [m, n] denotes the lowest common multiple (defined for m and n both non-zero).
The usual functions dr(n), µ(n), cp(n), and o,(n) of number theory are defined as follows:
Notation
xii
The number of ways of writing n as a product of r positive integers is called dr(n); we write d(n) for d2(n), the number of factors of n. The Mobius function µ(n) is zero if n has a repeated prime factor, 1 if n = 1 or if n has an even number of prime factors, all distinct, and is - 1 if n has an odd number of prime factors, all distinct. The Euler function cp(n) counts the number of expressions 1/n, 2/n,..., n/n which are in their lowest terms. The sum of the factors of n is called v(n). For a real number x, [x] is the integer part of x, the unique integer n in x - 1 < n <x, [[x]] is the nearest integer to x, the unique integer in x - z < n5 x + 2(W) is x - [[x]], depending only on x modulo one, and 114 is the
distance of x from the nearest integer, the absolute value of ((x)). The rounding error or row-of-teeth function is
p(x) = [x] -x+ i = - ((x - 2)). ANALYTIC NOTATION We write
e(x) = exp27rix = exp2iri((x)), a complex exponential that maps the reals modulo one to the unit circle. We use the following order-of-magnitude notation. The Vinogradov symbol
F << G means that G is positive and that the ratio F/G is bounded. The Landau symbol O(G) stands for any expression F satisfying F << G. The
relation F x G means that F << G and G «F, so that F and G share the same order of magnitude.
Introduction
This book is written from the standpoint of analytic number theory, a subject
whose name may seem contradictory. Number theory deals with the behaviour of the integers under addition and multiplication, and analysis with continuity and limit operations. The logicians have told us that any problem
in mathematics can be disguised as finding integer solutions of a set of simultaneous quadratic equations. This means that we must be prepared to use any method that comes to hand. We may classify problems in number theory only by the methods used, not by the form in which they are stated. An argument is algebraic if it uses only addition and multiplication, and analytic if it uses the ordering of the integers, or the ordering of the rational numbers within the real line, which is a consequence of the ordering of the integers. From this point of view `elementary number theory' is not a third category, but a description of the early levels. It is in this sense that the first part of this book is `elementary'. The typical question in analytic number theory is to give an approximate formula for the number of sets of integers satisfying certain conditions and
then to investigate how accurate the formula is, in terms of the size of the integers involved. A very simple example is the proof that there are infinitely many prime numbers which begins by showing that at least half the positive integers have no repeated prime factor. If there were only finitely many prime numbers, then there would be only a finite number of integers which were products of distinct prime numbers.
The central theme of this book is finding the area within a curve by `counting squares'. When a curve in the plane with area A is expanded by a factor M, on average there will be AM' points within the expanded curve with the x and y coordinates both integers. This property can be used as the
definition of area, or as a practical method of estimating the area. The accuracy certainly depends on the shape of the curve. For a smooth curve the root mean square accuracy is, to order of magnitude, the square root of M.
What is the maximum error of the `counting squares' method, for any position of the curve relative to the coordinate axes? Related questions are: how many integer points can lie on or very close to the curve, and whether the distances by which the curve misses the integer points are random. The integer points form a periodic system. Applying Fourier theory to the curve y=f(x) produces sums like Eexp2wif(n), the exponential sums of the title. There are three methods for estimating the size of the error in counting
2
Introduction
squares, and we shall touch on each of them. The geometric method is to
approximate the curve by a polygon, for which the error terms can be described explicitly. The symmetric analytic method is to take a two-dimensional Fourier series, whose convergence gave Hardy pause for thought even in the case of a circle. In the asymmetric analytic method we divide the curve
into quadrants. In each quadrant we sum over one coordinate, and use Poisson summation in the other (in the form of the Fourier series for the row-of-teeth function). All three methods draw our attention to the points on the curve where the gradient is rational. Just as the real line is the closure of the rational numbers, so a simple convex closed curve is the envelope of its tangents in rational directions.
The two analytic methods lead to exponential sums, that is, sums of complex numbers whose modulus is one (or slowly varying), but whose argument changes rapidly. In harmonic analysis a long sum corresponds to locating a real number in a short interval. A powerful technique in counting arguments is to `divide and conquer': to divide the region of search into smaller sets, then find an approximation to the conditions required that is valid on a small set, then combine the information from each small set, in some non-trivial manner. When this idea is applied to exponential sums, then
the approximation step uses Cauchy's inequality to pass from a complex number to its modulus, and the combining step involves expanding out the modulus squared as a multiple sum. Bombieri and Iwaniec (1986a), investigating the Riemann zeta function,
found a highly non-trivial divide-and-conquer method for estimating an exponential sum in one variable. The dividing and approximating steps were anticipated by Jutila (1985 et seq.), investigating the Dirichlet series whose
coefficients are the Fourier coefficients of modular forms. Iwaniec and Mozzochi (1988) extended the method to the double exponential sum in two special but famous lattice point problems of Gauss (counting squares for the circle) and Dirichlet (counting squares for the rectangular hyperbola). The author and Watt (1988) and Kolesnik (in Graham and Kolesnik 1991) saw that the method would generalize, and have since given various improvements. Heath-Brown and the author (1990) adapted the double sum version
to estimate the mean square of an exponential sum in one variable. The mean square of a lattice point sum can be treated this way also, but the symmetric analytic method (Chapter 6) gives better results. The new method is easiest to comprehend for the lattice point problems.
The curve is divided into short arcs, each containing a point at which the gradient is a suitable rational number. The approximating step is asymmetric in the coordinate directions, with Poisson summation in the second variable (y) followed by approximating y as a Taylor polynomial in x. The leading term corresponds to the rational gradient. Poisson summation now follows
first in x and then in y. The two Poisson summations in y do not simply cancel. There is a twisting factor corresponding to the rational gradient. The region of summation is awkward, and several steps are taken to neaten it.
Introduction
3
The combining step is beyond the wit of any one of us, and there is probably only one two-element set, the union of whose members could have discovered it, namely Bombieri and Iwaniec. The sums corresponding to the
short arcs can be estimated in mean fourth power by a four-fold integral. When the fourth-power modulus is expanded as a multiple sum, then terms in
which the summands are equal in pairs, or nearly so, dominate the sum. Bombieri and Iwaniec express this idea using the large sieve, an inequality which goes straight from the discrete sum of fourth powers to counting near-coincidences in combinations of summands (the integral in between occurs in some proofs of the sieve inequalities). Bombieri and Iwaniec need the large sieve in a form more general than any before. There are two problems with this method. The First Spacing Problem is to count the number of near-coincidences in the combinations of summands on the upper bound side of the larger sieve. It has a look about it of Waring's Problem on expressing positive integer as a sum of kth powers of integers, but here they are powers of square roots of integers. The Second Spacing Problem arises when we compare the fourth-power sum over short arcs with a fourth-power integral. There may be resonances among the short arcs, with different short arcs giving very similar exponential sums. To show that there are not too many resonances is like counting integer points close to a surface in several dimensions. The final answer, that the counting squares estimate for a nice smooth curve is accurate up to order O(M46/73+E) for any e> 0, is complicated, partly because the argument itself is complicated, and partly because the two main parameters H and N are each chosen to equalize two or more terms in the upper bound. There is even some moonshine: the Fourier integral in the last Poisson
summation is an incomplete Bessel function integral. For modular form coefficients it has integral order, for the lattice point problem it has order one-half, and for the simple exponential sum, order one-third. Bessel functions of integral order occur in the symmetric analytic method, but only for the special curves, the ellipse and hyperbola. Bessel functions arise in the representations of matrix groups, but order one-third is unexpected. This book is accessible to graduate students beginning research. The reader is assumed to know some real and complex analysis, and Fourier theory up to the function (sin 7rx)/Trx. Some ideas from plane geometry are used, such as the radius of curvature, and the duality of points and lines. For the applications to modular forms, prime numbers, and the Riemann zeta function, standard theorems are quoted. No special knowledge of number theory is required. All technical lemmas on exponential integrals or on the average number of factors of integers that are needed are proved. This book can be read as an introduction to research in analytic number theory. There
are no set exercises. The researcher should keep asking: `What is this about?'; `How is it done?'; `Why is it done this way?'; `Can't you use instead ... ?'; `Isn't this the same as ... T.
Introduction
4
The expert will find the components of the new method presented in one
place for the first time, with suggestions as to how it relates to other approaches, and what direction it might take in the future. The results for sums in one variable in Chapters 17, 18 and 19 have been summarized for ease of reference as tables of exponents. Chapter 5 collects the most useful lemmas on exponential sums and integrals of one variable. A quick impression of the new method may be taken by reading Chapter 7 on the simple exponential sum, and Chapters 11 and 14 which begin the discussion of the two spacing problems. The list of references includes all the recent work (to 1993), and all the sources consulted in preparing this book, whether cited in the text or not. Thanks to Voronoi, van der Corput, and I. M. Vinogradov, without whom twentieth-century number theory would not be the same; to Davenport, a better teacher than I will ever be; to Bombieri, Hooley, Rankin, Selberg, and Swinnerton-Dyer for help and inspiration; to K. Ramachandra and the Tata Institute of Fundamental Research for the opportunity to work on this book; and to S. W. Graham, D. R. Heath-Brown, A. Ivic, H. Iwaniec, M. Jutila, G. Kolesnik, E. Kratzel, H. L. Montgomery, Y. Motohashi, C. J. Mozzochi, W. G. Nowak, 0. Trifonov, N. Watt, and everyone else who supplied ideas and comments.
Part I Elementary methods,
1
The rational line 1.1
HEIGHT
The rational numbers form a countable set, dense in the real numbers. They are set in the real line like stars in the sky `to illuminate the mystery of the
continuum' (a remark attributed to Borel). In any region, if magnified sufficiently, faint stars appear. To explain this bright and faint analogy, we need the concept of height.
A rational number a is uniquely represented as a fraction a/q in its lowest terms, that is, by a pair of integers a and q, with q positive and the highest common factor (a, q) equal to one. The height is defined by
H(a) = max(IaJ, q) There are three rational numbers of height one: 0, 1 and - 1 (unless infinity is counted as 1/0), and for each H z 2, the number of fractions of height H is 4cp(H), where cp(H) is Euler's function. The rational numbers are partially
ordered by height, and totally ordered when we put an ordering on the rationale of fixed height, for instance their ordering as real numbers. Each rational number of height H appears in the ordering in a position H
53+4F, (h-1)=2H2-2H+3. h=1
Thus the rational numbers are countable. We use the expressions `small rational number' and `large rational number' to mean rational numbers whose height is small or large. This usage agrees with the common expression `a small integer'. The small rationale are the
bright stars in the analogy, the large rationale are the faint stars, and the irrational numbers form the background darkness of the continuum. The height gives a metric on the positive rational numbers with distance function defined by
d(a, /3) = log H(a//3 ), so that the translation identity
d(ay,lay)=d(a,a), and the triangle inequality d(a, y) < d(a, /3) + d(13, y)
The rational line
8
hold. Any two distinct rationals are a distance at least log 2 apart. It is fairly easy to see that any set of n distinct positive rationals contains a pair which is at least log p apart, where p is the largest prime up to n. Zaharescu (1987), answering a question of R. L. Graham, has shown that for large n there is a
pair which is at least log n apart. This result is probably true for all n (to verify it for small n means checking a finite list of possible exceptional sets). This chapter is about the interaction of the height and the metric I a - '61
on the real line. In later chapters we shall use the structure of sets of rationals of bounded height close to a given real number. To a first approximation the height distance and the real distance are independent, but there is an analogue of gravitational lensing, in which the images of faint stars are displaced away from a nearby bright star.
1.2 THE FAREY SEQUENCE The Farey sequence of order Q (in the strict sense) consists of the rational numbers of height at most Q between 0 and 1, arranged in their order as real numbers. We usually use the Farey sequence in contexts where only the fractional part of a real number matters. The extended Farey sequence of order Q consists of all the rational numbers with denominator less than or
equal to Q, the lifting of the strict Farey sequence from R/71, the reals modulo integers, to the real numbers R. We write .9(Q) for the extended Farey sequence.
Although .9(Q) is a subsequence of .9(Q + 1), the numbering of terms differs, and consecutive terms of .9(Q) may be separated in .9(Q + 1). Farey
was one of the first to notice the two properties in the next lemma, and brought them to the attention of the great Cauchy. Lemma 1.2.1 (Basic properties of the Farey sequence)
(a) If e/r and f/s are consecutive terms in .9(Q), then
f/s - e/r = 1/rs,
ft - es = 1.
(b) If e/r and f/s are consecutive terms in .9(Q), then they remain consecutive in .A(R) for Q ::g R < r + s, but are separated in .9(r + s) by the Farey
mediant (e + f)/(r + s). Proof First we show that property (a) implies property (b). If fr - es = 1, and
a/q lies between e/r and f/s, then
a/q - e/r = (ar - eq)/qr > 1/qr, and
f/s - a/q = (fq - as)/qs Z 1/qs.
The Farey sequence
9
Adding, we have
1/rs = f/s - e/r>- 1/qr+ 1/qs, so that
q>-r+s. The Farey mediant (e + f)/(r + s) does lie between e/r and f/s. We find that
f/s - (e + f)/(r+s) =1/s(r+s), (e +f )I(r + s) - e/r = 1/r(r + s), and thus no other rational number with denominator r + s can lie within the open interval (e/r, f/s). This calculation verifies property (a) for the pairs e/r, (e + f)/(r + s) and (e + f)/(r + s), f/s, which are consecutive in .3(r + s), and implies that (e + f)/(r + s) is in its lowest terms.
By induction we see that we can construct the whole Farey sequence between 0 and 1 from the consecutive pair 0/1, 1/1 in .9(1) by repeatedly taking Farey mediants, so property (a) holds throughout. If 1/0 counts as a D rational, then 1/1 is itself the Farey mediant of 0/1 and 1/0. Definition We say that e/r and f/s are consecutive Farey fractions if they are consecutive fractions of some Farey sequence .9(Q), for instance with
Q=r+s- 1.
Our next lemma describes the set of rationale between two consecutive Farey fractions. Lemma 1.2.2 (Self-similarity) If e/r, f/s are consecutive Farey fractions, then for Q r + s the members of .9(Q) lying between e/r and f/s are the fractions (eu + ft)/(ru + st) with t z 1, u z 1, (t, u) =1 and ru + st s Q.
Proof By the inductive construction of the Farey sequence in the proof of Lemma 1.2.1, all the members of .9(Q) between e/r and f/s can be written as (eu + ft)/(ru + st). Conversely for t > 1, u z 1
f/s - (eu +ft)/(ru + st) =u/s(ru + st), (eu + ft)/(ru + st) - e/r = t/r(ru + st). If (t, u) = 1, then there are integers v and w with Iw
ul-1'
so that
ew+fv rw+sv
eu+ftl ru+st -
f l
s
elly r 11w
t1=1 u
implying that the fraction (eu + ft)/(ru + st) is in its lowest terms.
The rational line
10
We often need an upper bound for the number of members of .g(Q) in a short interval. Lemma 1.2.3 (Short interval upper bounds) Let I be an interval of length A. Then
E
150Q2+1,
a/q e.9(Q)n 1 1
-
Proof The members of AQ) are at least 1/Q2 apart, so that if there are N Farey fractions in the interval, then
(N - 1)/Q2 5 A. Moreover the distance from a/q to the next Farey fraction is at least 1/qQ, so that
*
1
a/geY(Q)nl qQ
the star indicating that the last fraction in the interval is omitted. The missing 0 term is at most 1/Q. The extended Farey sequence makes a surprising appearance in the follow-
ing construction (Ford 1938). First take a straight line and a sequence of equal circles, each touching the straight line and the two adjacent circles in the sequence (Fig. 1.1). For each successive step, between each pair of circles
in the sequence, draw the circle that touches the straight line and the two circles externally, and insert this circle into the sequence.
If the original circles have unit diameter, then the typical circle has diameter 1/q2, and touches the straight line at the Farey point a/q.
FIG. 1.1
Lattices and the modular group 1.3
11
LATTICES AND THE MODULAR GROUP
A lattice in a finite-dimensional real vector space consists of all integer combinations of a fixed set of vectors which form a basis for the space. Algebraically the lattice vectors form a torsion-free 7L-module on d generators, where d is the dimension of the space. Geometrically the lattice points are the points whose position vectors are lattice vectors, and the lattice lines are the lines through these points in the direction of the lattice vectors. In two dimensions with basis vectors (1, 0) and (0,1), the lattice points are the integer points, and the lattice vectors are the integer vectors. We identify the rational numbers with the gradients of integer vectors,.and we call an integer vector primitive if it cannot be written as mw where m > 2 and w is also an integer vector. Each rational number a/q corresponds to a pair of primitive integer vectors (q, a) and (-q, -a), and each primitive integer vector (x, y), except (0, ± 1), corresponds to a rational number y/x. An infinite gradient can be regarded as the rational number 1/0. We consider the parallelogram in Figure 1.2 with vertices 0 (0, 0), A (r, e), B (r + s, e + f) and C (s, f ). The area of the parallelogram is fr - es. Lemma 1.2.1 corresponds to the fact that when OABC has area one, then there is no other lattice point within the closed parallelogram OABC, and we can tile the plane with copies of OABC translated by every integer vector (m, n). The
lattice with basis vectors OA, OC has the same lattice points, but its lattice lines are in different directions. Changing the basis without changing the lattice points corresponds to an automorphism of the group of vectors given
by the matrix I sf
er).). The full automorphism group consists of all integer
matrices with determinant ±1. We want areas to be preserved, not reflected, so the order of the basis vectors matters, and we use matrices of determinant +1, the group SL(2, Also, ,since rational numbers form the gradients of
vectors, the matrix I -o (
_° I acts trivially. The group which affects the
FIG. 1.2
The rational line
12
gradients of vectors is PSL(2, Z), called the modular group, the quotient of SL(2, Z) by its centre ( ±I), represented by 2 X 2 integer matrices of determinant one, but with I _d) and I a d )) both representing the same group element. ` ` The matrices of the modular group are classified by the trace a + d, which is invariant under conjugacy. Since we can change signs, we take the trace to be non-negative. Matrices with trace less than two are called elliptic, those with trace two parabolic, elations, shears or transuections, those with trace
greater than two hyperbolic. The modular group acts as rigid motions on two-dimensional hyperbolic space. In this action a hyperbolic matrix is a rigid motion along an axis, an elliptic matrix is a rotation about a finite point, and a transvection is a rotation about a point at infinity. Conjugacy corresponds to changing the origin of coordinates. This geometric action is parametrized by the action
ay
as+b ca+d
on the complex numbers. The real axis parametrizes the directions at infinity. We interpret Lemma 1.2.2 as mapping the positive rationals t/u one-to-one
onto the rationals between e/r and f/s by the matrix I Sf er ). In Lemma 1.2.1 we have
r+s) = (s r)(0 1)' e+f e f e 1 0). (r +s r) (s r)(1 1 (s
The semigroup of matrices in SL(2, 71) with non-negative entries is generated
by the transvections T =
(10
1) and U = (1
o
). These matrices T and U
generate the full modular group. The usual generators are S and T, where
S= (0
-10) =T_1UT-i
1.4 UNIFORM DISTRIBUTION The uniform distribution of a sequence of numbers is a basic concept in analytic number theory. Dirichlet proved his prime number theorem by showing that the sequence of prime numbers was uniformly distributed among the cp(q) possible arithmetic progressions modulo q. In this book we
usually consider the sequence of real numbers x =f(n), where f(x) is a smooth function. We expect x,, to be uniformly distributed modulo one.
Uniform distribution
13
Definition A sequence of real numbers xn is uniformly distributed modulo one if, for every e > 0, however small, there is an N(e) such that, for every M>_ N(e), and every subinterval I of the unit interval [0, 1], then the number M(1) of integers n in 1, ... , M, whose fractional parts x,, - [ xn ] lie in I, differs from M times the length of the interval I by at most Me. We call the difference M(I) - M I I I the discrepancy of the interval I. Uniformity conditions are required when one approximate process or limit operation occurs inside another one. If g(x) is a continuous function satisfying the periodic condition g(x + 1) = g(x), then the uniform distribution of x,, modulo implies that
M M n=1
1
g(x)
(1.4.1) 0
as M tends to infinity. Periodic functions belong to Fourier theory, and Hermann Weyl showed that a sequence x,, is uniformly distributed if and only if (1.4.1) holds for the trigonometric functions sin 21rhx and cos 2.7rhx for
each integer h. We use the notation e(x) = exp 21r i x
for complex exponentials (when x is a rational a/q, then e(a/q) is a qth root of unity). In this notation the Weyl criterion becomes M
F, e(hx,,)->0 n=1
as M tends to infinity for each non-zero integer h. Weyl's criterion for uniform distribution will be proved in Lemma 5.3.3. In practice Weyl bequeathed us the principle that one may average away a variable, provided that certain exponential sums are small enough; what
bound is actually required for the exponential sum, that depends on the context. In this section we test the Weyl criterion for the rational numbers in (0, 1], enumerated as in Section 1.1.
Lemma 1.4.1 (Ramanujan's sum)
F e Aqq
a=1 (a,q)=1
clh clq
cµ(
where µ(n) is the Mobius function. Proof Since
E µ(d)=0 din
The rational line
14
if n is greater than one, and the factors of the highest common factor (a, q) are just the common factors of a and q, the sum required is q
E
q1
a=1 din
q/d
b=1
dlq
where we have written a = bd. 1 he sum over b is zero unless h is a multiple of q/d so, writing c = q/d, we get /
F, µ(d) d = E F, cµ(d) _ > cp q clh
dIq
q/dlh
clh clq
d
cd=q
1 ).
`
Lemma 1.4.2 (Weyl's sum for the rational numbers) Let Q be a positive integer, and let x1, ... , x v be the rational numbers of height at most Q in (0,11. Then
N= 2Q2) +O(QlogQ)=
3Q2
+O(QlogQ)
and for each integer h N
dM(Q/d),
E e(hxa) dlh
1
where M(x) = Ea
x
µ(n) is the sum function of the Mobius function.
Proof For the first assertion, by Lemma 1.4.1 with h = 0, Q
q=1
Q
q
N= >
1= E F, F, cµ(d)
F, a=1 (a,q)=1
q=1
c
d
cd=q
Q
_ E E d µ(d) _ E
µd d)
d5Q
q=1 dIq
Q
E
q
q=1 q = 0 (mod d)
p(d) d
d5Q Q2
°r°
E
2 d=1
2 1
Q2 + O(Q) d
u(d) + d
2
O
I µ(d)1
Q2
d=Q+1
d
2
+ Q X-
dsQ
I µ(d)I d
The error terms are O(Q log Q) in order of magnitude. The infinite series is
1/i(s) at s = 2, and (2) takes the special value ire/6.
Approximating a given real number
15
For the second assertion, the sum can be rearranged to give
E
q=1
F, a=1 (a,q)=1
F, EdjA
elah q
d
q=1 dlh
dlq
Ed dlh
µ(r). r5Q/d
Lemma 1.4.2 gives the Weyl sum in the case when the enumeration x1, ..., XN corresponds to the complete set of rationals of height at most Q. If N is such that xN has height Q, but so does xN+1, then we must add O(Q) to the results of Lemma 1.4.2 to correct for the terms omitted. Since I M(x)I <x, we have, for non-zero h, N
Q << d(h)N1/2,
e(hx,,) << dlh
1
and Weyl's criterion holds.
The fact that M(x)/x tends to zero is one aspect of the prime number theorem. We recall that M(x) <<
xe+E
holds for any e> 0 if and only if the Dirichlet series for 1/i(s) converges for Re s > 0, that is, when i(s) is non-zero for Re s > 0. The Riemann hypothesis, that i(s) is non-zero for Re s > 2, gives N
F,
(Q Ihl)1/2+E
1
for any e> 0, when N is chosen so that x1,. .. , xN are the complete set of rationals in (0,1] of height at most Q. The extent to which the rationals are uniformly distributed depends on the truth or falsity of the Riemann hypothesis; perhaps this is what the Riemann hypothesis really means. 1.5
APPROXIMATING A GIVEN REAL NUMBER
When we wish to calculate with real numbers, we approximate them by rational numbers. There are three desirable criteria: (A) accuracy; (B) bounded denominator; (C) choice of a convenient rational number.
For hand calculation we represent a real number by a decimal expansion of finite length; within a computer we use a binary expansion of finite length. This is an example of (C), choosing a rational number whose denominator
The rational line
16
has only 2 and 5 as prime factors. For a quick calculation to one or two significant figures, (B) is important. All three criteria have been used to approximate ir. The Roman engineers
are said to have used 25/8, suited to the Roman numeral system in which multiplications by 5 and 2 are especially easy, and calculations up to 10 000 can be done quickly. The fraction 25/8 satisfies criterion (C). Schoolchildren before the age of the calculator used 22/7 in examinations, for speed and simplicity, criterion (B). The instructions for the early pocket calculators gave 17 as `355, multiply, 113, divide', chosen on criterion (A).
We use the notation [[x]] for the nearest integer to the real number x, ((x)) for x - [[x]], so that - z 5 ((x)) 5 Z, and Ilxll for the absolute value of ((x)). We now give Dirichlet's theorem, which answers criterion (B). Lemma 1.5.1 (Dirichlet's approximation theorem) Given a real numberx and a positive integer Q, there is a rational number a/q with q 5 Q and a
Ix
1
q15 q(Q+1)'
so that Ilgxll 51/(Q + 1).
First proof Consider the Q + 1 numbers 0, x, 2x,..., Qx modulo one. For some pair of integers m and n we must have Ilmx - nxll 51/(Q + 1). Take q = I m - nI, so that q 5 Q, and a = [[qx]].
Second proof We look for a/q in the extended Farey sequence AQ). If x is a rational of 9(Q) then a/q = x. Otherwise x lies between two consecutive Farey fractions e/r and f/s, with mediant (e + f)/(r + s). If x < (e + f)/(r + s), then we take a/q = e/r, a
X_ -I
e+f r+s
el
rI
1
r(r+s)
r(Q+1)'
we must have r + s > Q + 1 since the mediant is not in 9(Q). Similarly if x > (e + f)/(r + s), then we take a/q = f/s. 0 The first proof is a variation of Dirichlet's pigeon-hole principle, that if Q + 1 letters are delivered to Q persons, then somebody has at least two letters in their pigeon-hole. The pigeon-hole principle is used like compactness in analysis. The second proof can be made constructive by the continued fraction rule, which, given a real number x, constructs a sequence of rational numbers, alternately below and above x. If x is rational, then this algorithm finds the
Approximating a given real number
17
lowest-terms form of x, and then terminates. In particular, if x is given as a ratio m/n, then the continued fraction rule is simply Euclid's algorithm to find the highest common factor d = (m, n) and integers u, v with n m
ul =d.
In our notation r = m/d, q = n/d, and r/q, v/u are consecutive Farey fractions.
The continued fraction rule is an iteration. Let
ro=ao,
ao = [x},
qO = 1,
and, if x :k a0, then let
x1 = 1/(x - ao). Then 1
ro
0
qO
and
x=
rox1+l gox1 + 0
For the second step, let
a1= [xl],
(ql)
0)(
f qO
1
and, if x1 0 al, then let
x2 = 1/(x1 -a1). Then r1
ro
ql
qO
= 1.
and
x=
r1x2 + ro g1x2 + qO
For the general step, let
(rn) =rrn qn
I11 qn
1
rn-2
-1
qn-2
and, if xn # an, then let
xn+1 = l/(xn -a,,).
1 ( an)
The rational line
18
Lemma 1.5.2 (Continued fractions)
At each step in the iteration given above
we have
r_1
rn
qn- 1 qn
_
(-1)n,
(1.5.1)
and
rnxn+l
(1.5.2)
gnxn+l +qn-1 lies between rn/qn and rn_ /qn-1, which are consecutive fractions of A q,,). The fraction with the even suffix is the smaller of the two. If x is irrational, then the sequence r,,/qn tends to x with 1
1
gnqn+l
2n
Proof We see that I
rn-1
rn
rn-1
rn-2
qn-1
qn
qn-1
qn-2 I'
and the first assertion (1.5.1) follows by induction. Substituting
xn+1 -an+1 = 1/xn+2 in (1.5.2) gives
x=
(rnan+l +rn-1)xn+2+rn (gnan+l +qn-1)xn+2 +qn
proving (1.5.2) for all n >- 2 by induction. We find that rn-1
-x=
qn-1
(-1)nxn+1 qn-1(gnxn+l +qn-1)
and rn
(-1)nxn+1
qn
gn(gnxn+l + qn-1)
Both these differences have the same sign, which is that of (-1)", since xn + 1, q,, and qn _ 1 are positive. The fractions rn/qn and rn - 1/qn are consecutive (in the appropriate order) because of (1.5.1) and Lemma 1.2.1. For the third assertion we note that qn+1 ?qn +qn-1z2gn-1, gnqn+l ? 2gngn-1,
and by induction gnqn+l ? 2"gogl
? "2
Uniform Diophantine approximation
19
Since x lies between r,,/qn and
we have 1
The matrices rn-1 qn-1
rn
qnJ
which occur in the construction have determinant ± 1, and rn-1
r,,
qn-1
qn
=
rn-1 qn-1
1an)
1
q,, -2 I\0
rn-2 1Ta,,
_
l qn-1 (r 21
qn-2J
in the notation of Section 1.4. Similarly we have rn+1 qn+1
rn qn
_
rn - I
rn
qn-1
qn
1
(a,
0 1
_
n-1
- (qn-1
)s_1Ts. r
We note that, although S-1 = -S as matrices, in the action of the modular group PSL (2, Z), S-1 = S.
1.6 UNIFORM DIOPHANTINE APPROXIMATION Although the average gap between the fractions of .9'(Q) is in order of magnitude 1/Q2, there is a gap of length 1/Q between 0 and 1/Q, and there are long gaps about other small rational numbers. In Borel's analogy, a bright star is surrounded by an area in which the stars are fainter than usual, as if the images of nearby stars have been displaced away from the bright star. If criterion (A), accuracy of approximation, is the most important, then, given an accuracy S (with 0 < S < 1), we need a set of reference fractions such
that every real number is distant at most S from one of the reference fractions. One way to do this is to divide the real line into intervals of length at most 3, and to choose the smallest rational number (smallest in height) in each interval to be a reference fraction. If the extended Farey sequence .9'(Q) has SQ2 < 1, then no two fractions of 9(Q) fall into the same short interval, so that all the fractions of .9'(Q) are reference fractions. Some intervals will
give reference fractions with denominator greater than Q. Choosing the smallest rational in each interval is what is called a `greedy algorithm' in combinatorics. The following construction gives an explicit set of reference fractions. We
start with the Farey sequence .9'(Q) for Q = [2/1/S ] and refine it to a sequence of rationals at most S apart. If e/r and f/s are consecutive Farey fractions with
If/s-e/rI<S,
The rational line
20
then we leave the interval [e/r, f/s] alone, but if S < I f/s - e/rl = 1/rs, then we fill the gap. Suppose that r > s. We insert fractions e,
e+ft,
f
1
ri
r+st;
s
s(r+st;)
which are adjacent to f/s in their respective Farey sequences. We choose t; by to = 0, and for i z 1 r + st; = 2/isS + O(1), the sequence stopping at t; when 1
s(r+st;)
5 S.
We note that, since 2r - 1 z r + s > 2/15, we have
r+st;>rz1/,/S. We have enlarged the Farey sequence AQ) to a sequence of reference fractions. The gaps between consecutive fractions have lengths between 8/4 and S. Our next lemma, on the greedy construction and a variant of it, is taken from Huxley and Watt (1988). Lemma 1.6.1 (Uniform Diophantine approximation). Suppose that we are given a real interval J of length 0, divided into subintervals Jk of length at least S,
where 0 < 6 < 1. In each interval Jk we pick the rational number ak/qk with least denominator. Let Q be any positive integer. Then the inequality qk 5 Q holds for at most OQ2+1 values of k, and qk Z Q holds for at most 80
8
62Q2 + SQ values of k.
If we pick instead the rational number ak/qk in Jk with qk least, subject to qk z 1/3, then qk z Q holds for at most 240
12
S2Q2 + SQ
values of k.
Proof The first assertion is part of Lemma 1.2.3. Intervals for which qk > Q contain no fraction of .1(Q - 1), so they lie between some pair of consecutive
Uniform Diophantine approximation
21
fractions e/r and f/s of 9(Q - 1). Suppose that s is the smaller donominator; then s < r, r >- Q/2. If there are n values of k for which the interval Jk lies strictly betwen e/r and f/s, then
n6
s< 2/SQ.
We define these n intervals to belong to the rational f/s. For each rational f/s of .9(Q - 1) there are at most 2/ 6sQ intervals in the half-line (f/s, co) and 2/ SsQ intervals in (- -, f/s) that belong to f/s. At each end of the long interval J there are at most 2/8sQ short intervals Jk with qk >- Q that belong to a fraction f/s outside J. All other intervals Jk with qk >- Q belong to some fraction f/s within J. Thus by Lemma 1.2.3 we find that qk >- Q occurs for at most 2 2-+2 E 6Q ss2/sQ
E f
2 SsQ
8i
< S2 Q 2
+8
SQ
values of k. In the case when choices of ak/qk with qk < 1/6 are not allowed, we may suppose that Q2 z 4/6, or the result follows from the trivial bound L/S + 1
for the number of subintervals. We must add the number of intervals Jk containing a fraction a/q in 9(1/%6) (which must be unique) and no other member of 9(Q - 1). Then q 5 Q/2. Suppose that a/q lies between e/r and f/s in .9(Q - 1). Since the mediant (a + e)/(q + r) is not in F(Q - 1), we have q + r >- Q, so that r >- Q/2. Similarly we have s z Q/2. We deduce that
SSf/s - e/r= 1/qr+ 1/qs <4/qQ, and thus q< 4/6Q. Hence we must add < 9(4/SQ)nJ
160 S2Q2
+ '<
4 160 S2Q2 + SQ
0
extra intervals with qk >- Q.
In the next lemmas we obtain two lower bounds for the number of Farey fractions in a short interval J (corresponding to a subinterval Jk in Lemma 1.6.1). For a long interval there is an asymptotic formula as in Lemma 1.4.2. Lemma 1.6.2 (Large rationals in an interval) Let J be an interval on the real line of length S. Then
E 1- E 1 > SQ2 - 4Q. 9(4Q)nJ
9(Q)nJ
The rational line
22
Proof We number the Farey fractions of A4Q) as fi = a./qi, and we suppose that I = [ fr, fs] is the shortest interval containing J whose endpoints
are in .7(4Q). Then I has length at most S + 1/2Q. We recall that the interval [fi, fi+1] has length 1/qiqi+1. If fi is in Y(Q), then qi+ 1, qi-1 > 4Q - qi > 3Q.
Hence the total length of the intervals [ fi, fi+ 1 ] contained in I with either fi or fi+ 1 in 9(Q) is 2
1
2( ++12Q +1 = 2S 3+1 1
-53Q
<3Q
Jq by Lemma 1.2.3. The remaining intervals [fi, fi+ ] contained in I with neither endpoint in 9(Q) have length at least 1/3Q2, and total length at most 8/3 - 1/Q. The number of these intervals is 1
S
1
> 3Q2 3 - Q = 6Q2 - 3Q, so that there are at least SQ2 - 3Q + 1 distinct endpoints. Two of these endpoints may be inside I but outside J, so the number of rationals of
94Q) -.((Q) in J is O
To obtain a lower bound for shorter intervals, we must take the location of the interval into account.
Lemma 1.6.3 (Large rationals in a short interval) Let e/r and f/s be consecutive fractions in some Farey sequence, and let J be an interval of length 6
lying between e/r and the midpoint (e/r + f/s)/2. For Q > r we have
1- E 1>6Q2
E .9(16Q)nJ
.S(Q)nJ
--. 8Q
r
Proof Without loss of generality we can suppose that e/r
1(e
rSa<(35z
1
(1.6.1)
r+S
The rationals (et + fu)/(rt + su) with t and u relatively prime positive integers are reduced to lowest terms, and lie between e/r and f/s. We have et +fu
a5
when
r s
a-e/r
rt+su u
f/s - a - t
r s
/3-e/r
Ps-R'
(1.6.2)
Uniform Diophantine approximation
23
so that u/t lies in an interval of length
r(/3 - a)(f/s - e/r)
s(f/s-a)(f/s-/3)
S
s2(f/s-a)(f/s-/3)'
By (1.6.1) we have
Sr2 5
< 46r2.
We take an integer T > 1 and consider the rationals u/t satisfying (1.6.2) with T < t s 4T. By Lemma 1.6.2 there are at least
,qT2-4T>-Sr2T2-4T of them. If T is chosen so that
QSrT<2Q, then the lower bound gives the result of the lemma, when we observe that (1.6.1) ensures that /3 - e/r 5 f/s - /3, and that by (1.6.2)
Q
2 Polygons and area 2.1
COUNTING SQUARES
Given a simple closed curve, what is the area inside it? This is the problem of quadrature, important in the history of mathematics. At first to find the area
meant to construct a square of equal area; later it meant to find a real number, the ratio of the area of the curve to the area of a unit square. To find a real number can mean one of four things.
1. Prove that the area exists and can be represented by a real number. For example, all circles of radius p have area -rrp2 for some real number IT independent of the radius of the circle. 2. Find an alternative expression for this real number. For example, IT = 4tan-11, where the inverse tangent function must be defined in a noncircular fashion. 3. Give a theoretic procedure for calculating the real number to any accuracy desired. For example
- + r=3+57 44
1
4
4 Give a practical procedure for calculating the real number rapidly to any accuracy desired. For example
it=4/tan-1
z+4tan-13
1
=4(2+334
I
We use the word curve in the sense of Euclidean geometry: /the locus (trajectory) of a point moving in the Euclidean plane according to some rule.
The position of the moving point is a continuous function P(t) of a real parameter t. We may think of t as the time. The curve P(t) is closed on the interval 0:!g t :!g t1 if P(t1) is the same point as P(O), and simple closed if the points P(t) are all different for 0:!g t < t1. By a smooth curve we mean a curve
that is sufficiently differentiable; in particular, that the position P(t) is piecewise differentiable in t, so that there is a velocity vector defined at all but finitely many points on the curve. The existence of a velocity vector means that the arc length s is well defined, and so is the tangent angle except at finitely many points (Fig. 2.1). For a smooth curve we can (in principle) eliminate t and describe the curve
Counting squares
25
by an equation between s and 4i, called the intrinsic equation of the curve. The intrinsic equation does not depend on the coordinate system chosen for the Euclidean plane. For example, s = ai/r describes any circle of radius a. The derivative p = ds/d+/ (when it exists) is called the radius of curvature; for
a circle, ds/d/i is constant and equal to the radius. Its inverse d/i/ds is called the curvature. We prefer to work with the radius of curvature, which has the dimensions of length. The Jordan curve theorem says that, for a simple closed curve, all points of the plane which are not on the curve are either inside it or outside it, but not both. This theorem is most easily proved for convex closed curves. We define a closed curve to be convex if it behaves like the boundary of a convex set, that is, if any straight line that meets the curve, meets it either in two distinct points or in a closed interval on the line, which may be one point only. The outside of a convex closed curve is the union of all straight lines which do not meet the curve. The inside consists of all points not on the curve such that every straight line through them meets the curve. We have taken some trouble to explain what `inside a curve' means. If it is difficult to prove that a number exists, then it is generally difficult to calculate it also. The method of counting squares, as explained to infant schoolchildren at age seven, runs as follows. Given the curve C in the Euclidean plane, take a square lattice in the plane, generated by two vectors at right angles of the
same length. If the side of the squares is 1/M, and you count N squares inside the curve, then the area A is approximately N/M2. As we prefer to deal with integer points, we make the equivalent construction of enlarging the curve C by a factor M, then choosing a Cartesian coordinate system. We
Polygons and area
26
p
FIG. 2.2
take the lattice to be the lattice of integer vectors, generated by the two orthogonal unit vectors in the x- and y-directions (Fig. 2.2).
There is an ambiguity about squares cut by the curve. Suppose that K squares are inside the curve, and that L squares are cut by the curve. Then
K/M2 is a lower estimate for the area, and (K + L)/M2 is an upper estimate. To fix our ideas, we take the origin of coordinates to be a point inside the curve, and construct the enlarged curve MC by multiplying all the position vectors of points on C by M. We consider K and L as functions of M. If M2 > M1, then all points inside M1C are inside M2C. Thus K(M) and K(M) + L(M) are increasing functions of M. We show that the limit of K(M)/M2 exists as follows. Let n be a positive integer. Then each unit square inside the curve MC corresponds to n2 unit squares inside nMC, and each unit square cut by MC corresponds to n2 unit squares, some of which are cut by nMC. Thus we have K(nM) >_ n2K(M),
L(nM) 5 n2L(M).
If L(nM)/n2M2 tends to zero as n tends to infinity, then K(nM)/n2M2 tends to a limit, the area inside the curve. We show that the area exists when C is a convex curve, of length A say. The curve nMC has length nMA, and the x-coordinate changes by at most nMA/2 on the curve nMC. Hence at most nMA/2 + 1 lattice lines parallel to the y-axis are cut by the curve nMC (Fig. 2.3), since the curve has to go there and back again. Similarly there are at most nMA/2 + 1 lattice lines parallel to the x-axis which are cut by the curve.
As the curve nMC is described in the sense of t increasing, it passes
Counting squares
27
N
nM.U2 FIG. 2.3
through L(nM) lattice squares. If the curve never cuts a lattice line, then L(nM) = 1. If L(nM) > 1, then the curve nMC enters each new square by cutting a lattice line. Since nMC is a convex curve, it meets lattice lines in two distinct points, or possibly in a closed interval. In the case of a closed interval, we take the endpoints of the interval to be the two points in which the lattice line is cut by the curve; they may coincide. Then L(nM), the number of squares cut, is at most the number of times that the curve nMC cuts a lattice line in this sense, and L(nM) < 4(nMA/2 + 1) = 2nMA + 4.
We see that L(nM)/n2M2 tends to zero, and the common limit, A, of K(nM)/n2M2 and (K(nM)+L(nM))/n2M2 does exist. To show that the limit is independent of M, we use the fact that if n and r are integers with rM
K(rM)
K(nM2) =Ar2M2 + 0(rM) =A(rM + O(1))2 =A(nM2 + O(M))2 =An 2M22 + O(nM2).
Hence K(nM2)/n2M2 also tends to A, and the area A is well defined. We summarize the result of this discussion as a lemma. Lemma 2.1.1 (Counting squares) A simple convex closed curve C of finite length A has an area A. Let MC denote the curve C enlarged by a factor M. For any lattice of unit squares, K(M), the number of unit squares inside the curve MC, satisfies
K(M)
Polygons and area
28
FIG. 2.4
We can also treat non-convex curves C, provided that the point set consisting of C with its interior can be expressed as a sum or difference of convex regions bounded by simple closed curves. We apply Lemma 2.1.1 to each convex region. As an example, we can treat the cardioid curve whose equation in polar coordinates is r = 1 - cos 0 (Fig. 2.4). A second example is the area between the rectangular hyperbola xy =1 and the lines x = 0, x = 0, y = 0 and y = 0 (Fig. 2.5). In Figs. 2.4 and 2.5 the dotted lines complete the outline of two convex sets. In Fig. 2.4 they are the upper and lower halves of the region, and in Fig. 2.5 they are a square of side 0 and the part of the square that is separated from the origin by the hyperbola. Figure 2.5 corresponds to Dirichlet's divisor problem, to estimate the sum
M
Y, d(m) = mE F, 1. m5M
r-1 s=1 rsSM
A FIG. 2.5
z
29
Counting squares
Here d(m) denotes the divisor function, the number of ways of writing m as a product of two positive integers.
Lemma 2.1.1 sets out the counting squares method theoretically. For a practical method, we need a rule for counting the squares cut by the curve. We want a number N(M), greater than K(M), for which N(M)/M2 should tend more rapidly to the limit A, at any rate for smooth curves. There are three such rules in common use. Rule 1. Count a square if its centre lies inside the curve. Rule 2. Count a square if its lower left corner lies inside. Rule 3. Count all incomplete squares as half a square. Rules 1 and 2 are really the same rule. If we shift the enlarged curve MC by the vector (2, Z), then the lattice points occur where the centres of lattice squares were before. Rule 2 is easier to apply graphically. For a convex curve, Rule 3 corresponds to splitting the curve into four quadrants, in which the tangent angle iP goes from 0 to 7r/2, from Ir/2 to 7r, from it to 31r/2 and from 31r/2 to 27r. In the quadrant in which 4 goes from 0 to it/2, we count a square as one half unit if its upper left corner lies inside the curve MC and its lower right corner lies outside. This is an average of the Rule 2 estimates for two curves, which are formed by shifting the quadrant of the curve MC by a unit vector in the x- and the y-directions. In the next quadrant we again have an average of the estimates given by Rule 2 for two curves, one being the quadrant of MC, the other being the same quadrant shifted by the vector
(1,1). The shifted quadrants do not join up at their endpoints. When one allows for this, it appears that Rule 3 counts more squares than Rule 2, but at most four more squares. We take Rule 2 as the practical form of the method of counting squares. There is still an unavoidable ambiguity in Rule 2 over lattice points on the curve MC, whether to count them as inside or outside. For any e > 0, these points are outside the curve (M - e)C, which has almost the same area as the curve MC if a is small, and they are inside the curve (M + e)C. Let D(M) denote the discrepancy, I N(M) -AM21, where N(M) is the number of lattice points counted under Rule 2. If there are R points on the curve MC, then for e small enough
N(M+ E) =N(M- e) +R, so that
D(M+ E) +D(M- E) ZR - O(eM). The supremum of the discrepancy is at least R/2. If C is the square with corners (±1, ± 1), then there are 8M lattice points on MC whenever M is an integer. This square gives the worst case, in which R = AM. Since lattice
points are a distance at least 1 away from one another, we must have R< AM. At the other extreme, when 2M is an odd integer, then there are exactly 4M2 lattice points inside MC, and the discrepancy D(M) is zero.
Polygons and area
30
There is a corresponding practical problem with lattice points which are close enough to the curve MC to be indistinguishable to the eye. If the eye cannot separate the curves (M - 6 )C and (M + S )C, then
N(M+ S) =N(M- S) +R, so that
D(M+S)+D(M- 6)>-R-O(SM). So if R is greater than 6M in order of magnitude, then there is a discrepancy as large as R in order of magnitude. If the curve MC has a large radius of
curvature (greater than M in order of magnitude), and if there is a lattice line which is almost a tangent, then there are too many lattice points very close to the curve. The accuracy of the counting squares method is certainly limited by the
number of lattice points on or very close to the curve MC. We consider smooth curves C for which the radius of curvature p = ds/d tai exists and is continuous at all but finitely many points (thus the curvature dt/r/ds is piecewise continuous with at most a finite number of zeros). The curve length
A= f pdclJ
is bounded by 27T times the maximum radius of curvature. Under these hypotheses the number of lattice points on C is << A2/3 (Jarnik 1925), and this bound can be attained; we sketch a variant of Jarnik's construction in the next section. Jarnik constructed a curve C going through many lattice points, but the construction does not give extra lattice points on the curves MC as M tends to infinity. Swinnerton-Dyer (1974) showed that, if C has a continuous
radius of curvature bounded away from zero and infinity, then R(M), the number of lattice points on the enlarged curve MC, is in order of magnitude R(M) <<M315+E
Arcs of the curves y = xd with d = 2,3,4,..., provide examples of curves with R(M) >> M1"d+E and a certain number of derivatives not vanishing. Bombieri and Pila (1989) took these ideas further, getting the bounds
R(M) <<Mlld+E for some positive integer d when C is an arc of an algebraic curve, and R(M) <<ME when C is an arc of a transcendental curve. Here a can be arbitrarily small
and positive, but the constant of proportionality depends on a and on the curve in a manner difficult to control.
In Chapter 3 we shall estimate the number of lattice points close to the expanded curve MC by Swinnerton-Dyer's method. We shall show in Chapter 6 that the discrepancy of the closed curve MC has root mean square value at least M1/2 for a sufficiently smooth curve. The
Jarnik s polygon
31
Gauss circle problem is to estimate the discrepancy for the circle radius M centred at the origin. Special methods (Hardy 1915, Landau 1915) show that
the discrepancy for the circle is sometimes as large as M1/2f(M), where f(M) is a function tending slowly to infinity. Discrepancy is so subtle that we
do not know whether the circle is likely to have a larger or a smaller discrepancy than a general smooth curve. The boldest conjecture would be that if C is sufficiently smooth, then D(M) <<M1/2+E
In numerical integration the curve or arc C is given by an equation y = F(x). The basic method (trapezium rule) is to evaluate F(x) at a sequence of equally spaced points 1/M apart, and to divide the sum by M. In practice the function F(x) is evaluated to some accuracy 1/T, ignoring a
rounding error. We interpret this rule as counting squares for the curve y = TF(m/M) expanded by a factor M in the x-direction and by a larger factor T in the y-direction. 2.2
JARNIK'S POLYGON
Jarnik's polygon is formed by choosing a height H, then taking all the primitive integer vectors (q, a) corresponding to rationals a/q with height at most H as sides of the polygon. The construction starts at an integer point, then takes all the primitive integer vectors with q > 0 in order of the gradient
a/q increasing, then the vector (0, 1) with infinite gradient, then all the primitive integer vectors with q < 0 in order of a/q increasing, then the vector (0, - 1). The polygon is closed and, if the starting point is suitably chosen, symmetric about the coordinate axes and about the lines y = ±x. Lemma 2.2.1(Jarnik's polygon) The number of vertices of the Jarnik polygon is H
H
4+4 F, 5' 1= a=1 q=1
24H2 2
ar
+O(HlogH).
(a,q)=1
The diameter of Jarnik's polygon along the x-axis has length H
H
1+2 E F, q= a=1 q=1
6H3
+O(H2logH).
(a,q)=1
Proof The first assertion follows at once from the first part of Lemma 1.4.2 and the second assertion follows by the same method. o Lemma 2.2.2 (Jarnik's curve) There is a simple closed curve passing through
Polygons and area
32
Pr-1
FIG. 2.6
the vertices of Jarnik's polygon, whose radius of curvature is continuous and O(H3) in order of magnitude. Corollary Let the Jarnik curve have area A and length A. Then the number of points on the Jarnik curve is >> A2/3 >> A1/3.
Proof Let the vertices of the Jarnik curve be PI,..., PR. Let Cr be the circumcircle of the triangle Pr-1PrPr+1 (Pa is identified with PR, PR+1 with P1). The triangle Pr_1PrF.+I has area i by Lemma 1.2.1, and sides of length
between 1 and HV. Either Pi_1Pr or PrPi+1 is a vector with one entry at least H/2. Let Or be the angle 1 _1PrPr+1 (Fig. 2.6), K, be the length of Pr Pr+1, la'r be the length of Pr_1'r+1, and let pr be the radius of the circle Cr. By elementary geometry 1
2
1A'r
= Kr- 1 K,sln 6r,
sin Or e = 2Pr'
so that Pr = Kr-1 KrPA'r s Kr-1Kr(Kr-I + Kr) 5 4H3V.
The circles Cr and Cr+1 meet at P. and at Pr+1. We now take circles Dr of radius 2 pr, such that Cr and Dr touch internally at P. (Fig. 2.7). The circles
Dr and Dr+ 1 meet outside the circles Cr and Cr, 1, so that there is an intersection between P. and Pr+ I and outside the Jarnik polygon. We can
Dr+l
FIG. 2.7
The discrepancy of a polygon
33
join P and P,+ 1 by a smooth curve E, which has the same tangent and radius of curvature at the point P, as the circle D, and has the same tangent and radius of curvature at the point Pr+ 1 as the circle D,+ 1, and has radius of curvature at most 4HVV. When we subtract the equation of Dr from the equation of D,+ and change coordinates, then this corresponds to finding a 1
function g(x) with g(0), g'(0) and g"(0) all zero, g(x), g'(x) and g"(x) non-negative and bounded on 0 < x 51, and taking prescribed values at x = 1. The length of Er is less than the length of the polygonal path from P, to P,+ 1 formed by the tangents to the circles Cr and Cr, 1 at Pr and P,+ 1,
respectively. The angle between the tangent at Pr and the line P,P,+1 is equal to the angle P,Pr_1Pi+1, which is less than IT- Or. If H is large, then Ors 31x/4, the tangents cut at an angle greater than 7r/2, and the length of Er is Ar 5 (sec 0, + sec 0,.+1)K,:!,r+ 1) K, S 2K,'!1.
The Jarnik curve is formed from the curves Er, having continuous radius of curvature, and length
A= EA,<
O
Jarnik's work (1925) is much more accurate than this account suggests. He finds the best constant in the inequality R >> A213 of the corollary.
2.3 THE DISCREPANCY OF A POLYGON We define the discrepancy of a plane set to be the difference between the number of integer points within the set and its area. This is a generalization of the discrepancy of an interval described in Chapter 1. The rounding error or row-of-teeth function is
p(t)=[t]-t+2, named from its graph (Fig. 2.8). The number of integers n with a < n < 6 is
/3- a+ p(/3) - P(a).
(2.3.1)
It satisfies an addition formula. Lemma 2.3.1 (Addition formula) For any positive integer q q-1
a
F,P t+ - ) = p(qt). a=0 q Proof The left-hand side is periodic with period 1/q. For 0 5 t < 1/q, the integer parts in the definition of p(t + a/q) are all zero, and we verify the result by direct calculation.
0
34
Polygons and area A
o
2
1
FIG. 2.8
Lemma 2.3.2 (Explicit discrepancy formula) Let a be a real number with continued f r a c t i o n ao + 1/(al + ) and convergents r;/qi. Let N be a positive integer. Define the a-expansion of N as follows. Let k be the largest integer with qk 5 N, and let bk = [ N/qk ] ,
N = bk qk + Nk -1
so that Nk_ 1 < qk. Now let
bk-1 ; [Nk-1/qk-1
Nk-1 =bk-]qk-1+Nk-2,
and so on. Since qo = 1, we have
+blg1 + b0.
N = bkgk + bk- lqk- 1 + Then
N 1
p(an) = 2(bo-bl + ... +(_1)kbk) 1
E i=1 n=N,_1+1
r! qi (2.3.2)
where a sum from 1 to 0 is interpreted as zero. Hence 1
Also, for any real 6, 5
N
p(an+(3)I 1
2(bo+b1 +
+bk)+k+ 1.
The discrepancy of a polygon
35
Proof We use induction on k. If k is even, then a > rk/qk, and for n < N we have nrk
p(an)=p
qk
+n a-
rk
nrk
---
)( qk)
/
-n(a-
rk qk(2.3.3)
since, if a = rk/qk, then the difference is zero, and, if not, then N < qk+1, whilst
n a-rk
1
1
qk
gkgk+1
qk
If k is odd, then a - rk/qk is negative, and then (2.3.3) holds unless qk I n. For qk I n we have
p(
+n(a9k
rk
-1
2
))
9k
-n(a-
nrk
gk
-n(a-
=p( gk)
- 1.
qk
For k even we have, by Lemma 2.2.2, bxgk+Nk-,
N
nrk
p(an) =
N
p(gk)-_n(a
1
rk
-
=bkP(gnrk
qA:
gk Nk-1
bk
2+
r
Nk-I
nr
p(9k)
Nk
Y
n(a -gklvk1+1n(a-9k
For k odd there is an extra term -bk, which combines with the term bk/2 to give (-1)kbk/2. For k = 0, the proof of (2.3.2) is now complete. For k >_ 1, we have established the induction step, continuing with Pk/qk in place of a. For the estimate we have Ni
n(a-
E
rr
<(Nr-Nr-1)Nrl a-
qr
n=N;+,+1
qr
b;q,gr+l
-b.
gr qr+ 1
This term has the sign of (-1)r+ 1, so that bi
2
Nj
+
F, n=N;-,+1
n b; n(a--2 qi
(2.3.4)
Polygons and area
36
In the shifted sum E plan + /3) we replace (2.3.3) by nrk rk p(an+/3)=p -+/3 -n a-+ qk
En ,
qk
where en = 1 if
[an+/3]> f nk +(3l, [ qk
f
which can happen only if a > rk/qk and
nrk+[aqk]= -1(modgk) and
(aqk - rk)n Z p( l3gk) + 1/2;
and En = -1 if
[an+/3]
J
which can happen only if a < rk/qk and nrk + [ 13gk ] = O(mod qk ) and
(rk - agk)n > 1/2 - p( aqk).
The congruence condition implies that en is non-zero for at most bk + 1 values of n in the range 1,..., N. We have
r'N pIr nrk +1) '+
1
r
=bk E pI
1 qk
+
` qk
1
Nk'1
nrk +1)
EI
nrk
p(- +1) qk
The first term is bk p(qk (3) by Lemma 2.3.1, since the numbers rk, 2rk,... , qk rk
are congruent to 1, ... , qk (mod qk) in some order. We do not try to cancel this term against the sum in (2.3.4) or against the sum of the e,,, so that we have N
Nk-t
p(an+(3)I 1
-E 1
nrk qk
5bk
< 2
+ 1.
The bound of the lemma follows by induction.
We are now ready to compare the area of a convex polygon P1 P2 ... PN
with the number of lattice points inside. We also refer to P. as P. in the following construction. For each vertex P,,, let Qn be the centre of the lattice
square containing P. For n =1, ... , N, let Rn be that corner of the rectangle, with diagonal Qn - 1Q,, and sides parallel to the axes, which is on the side of Qn_1Qn corresponding to the inside of the polygon. Then R1Q1R2Q2 . . . RNQN forms a 2N-sided polygon with sides parallel to the
The discrepancy of a polygon
37 Pn n
Qn-1
Rn
Pn-1
FIG. 2.9
axes. If this polygon is not self-intersecting, then its interior is a union of unit squares with centres at lattice points, and its area is equal to the number of lattice points contained. If the polygon does intersect itself, then we have to count unit squares and lattice points with signed multiplicity.
We now compare lattice points and the area within the pentagon D: (Fig. 2.9). Suppose that the line P,,_1Pn has equation Q,, is the point y = ax + /3, and that Q,,-, is the point (t + Z, v + Z),
(u + 1, w + ), where t, u, v, w are integers. In the calculation that follows, i I a 151. If I a I > 1, or if the equation has the form x = y, then we assume that
we treat y as the independent variable. Let E be the region between the lines x = t + 2, x = u + 2, y = v + i and y = ax + /3, a quadrilateral whose R, and Pn_1Pn, extended if necessary. The quadrilateral may be self-intersecting. We take area and sides lie along the lines Rn_1Qn_1, RnQn, Q,,
lattice points above the line y = v + i as counting positively, those below as counting negatively. Then the signed number of lattice points in D differs from the signed number of lattice points in E by at most two, and the area within D differs from the area within E by at most four, the difference being
contained within lattice squares containing Q,,-, and Q, and adjacent squares above or below these two. The area of E is 2 a-v-)dx=(u-t)
(a(t+ u+1) 2
1
t+
1
+/3-v-2Ji
and the number of lattice points is u
u
m=t+1
1+1
F, ([am+/3]-v)= L(p(am+/3)+am+/3-v-Z) U
_ L p(am+/3) t+1
+(u -t)
(a(t+u+1) 2
+/3) -v-Z
Polygons and area
38
Hence the signed number of lattice points in the pentagon D differs from the area of D by u
E p(am+/3)+O(1). r+1
We put together the bounds given by Lemma 2.3.2 for each such pentagon to obtain the error estimate in counting squares for a convex polygon. Theorem 2.3.3 Let D be the region inside a convex plane polygon P1P2 ... P. as follows. 1P,, (with P0 = PN) define the bound 1, For each edge
Let L be the length of the edge P1P,, and a be its gradient, with continued fraction a = ao + 1/(a1 + ), and convergents r,/q;. Let k be the largest integer with qk 5 L + 1. Then
B(Pi-1,
=ao+a1 + ... +ak + (L + 1)/qk
In this notation the discrepancy 0 of D, the difference between the number of integer points in D and the area of D, satisfies N
0 <<
2.4
B(Pn-1,P.).
FITTING A POLYGON TO A SMOOTH CURVE
In this section we assume that the simple closed curve C is piecewise twice differentiable, and that on each piece C, the radius of curvature is bounded.
The enlarged curve MC is composed of arcs MC1, each with radius of curvature p5 BM for some constant B. Each arc MC, corresponds to a closed interval of values of the tangent angle 41. We choose a small positive 6(< it/8), and pick a sequence of angles 41i lying in this interval, such that each value of 4 on the arc MC, lies within 8 of some fit. Let P, be the point on MC1 at which rG = 41,. The outer polygon is formed by the tangents at the
points P. Let the point T be the intersection of the tangents at F, and P, 1, and let Sj be the intersection of the corresponding normals (Fig. 2.10). The angle at S, is at most 2 S. The lengths SPA and SP + 1 are longest when the radius of curvature is largest, so they are at most BM. Hence PST SBMtan2853B8M, and by symmetry the side T _ T of the outer polygon has length T j- 'T j5 6BSM.
Now let Q, be the foot of the perpendicular from P j, 1 to the normal at P. Then (2.4.1) QQP1=P+1T sin28<6BS2M.
We form the inner polygon from lines parallel to the tangents 7_ 1T, a
Fitting a polygon to a smooth curve
39 Pj+1
Qj
Si
Pi
FIG. 2.10
distance 6B62M inside the curve. In Fig. 2.10 the inner polygon lies to the left of the line Pj,1Qj, so the arc PjP+1 lies outside the inner polygon. The perimeter of the inner polygon is less than the length of the curve MC, which is at most 2irBM. Let U be the vertex of the inner polygon corresponding to T. The area between the inner and outer polygons is 6B62M times the perimeter of the inner polygon plus the sum of the areas of kite-shaped regions with opposite corners U. and T. (Fig. 2.11). These kites can fit without overlapping inside a circle of radius Hence the total area between the inner and outer polygons is 5 127rB28 2M2 +721TB284M2 5 30vrB282M2.
Uj
FIG. 2.11
(2.4.2)
40
Polygons and area
The number of lattice points within the curve MC minus the area within MC is less than or equal to the number of lattice points within the outer polygon minus the area of the inner polygon, and greater than or equal to the number of lattice points within the inner polygon minus the area of the outer polygon. Hence the discrepancy D(M) of the curve MC is at most the maximum of the two discrepancies of the inner and outer polygons, plus the difference in area.
Lemma 2.4.1 (A suitable polygon) Given a number S in 0 < 5 5 z and a length L, then there is a sequence of angles 0 'YR = 27r with 4'r+ 1 - 'Pr < 5 and tan 41r rational for each r, such that any convex polygon with edges in the direction of 'lr , ... , r/IR _ 1, each of length at most L, has discrepancy
L
1
1
<<-+s-log 2s Proof First we pick the angles I/rr in the interval 0 S 415 Ir/4, and then we extend the sequence by reflections first in the point it/4, then in the point
or/2, and then in the point ir. We pick an integer Q with Q2 z 2/8, and expand the Farey sequence .F(Q) between 0 and 1 by the reference fraction construction of Chapter 1 to a sequence of rationals 61 = 0, /32, ... , ,6m = 1 with 8/4 S 8r+1 - ar 5 S for each r. We define 1/r as tan-1 Nr In any continued fraction, the convergents rn/qn satisfy qn+2 > 2qn If qn 5 Q, then 2log Q
ns
log 2
+1.
The extra rationals introduced by the construction have continued fractions of the same length as some fraction f/s in .9(Q), so that we have the same bound for the length of the continued fraction as before.
Now let a= ao + 1/(al + - ) be a particular continued fraction with convergents rn/qn. By Lemma 1.5.2 the real numbers R with the same partial
quotients ao,...,an as a form the interval from rn/qn to (rn+rn-1)/ (qn +qn-), of length 1/gn(gn +qn-1). Of these, the numbers with the next partial quotient an+1 taking the value an+ = a form a subinterval of length
1/(aqn +qn-1)((a + 1)qn +q,, 1) The ratio of lengths lies in the range 1
(a + 1)(a + 2)
gn(gn + qn-1)
2
(aqn + qn-1)((a + 1)qn + qn-1)
a(a + 1)
<
For each k >- 0, the proportion of the interval on which an+ 1 z 2k is 00/2 k). If y; are a sequence of real numbers within this interval, satisfying rn
4
(
qn--1) 1
4
Fitting a polygon to a smooth curve
41
for each i, then the sum of the (n + 1)th partial quotients of the numbers y; is
«
2k 71
k4logl/S 2 k Sq.(gn +qn-1) ( 11 log 1/5
<< log -
<<
S
Sgn(gn + qn_1)
1.
When we sum the bound of Theorem 2.3.3 over edges Pn-1P,, with gradients f31, ... , (3,,,, then, for each n, the partial quotient a sums to 1
<<m logs <<
1
1
logs .
s
The sum of the terms (L + 1)/qk in Theorem 2.3.3 is
L+1
m(L+1) Q+ 1
+
9tg)n [0,11
q
m
l
5(L+1)( Q+1 +Q+1J <<
where we have used Lemma 1.2.3. We deduce the result.
L+1 ,
AS Cl
We use Lemma 2.4.1 with L = 4BSM and log BM 2/3
S= ( BM
)
when the bound becomes << (BM log
2BM)213,
the same order of magnitude as the area 301rB2S2M2 in (2.4.2). The bound depends on the curve only by way of the maximum radius of curvature. We have now proved a version of van der Corput's theorem. Theorem 2.4.2 Let D be the region inside a convex simple closed curve, piecewise twice differentiable, with radius of curvature bounded above by R. The discrepancy A of D, the difference between the number of integer points in D and the area of D, satisfies
0 « (R log 2R)213.
We have given a proof of the theorem which echoes the themes of this book. The first results of this form were those of Voronoi (1903) for the divisor problem (where the region must be subdivided, because the radius of curvature changes its order of magnitude), and Sierpinski (1906) for the circle problem. General results were found by Vinogradov in 1917 (see Gel'fond and Linnik 1965) and by van der Corput in his thesis of 1919 (see van der Corput 1920). Van der Corput's bound was just O(R2/3). Vinogradov had an extra factor (log R)2/3, which can be removed by elaborating the argument
(Chaix 1972). The modern proof, also due to van der Corput, uses the exponential sum estimates of Chapter 5.
3 The integer points close to a curve INTRODUCTION
3.1
We give upper estimates for the number of integer points (m, n) lying close to the curve
y/T=F(x/M), 0<x
(3.1.1)
IITF(m/M)II < S.
This represents a fixed curve C with equation y = F(x) expanded by factors M in the x-direction and T in the y-direction, and corresponds to a generalization of the counting squares algorithm in which one counts rectangles of
size 1/M by 1/T relative to the size of C. We shall always suppose that L <M, and that F(x) is at least twice continuously differentiable. Let R be the number of solutions of (3.1.1) with m an integer, 0 < m 5 L. The obvious way to estimate R is to use the row-of-teeth or rounding error function
p(t)= [t] - t+ 2 where t is a real variable. The rounding error function satisfies
p(t+S)-p(t-S)+25=
I1 l0
if t+6zn>t-8forsome integer n, otherwise.
Then L
R=2S(L+1)+
/
r
OI pITF(
)+S)-p(TF( )-6)), 1
(3.1.2)
which gives an asymptotic equality for R if we have a good estimate for the rounding error sums on the right of (3.1.2). When S is small, then the estimates for rounding error sums are larger than SL, and we can hope only to find an upper bound for R. When 6 = 0 we are back to counting points on the curve.
Suppose that a set of points contains far more integer points than one would expect. Does it have an arithmetic structure? The divide and conquer method consists of three steps: 1. Dirichlet's pigeon-hole principle. There is some smaller set containing many of the integer points.
Introduction
43
2. Approximation. The point set is defined by certain inequalities. If these involve smooth functions, then the functions can be approximated on the smaller set by Taylor polynomials, preferably linear.
3. Combination. For each integer point, there is some integer constructed from its coordinates. If this integer is numerically less than unity, then it must be zero, and we have shown that the integer points in the smaller set all satisfy some relation.
If the divide and conquer method works, then it tells us that the excess integer points in the smaller set lie in some set defined in an arithmetic way, such as a rational line. Although the method is powerful, it is limited by the pigeon-hole step. It is usually impossible to use the fact that the smaller set of
integer points selected at this step is a subset of a larger set with similar properties. For the method to give best-possible results, there must be some uniformity in the original set of integer points, so that when we allocate them to pigeon-holes then no subset is very much larger than the average subset. More often we can prove that each pigeon-hole contains a bounded number of integer points at most, but we can neither enlarge the pigeon-holes nor find a different argument to show that most pigeon-holes must be empty. We give two methods (Huxley 1989b) for estimating the number of integer points close to a curve. The first is an elementary analogue of the van der Corput iteration (1922; see also Graham and Kolesnik 1991 and Kratzel 1988). The second is Swinnerton-Dyer's method (1974), which works when M and T are about the same order of magnitude. Both methods use `divide and conquer'. Let the integer points close to the curve be P1, P2, ... in order. In the first method Dirichlet's pigeon-hole principle is applied to the projections of the sides P,P, onto the x-axis. In the second method the pigeon-hole principle is applied to the areas of the triangles P;Pt+1P;+2 The simplest idea for estimating the number of integer points close to a curve is to interchange the x- and y-axes. If the gradient is,small, then the curve cuts fewer lattice lines parallel to the x-axis than lattice lines parallel to the y-axis (Fig. 3.1). We pass from the function y = F(x) defining the curve to its inverse function. Inverse functions appear whenever we do an interchange or inversion step. Lemma 3.1.1 (Interchanging variables)
Let F(x) be a monotone real function
with a continuous non-zero derivative on the interval 05 x5 L/M. For 8 in 0 < 8 < 1, let R as usual denote the number of solutions of (3.1.1) with 0:!-, m < L. Let x = G(y) be the inverse function of y = F(x), and let R' be the number of pairs (m, n) of integers with
Im -MG(n/T)I,_< S', where
8' = 6M/(T min JP(x)J),
The integer points close to a curve
44 n
N
0
L
m
FIG. 3.1
and n/T is in the domain of G(y); thus n lies in an interval of length LT
K<MmaxIF'(x)I.
Then
R5R'+2(1+8 '). Proof Let I be the interval taken by the values of TF(x) for 0::g x:!-, L/M. Then I has length LT L/M K= T fo F '(x) dx 5 M max IF (x)I. If m is an integer counted by R, then
TF(m/M) = n + a, I al < S. Either n itself lies within the interval I, or n lies within I extended by a distance S at each end. If n lies within I, then
MG(n/T) = m - f3, where aM/ f3T is a value of F'(x). When I a 1:5 S, then we have I /3I S S' = SM/(Tmin IF'(x)I). If n lies outside I, then let v be the endpoint of I which is nearer to n. Since the endpoint v lies between n and TF(m/M), then we have
TF(m/M) = v+ a', and
MG(v/T)=m-,l3', where a'M/(3'T is a value of F'(x). Again we have 8'. Thus m lies within a distance S' of one of the endpoints 0 or L. At most 2(1 + S') integers m give values of n outside the interval I.
Estimating R' trivially, we get our first result.
0
The reduction step
45
Lemma 3.1.2 (A gently sloping curve) Let F(x) be continuously differentiable, with F'(x) non zero, on 0:5 x< L/M(< 1). Let R be the number of solutions of (3.1.1) with 0< m 5 L. Then R << 1 + SL + LT/M + 6M/T. (3.1.3) The constant implied in the upper bound depends on the range of values of F'(x),
but not on S, L, M, or T. Proof The trivial estimate for R' in Lemma 3.1.1 is (K + 1)(2 S' + 1), since there are at most K + 1 integers n, and for each n there are at most 2 S' + 1 integers m. Substituting the estimates for K and 6' given in Lemma 3.1.1 gives the result.
Lemma 3.1.2 is useful if T < M, and best possible if T << SM.
3.2 THE REDUCTION STEP In this section we go from a steeply sloping curve to a gently sloping curve. We begin with the combinatorial lemma. The use of n-tuples is based on Filaseta's use of second differences (1988), but it can also be regarded as an iteration of one of Swinnerton-Dyer's innovations (1974). Lemma 3.2.1 (Repeating multiplet patterns) Let m1,..., mR be a sequence of positive integers with
<mR L.
1 <m1 <m2 <
Let r
m5-m13_1 = us for s = 1, ... , r. Then
U
U
u1=1
u,=1
R<2 F, ... E R(u1,...,u,), where
U = [2rL/(R - 2r)]. Proof The number of values of j for which
mj+1-mj ZU+1 is at most
L
U+1
R
<2r-l
Thus the number of values of i for which
mi+s-m;+s-1
R-r+1-r(R/2r-1)>R/2,
(3.2.1)
The integer points close to a curve
46
and the result follows on classifying according to the values of the r consecutive differences.
Our next lemma, taken from Swinnerton-Dyer (1974), is the analogue for r + 1 points of the usual mean value theorem. Lemma 3.2.2 (Mean value theorem) For 0 = to < tl < ... < t, real and f(x) a real function r times differentiable we have
f(r)q)
f(x+tj)
r 57
r
j=0
H
k=0
=
r!
(t;-tk)
(3.2.2)
k#j
for some
depending on x in the open interval
Proof Rolle's theorem states that if h(x) is a differentiable function which is
zero at distinct points xo and x1, then its derivative h'(x) is zero at some point between xo and x1. By induction, if h(x) is r times differentiable and X0,. .. , x, are distinct zeros of h(x), then h(r)( ) is zero for some between the minimum and the maximum of the points xi. Let g(x) be the interpolation polynomial r
g( x) =
1
r
(x - xk)
E f( xj ) k F71 (x. -xk) k #j
with g(xj) = f(xj) for j = 0,..., r. We put h(x) = f(x) -g(x) and xj =x + tj to deduce the lemma.
To apply Lemma 3.2.1, we let u = (ul, ..., ur) be a vector with positive integer entries, and write G(u, x) for the expression on the left of (3.2.2) with
tk = (ul + ... +uk)/M.
f (x) = F(x),
The function G(u, x) is called a divided difference of F(x). We have
G'i)(u, x) =f (r+ I)q)/(r+ 1)!
(3.2.3)
for some , so that the values taken by the derivatives of G(u, x) correspond to those taken by the derivatives of F(x). The expressions r
j-1
A(u) = fl I I (Mtj-Mtk), j=1 k=0 r
r
j-1
B(u) _ E fl fl (Mtj -Mtk) i=0 j=1 k=0 j#i k#i
47
The reduction step
are positive integers. If now i is an index for which (3.2.1) holds, then we have
T
MrA(u)G u, M ) II < SB(u).
(3.2.4)
m The inequality (3.2.4) has the same `form as (3.1.1).
When we apply bounds for R(u), then terms in which T occurs to a negative power lead to sums over B(u)/A(u)o for some 0 < 1. We need another lemma to handle these. Lemma 3.2.3 (Counting multiplet patterns) In the notation above, for any W>>-1
E B(u) <<2'rr+1W log'rU. A(u)5 W
Moreover for 0 < 0 < 1
E B(u)(A(u))- o << (2rrr+1 log'
rU)0(rr2-r+2Ur2+r)(1- o)/2,
it
with an implied constant depending on 0, and
L B(u)/A(u) << 2'rs+1 logi+1rU. U
Proof We have
A(u)/B(u)Z
1 r+1
min flIMtt - Mt,I. i t#t
The number of sets of r positive integers v1, ..., yr with viv2 ... yr S V is
O(V logr-1 V). There are 2r ways of choosing the sign of the differences Mtt - Mt;; i is then the number of negative terms, and the integers Mt1,... , Mtr are determined. Thus for any V Z 2 A(u)
B(u)
2A(u)
V
r+1
V
holds for at most
0(2'V logr-1 V sets u. The first sum of the lemma is
O(WF, 2'rlogr-1V), `
V
where V is summed over powers of 2 not exceeding W)'.
The integer points close to a curve
48
The second assertion comes from dividing into cases
W/2
EB(u) <<
rUr(rU)r(r-1)/2
so that the second sum is
<< 1 W-e min(2'ri+1W log'rU,
rUr(rU)'('-1)/2),
W
where W is summed over powers of 2 not exceeding (rU)r(r+1)/z
ci
3.3 THE ITERATION We give a `divide and conquer' method for estimating the number of integer
points close to a steep curve y = TF(m/M) with T larger than M. Lemma 3.2.1 provides the pigeon-hole step. The pigeon-holes correspond to multiplet
patterns u. The combinatory step lies in forming the linear combinations G(u, x), which give gently sloping curves suitable for Lemma 3.1.2. The approximation step is Lemma 3.2.2 in which we compare an rth divided difference with the rth derivative. Theorem 3.3.1 Let n z 1 and let F(x) be n times continuously differentiable, with F(")(x) non-zero, on 0 <x < L/M(< 1). Then R, the number of solutions
of IITF(m/M)II <_ S
with 0 < m < L, satisfies R L-1/" + Q2/"(n+1) <<
+ S2/(nz-"+2) + QL log" L,
(3.3.1)
where
0 = TIM". (3.3.2) The implied constant depends on n and on the range of values of the derivative
F(,")(x)
Corollary
When S = 0 we have R <<M1-1/"
for T«M("-1)/2.
Proof We can suppose that S < Z, as the result is otherwise trivial. We write (3.2.1) as
IIT'G(u, m,/M)II < S', with
S' = SB(u),
T' = AMA (u).
The iteration
49
By (3.2.3) we have
G'(u, x) =f
1)!
for some , so that G' is bounded below uniformly in u. By Lemma 3.1.2 we have
R(u) << 1 +LOA(u) + SLB(u) + SB(u)/AA(u). Lemma 3.2.3 now gives L
UxR<< ER(u) << Ur + ALUr(r+3)/2 + 8LUr(r+1)/2 + SO-1Vlogr+1 L.
There are four cases according to which term on the right of this inequality is largest. For example, if the third term dominates, then L/U << SLUr(r+ 1)/2,
so that R/L << 1/U << S 2/(r2+2 +2)
which is the third term in the upper bound on the theorem when we put n = r + 1. The other cases give the other three terms in the upper bound. The 0 case n = 1 of the theorem is Lemma 3.1.2.
Theorem 3.3.1 is not the only way of combining Lemmas 3.1.1 and 3.2.1.
These two lemmas enable us to compare the problem (3.1.1) with other problems of the same type, and they can be used repeatedly. We call Lemma
3.2.1 the Ar process. In effect the Ar process replaces R/L and T in order of magnitude by (R/L)r+1 and (L/R)r(r+1)12T/Mr, keeping L and M the in order of magnitude. There are same, and replacing S by two drawbacks. First, because R/L has been raised to a high power, we take the corresponding (r + 1)th root of the upper bound in the comparison problem. Second, if we take r > 1 to reduce T greatly, then S increases, to S' (L/R)r(r-1)/23
say. If S' is too large, then the comparison problem will have many solutions.
The B process interchanges M and T, and adds some trivial terms. It increases S to a number S', which may be greater than unity, in which case we estimate the number of solutions of the comparison problem trivially. The effect of Ar followed by AS is different from that of Ar+s. All the processes except Al increase S. If S is large, but still smaller than unity, it is usually best to use estimates for rounding error sums, which we consider in Chapter 18. If S is small, and the size parameters T and M are of similar order of magnitude, then Swinnerton-Dyer's argument is more powerful than the A or B comparisons. If S is very small, then we may use a long sequence of A and B steps, iterating between ranges where M is much larger than T and ranges where T is much larger than M. A scheme for estimating R can be presented as follows.
The integer points close to a curve
50
Case 1. T large, S small. Use the Ar process with r large to reduce T but increase S.
Case 2. T large, S large. Use the Al process repeatedly to reduce T. Case 3. T near M, S small. Use Swinnerton-Dyer's method.
Case 4. T near M, 6 large. Use exponential sums. Case S. T small. Interchange M and T by the B process. The number of possible cases, depending on the sizes of the four parameters, and also on the possible vanishing of derivatives, is so large that we do not attempt to give a comprehensive set of bounds.
When the B process is used, and when we use an estimate in the last comparison problem, then we need certain derivatives to be non-zero. When
the B process is applied to y = TF(m/M), then we need F'(x) to be nonzero. Swinnerton-Dyer's method requires F" (x) not to vanish, and bounds for
rounding error sums using exponential sum methods also require certain derivatives or determinants of derivatives to be non-zero. The derivatives of the function defining the curve in the final comparison problem will be rational functions in the derivatives F(T)( ) of the function F(x) defining the curve in the original problem, evaluated at one or more nearby points . There is a useful class of functions for which we can control all these derivatives.
First we make a small change in the notation. In Theorem 3.3.1 the function F(x) was defined for 0 5 x 51. We replace x by x - 1, so that the arc of the curve corresponds to a subinterval of 1 <x 5 2. Functions of a complex variable are usually defined so that 0 is a singular point and 1 is a regular point. Let E(a, /3, p) denote the union of all closed discs in the complex plane with radius p and centres in the interval a 5 x5 /3 of the positive real axis. Definition The function F(z) is called a monomial function of degree s on E(a, /3, p) to accuracy S when
1. F(z) is regular on a domain D containing the set E(a, /3, p), 2. For some constant A we have
F(z) =A(zs +f(z)), with I f(z)I 5 S on E(a, /3, p). We call Az' the monomial part of F(z). Lemma 3.3.2 (Monomial functions) Let F(z) be a monomial function on E(a, /3, p) to accuracy S, with monomial part Az', where A is positive and s is
The iteration
51
< tr, and
real. Then for real numbers to, t1, ... , tr with to=O, to < tl < T = tr S p/4, i f s i s not 0, 1, ... , or r - 1, then the function r
r
H(z)F, F(z+tj) f 1
k*j
1
(tl-tk)
is a monomial function of degree s - r on E(a, /3, p/2) to accuracy A with S
4
r
P
I s(s -1) ... (s - r + 1)I
-rImax(as-r+1,()q+ r)s-r+1).
+TIs
Also, if 3 < (a - p)s and the expression
-AS Pi =Ap lslmin((a- p)S-1,(,8 is positive, then the inverse function F-1(z) is a monomial function with +p)s-11
monomial part (z/A)1"s on E(Aa', A/33, pl) fors > 0, or on E(A/3s, Aa', pl) for s < 0, to accuracy )1/S- 1
S1 =A1/$ Smax(a - p)S - 6)1/5-1(13 + p)S + S B
Proof The function H(z) is regular at any point zo where each F(zo + t1) is regular, and by Lemma 3.2.2 applied twice, to Re F(zo) and to Im F(zo), we have
H(zo) = i (Re F(r)(lj) + i Im Pr) (,q)) r. for some , rt of the form zo + t with t real in the range 0 < t < p/4. Let C be the circle centre zo, radius p/2. If zo lies in E(a, /3, p/2), then C lies within E(a, /3, p), and we have
F(')O r!
1
F(z)
1
Azs
2ai c (z - e)r+1 dz
tai c (z -
Af(z) dz+-(d )r+1 tai c (z 1
-).+1
=s(s- 1)...(s-r+1)As-'+ tai rc The first term differs from As - 1)
IAs(s - 1)
(s - r + 1)It
dz (zAf))r+1
(s - r + 1)Az0, -' by at most T)s-r+1) max(as-r+l(
The second term has modulus
S Za 21rA3 41 P
a+
dz.
The integer points close to a curve
52
The inverse function is defined on a domain D1 which contains the conformal image of E(a, (3, p). Under the simpler function Azs, the points a and f3 are mapped to A a s and A P'. The circle centre x, radius p is mapped to a curve C given by Ps(x+te`o)s-1ei°dt,
A(x+peie)5=Axs+A f 0
with
fPlsl(x f Ps(x + te'B)s-
Ieie
P)'- 1
dt
ifs >-1,
f Pl sl(x + p)s-1 dt
ifs < 1.
0
dt I >
o
0
Thus the interior of C contains a disc centre Ax', radius po=ApIslmin((a-p)s-1,(a+p)s-1)
The function F(z) =Azs +f(z) maps the circle centre x, radius p to a curve C1 which contains a disc centre x radius po -AS. If C = F(z), then
z = (i/A -f(z))l"s, so that
z - CIA )I/$
'IS
t'/A-f(z) 1 s 3/A
o)1/s-1 do)
=
1 w1/s-1 d (o,
z'+f(z) S
and
Iz-Q/A)1/sls 3
maxIwII/s-isSl.
The iteration steps take us from the original problem to an easier comparison problem for which we know a bound. To calculate the form of the bound
in the original problem, we trace the steps in reverse order, using the following Lemma.
Lemma 3.3.3 (Exponents in the iteration) Suppose that the function G(x) satisfies the conditions for the following bound for the number of solutions of (3.1.1) with F(x) = G(')(x): L A M"T"(log M)r, (3.3.3) M the sum being over certain vectors (K, A, µ) v, ) of exponents, with O5 K< 1 and K + v >- 0. Then the number of solutions of (3.1.1) with F(x) = G(x) R << E S"
satisfies r
L
R << EI S"( M)
x
1/C
1 Mµ-
1T °(Iog
L,
(3.3.4)
The iteration
53
summed over the same vectors of exponents, with 0
if v>0,
-1-1/r
if -K
-1
and
C=r+ 1 + Kr(r- 1)/2+ vr(r+ 1)/2. Also if G(x) satisfies the conditions for the validity of (3.3.3) with F(x) = G-1(x), then the number of solutions of (3.1.1) with F(x) = G(x) satisfies
lx
My+KTµ-K(log
R«1+SL+ESK(M
M)
(3.3.5)
summed over the same vectors of exponents.
Proof We use the same process as in the proof of Theorem 3.3.1 with R(u) << E S'K (
L\A
M"T"(log M)
)
where
S' = SB(u),
T' = AMA(u).
By Lemma 3.2.2 we have
L U
L 1a xR« F, R(u) << ESK I MJ M'T°(logM)6E(K,v), 1
U
where
E(K, V) = F, (B(u))K(A(u))v. U
If v is positive, then we use
A(u) 5
(rU)r(r+1)/2'
B(u) 5
(rU)r(r- 1)/2'
to get
E(K, V) << Ur(rU)
r(r- 1)/2+ vr(r+ 1)/2
If v is negative, but K + v > 0, then we apply Holder's inequality and Lemma 3.2.3 to get 1-K
lK
(B(u))K(A(u))v <- (U 1) ( B(u)(A(u))v/K1 << Ur(1- K)(2rrr+ 1 logr rU)-
v(rr2-r+2Ur2+r)(K+ v)/2 ,
the power of the logarithm being raised from r to r + 1 if v = - K.
J
o
54
The integer points close to a curve
3.4 SWINNERTON-DYER'S METHOD: THE CONVEX POLYGON We suppose in this section that F(x) is three times continuously differentiable and that F"(x) has constant sign. By changing the sign of y we can suppose that F"(x) < 0 for 0 <x5 L/M. We suppose that
0=T/MZS1, and that S < z with
0
82 < -
min
IF" (x)I.
4 05x5L/M We consider the points (mi,Y) on the curve with m; integers and
(3.4.1)
IIYII <- 8.
Let n; = [[Y]] denote the nearest integer to the real number Y. We put
ui=m;-mi-1, and
vi =ni-ni_1=Y-Y_1+9,8, where I9iI S 2. The heart of the method will be to show that the determinants di! = uiv1 - u1vi
are non-negative, and that the determinants almost determine the points concerned; this requires ingenious divisibility arguments. By the mean value theorem (Lemma 3.2.2 in its simplest case r = 1) v,
uejs
=
(3.4.2)
for some i in
mi_1/M< ; <mi/M. For a lower bound we use Lemma 3.2.2 with r = 2: v1 - 018 v2 - 028 T (u1 +u2) F" u1 U2 M 2M
()
(3.4.3)
for some i in
m0/M<e<m2/M. We see from (3.4.1) and (3.4.3) that
d21 +28(u1 +u2) u1u2
> 28 (u1 + U2).
Since u1 and u2 are positive integers, u1u2 > (u1 + u2)/2, and
d21>(S(u1+u2)-1)2-1.
(3.4.4)
Swinnerton-Dyer's method: the convex polygon
55
We see that d21 (and similarly d;+1,i) cannot be negative. The integer points Pi, (mi, ni), close to the curve form a convex polygonal line. The gradients vi/ui form a decreasing sequence. No four vertices form a
parallelogram, so the differences m, - mi, n1 - ni together determine the points Pi and P,, we use this in the next section. We call PiPi+1 a minor side
of the polygon if neither P;_1 nor P;+2 lies on the straight line PiPi+1. A major side of the polygon consists of a maximal sequence Pi, Pi+ 1,
, Pi+, of
collinear consecutive vertices, along some line y = (ax + b)/q with a, b, q integers, and a/q in its lowest terms. By (3.4.2) the gradient a/q lies in an interval I with length j
AL+
AL
There are at most two intervals of x on which TFlMax+blS6.
(3.4.5)
q
The integer points Pi,..., Pi+, fall into one or other of these intervals. A proper major side is a maximal sequence Pi, Pi+ 1, ... , Pi +, of at least three collinear consecutive vertices on a line y = (ax + b)/q, lying within an interval of x on which (3.4.5) holds. The integers mi, ... , mi+r are in arithmetic progression with common difference q. We see from (3.4.3) that a proper major side has length at most K, with qt<-K<< By Lemma 1.2.3 the total number of integer points lying on proper major sides is 1
a/ge.9'(K)ni q
1/2
1/2
-( 6
<<
z6L+I 1
l
.
(3.4.6)
On any major side there are at most four integer points which do not belong to some proper major side. Hence the number of integer points not on proper
major sides is no more than three times the number of vertices of the polygon.
We now pick a parameter N and show that not too many sides of the convex polygon have length less than N. Let Q be the number of quadruples
of integer points, indexed by i, j, k, 1 say, which are true vertices of the polygon (in order, but not necessarily consecutive) with
mi<mj <mk<mi5mi+N.
(3.4.7)
Lemma 3.4.1 (The polygon method) Let F(x) be three times continuously differentiable on 0 5 x 5 L/M with F" (x) 0 0. Let R be the number of solutions of (3.1.1). For 6 < z with 62 <
0 min IF"(x)I, 4 osxsL/,u
The integer points close to a curve
56
for N < L an integer parameter, and Q as defined above, we have S
1/2
R<<SL+( -)
L L 3/4 +N_+( N) Q114
Proof We divide the interval 0 < m <- L into at most L/N + 1 subintervals of length at most N. Let Ri be the number of vertices of the convex polygon
which fall into the jth interval. If R. Z 4, then the number of ways of choosing four vertices is Ri!
Ri
4!(Ri - 4)!
256
We deduce that
E R!
Q,
Rjz4
so that
E Ri <
The lemma now follows on adding the bound (3.4.6) for the number of integer points which are not vertices of the convex polygon.
SWINNERTON-DYER'S METHOD: COUNTING QUADRUPLETS
3.5
Swinnerton-Dyer's argument (1974) for bounding Q uses `divide and conquer' again. Dirichlet's pigeon-hole principle is applied to the determinants formed with the coordinates of consecutive vertices. The next lemma corresponds to
the First Spacing Problem discussed in Part III with its ingredients of inequalities, symmetric equations in integers, and sums over common factors.
Lemma 3.5.1 (The quadruplet bound) With the notation and under the assumptions of Section 3.4, for A < 1 we have Q«A3N9log2M1+AL+(loAM
+
og3 LlM M1+
OM
(3.5.1)
Proof Q counts the number of sets of four vertices of the convex polygon satisfying (3.4.7). To simplify the notation, we put
a1=mi- m;, b1 =ni-n,,
a2=mk-mi, b2=nk-ni,
a3=m1-mk,
b3=nt-nk.
Since the vertices are in order, the gradients b1/al, b2/a2, and b3/a3 are in decreasing order, and the determinants Crs = arbs - asbr
(3.5.2)
Swinnerton-Dyer's method: counting quadruplets
57
are positive integers for r > s. In this notation (3.4.2) is b,
-T
a(prS
=
PQ)
for some cp, with I cp,I < 2 and some 1 in
m ./M < 1 < Mk/M,
(3.5.3)
and differencing two such expressions gives b, - (pr S
bs, - cps 8
a,
as
<< ANmax IF" (x)I.
We deduce that
zAN3+3N.
Crs
(3.5.4)
Since c,s is a positive integer, the term SN makes no difference unless N>> 1/S, in which case by (3.4.1)
AN3 >> N» SN. Hence we can sharpen (3.5.4) to
c,, «ON3.
(3.5.5)
We restate (3.4.3) as
bl-cp16
b2-(P26
al
a2
T (a1+a2) 2M
M
F"(e12)
for some 1;12 in the range (3.5.3). Subtracting the corresponding expression with suffices 2 and 3 gives a third divided difference, with
a2(bl - cp16) -al(62 ala2(al +a2) =
cp28)
a3(b2 - c025) -a2(b3 - (P36) a2a3(a2 +a3)
2(F"(623)-F"(612)) z
M
maxIF (3)(x)I«ON/M.
Clearing fractions and rearranging gives
a3(a2 +a3)C21-al(al +a2)c32 ' AN6/M+ SN3.
(3.5.6)
Our aim is to show that the three determinants c21, C311 and c32 almost
determine al, a2, a3, b1, b2, and b3. If we choose N so small that the right-hand side of (3.5.6) is less than unity, then we recover the exponent 2/3 that occurs in Theorem 2.4.2 and in Jarnik (1925). Instead we eliminate a2 between (3.5.6) and the determinant identity a2C31 = a1C32 + a3C21,
(3.5.7)
C32(C31 +C32)aj -C21(C21 +C31)a3 z c31(AN6/M+ SN3).
(3.5.8)
to get
The integer points close to a curve
58
We take c21, c31, and c32 as fixed, a1, a2, and a3 as variable; more precisely
we consider triples of integers k1, k2, and k3 for which (3.5.6) and (3.5.7) remain true with a,. replaced by kr and for which there is a choice of 11, 12, and 13 with Crs = krls - kslr.
(3.5.9)
The argument is based on (3.5.8). We must be careful with common factors. Let a = (a1, a2, a3).
Since a is a factor of each crs, we can write
(a1,a3) =ad.
(C21,C31,C32) -ac,
From (3.5.7) we see that acd I c31. We take the common factors a, c, and d as fixed also. We also need an order-of-magnitude subdivision. We say that k1, k2, k3 is a leading triple if k1 z k3, otherwise a trailing triple. We divide the leading triples into classes indexed by the integer r >- 0 for which 2` > k1 > 2`-1
(3.5.10)
Here r takes at most log N/ log2 + 1 values. We treat the trailing triples similarly, according to the size of k3. Suppose that a1, a2, a3 and k1, k2, k3 are leading triples in the same class (3.5.10). From (3.5.7) and the analogous determinant formed with the k, we have c31(a2k1 - a1k2) = c21(a3k1- a1k3), c31(a2k3 - a3k2) = c32(a1k3 - a3k1). We see that the integer
e = a1k3 - a3k1
(3.5.11)
is divisible by
_
C31
C31
(C211 C31) ' (C31, C32)
_
C31
C31
(c21, C311 C32)
ac
Eliminating c21 from (3.5.8) and its analogue for the ki gives C32(C31+C32)(a2k2-a2k;) << C31(AN6/M+SN3)(a2+k3). (3.5.12)
Since both triples are leading, and (3.5.10) holds with the same r, we have
a3 +k2 5 a1a3 + k1k3 S 2a3k1 + 2a1k3. Substituting in (3.5.12) and cancelling the positive integer a1k3 + a3k1 gives e<<
AN6
c31 I1
(C31 + C32)C32
M
+SN2 «1 C32
M (iN6
+ SN 2
.
(3.5.13)
Swinnerton-Dyer's method: counting quadruplets
59
The most delicate part of Swinnerton-Dyer's argument is to show that e determines k1, k2, and k3. Since (a1, a3) = ad, any triple giving the same e in
(3.5.11) has the form k1 + als/ad, k2 + a2s/ad, k3 + a3s/ad for some integer s. For the first and third members to be divisible by ad, we must have a 2 d 2 I s(a1, a3).
Thus ad I s. However, if (3.5.10) is to hold for k1 + als/ad, then I sI < ad, so s = 0 and the triple k1, k2, k3 is uniquely determined by a1, a2, a3, and e. We argue similarly with b1, b2, and b3, assuming that a; and c;j are all
fixed. The only integer triples which can replace the b, have the form b. + ais/a, where s is an integer. To bound s we return to first principles. By (3.4.2)
a
=
bl+als/a
-
ab
MT
Q)
1=
1
`
1
for some 6 and q in 05
,
111
5 L/M. Thus
S
-+ALmaxIF"(x)I «S+a AL.
s<< a
(3.5.14)
a1
Again if L/M is small and a not too large, then we have s = 0, and the b1 are uniquely determined. Thus Q, the number of sets of four vertices of the convex polygon satisfying (3.4.6), is bounded by
E
Q<< E E E a
C
E
F,
F,
d c21=0(modac) c31=0(modacd)c32=0(modac) e=0(modc31/ac)
F, log N, S
where the ranges for c,1, e, and s satisfy (3.5.5), (3.5.13) and (3.5.14). The length of the sum over s depends only on a, and we can write
Q<< F, (1 + a OL)log N E E S(c, d), a
d
c
where (with the same conditions of summation for cr1) ac
Ac, d) C21
C31
03N9
C31C32
C32
(AN6
M
+ 8N 3
AN3 ac loge M AN6
<< a3C3d + ac
a2c2d
M
))
+ SN 3
where we have estimated the sums by comparison with the corresponding integrals. Thus 2
9
Q «&3N9(1 + AL)log2 M+ AM (1 + LL log M)I 1 + N3 )1og4 M, and the lemma follows.
1
The integer points close to a curve
60
3.6 EXPANDING THE RESULT Because of the multiplicity of cases, we build up our result by stages. Lemma 3.6.1 (Swinnerton-Dyer's bound for small S) Let F(x) be three times continuously differentiable with F" (x) non-zero on 0< x 5 L/M(< 1). Suppose that S2 < A
min
4 05x5L/M
IF"(x)I,
(3.6.1)
where
0=T/M2<1.
(3.6.2)
Let R be the number of solutions of IITF(m/M)II < S with 0<_ m < L. Then
R << 1+5 +SL+ r
+ z3/10L9/lo(log M)1/5I 1 + AL + `
3
L1
M)
1/10
M l 1/'
1
(
+ S1/7&/'L1/'(log M)5/7I 1 + 1
2 1 ggM +
AL log M 1
(3.6.3)
.
Proof From the form of the bound in Lemma 3.4.1, we see that we should choose parameters so that Q has the same order of magnitude as L/N. If this leads to a choice of N less than unity, then the bound that we get will be greater than L, and so valid trivially. If the choice of N becomes greater than L, then Lemma 3.5.1 implies that there cannot be four vertices within an interval of length L. There are two critical values of N: L
N1=(03l0g2M)
1/10
Llog' QM + M (1+L log2M 1
N2
1+
M1-1/10
,
1/7
1
(3.6.5)
ALlogM (S02log5 M)1/7 For small S, we have N1 < N2, and we pick N x N1, with 2
Qx N xA3/10L9/1o(log M)1/5(1+AL+
L1M3 M)
1
For large S, we have N2 < N1, and we pick N x N2, with 1/7
X S1/'02/'L1/'(log M)5/7(1 + LL log M ) Lemma 3.4.1 provides the further terms QX
NZ
1 + SL +
(3.6.4)
J
1/10
Expanding the result
61
If 6 > 1/2, then the bound
S+6L is trivial.
Next we use the interchange argument, and apply Lemma 3.6.1 to (3.3.1) instead of to (3.1.1). In the notation of Section 3.3, min IF"(x)I IF"(x)I . IG (y)I = IF-(x)12 max IF'(x)12 If S' >_ Z, then we use instead the trivial bound 6'(K + 1) and, if K < 1, then we use the trivial bound S' + 1. Lemma 3.6.2 (Small d, small T)
Let F(x) be three times continuously differen-
tiable with F'(x) and F"(x) non-zero on 0 <x < L/M(s 1). Suppose that 1 62< 4M
(3.6.6) max IF'(x)I2 where the extrema are over the range 0 5 x 5 L/M. In the notation of Lemma 3.6.1
R << 1 + SL + S/AM+ OT) 1/2 +
0s/1oLs/1o (log
+ S1/7&/7L(log
M)1/5 1+ M)5/7
1+
1/10
L
g2
T +IM loM+ T
L log3 M 1
M
)
1/7
(3.6.7)
L log M
We can remove the restriction on S in Lemmas 3.6.1 and 3.6.2 by choosing an integer r z 0 so that 8/(2r + 1) does satisfy the conditions of the theorem, and applying the theorem 2r + 1 times with F(x) replaced by f (x) + 2s3/(2r + 1)T, for each s in -r s s < r. However, if L and S are both large, then exponential sum methods are probably better. Theorem 3.6.3 Let F(x) be three times continuously differentiable with F"(x) non-zero on 0 5 x 5 L/M(< 1). In the notation of Lemma 3.6.1 we have
R<< 1+3 +6L+S/03/4+(S/A)1/2 + 03/1°Lv/1o(log M)115(1 + S2/0)1/2 I
x11+AL+
log2 M L log3 M OM
+
+ S1/7y/7L(log M)5/'(1 +
1/10
M
62
A)
3/7
1/7
1
(1 + AL tog
M)
.
(3.6.8)
The integer points close to a curve
62
If, in addition, F(x)
0 for 0 5 x S L/M, then
R << 1+8 +SL+S/AM+6M1/2+(6T)'/2(1+82M)'/4 52M)1/2
+ Q3/10L9/10(log M)115(1 +
X(1+
L T
+OMlog2M+
L 1og3 M
1/10
M
+ S1/702/7L(log M)5/1(1 + 82M)3/7 1 +
3.7
T
\1/7 I
LlogM)
.
(3.6.9)
POINTS ON THE CURVE
When we consider points on the curve, then S = 0, and most of the terms in Theorems 3.3.1 and 3.6.3 disappear. Theorem 3.3.1 gives R << M("-1)/(n+1)T2/n(n+1)
(3.7.1)
for (3.7.2) (n - 1)/2 5 t5 n/2, where t = log T/log M. For n > 3 (3.7.1) is a very good result. Lemma 3.6.1 gives
R << (M2/5T1/5 +M1/5T2/5)(log M)1/2,
(3.7.3)
which is better for 1 5 t < 9/7. For t < 1 we interchange M and T and we can apply Theorem 3.3.1 with n >- 3 or Lemma 3.6.2. The bound of Theorem 3.3.1 is obtained by reducing to a range where T is small, interchanging and estimating trivially. Hence, for
t > 1 we can always improve the result of Theorem 3.3.1 by reducing to a range where t is small, interchanging and applying Theorem 3.3.1. If we denote the reduction step by An with n >_ 2, and the interchange step by B, then Theorem 3.3.1 uses the steps A,, B, and can always be improved by using A,. BASB, where r = n or n - 1. For instance, A3BA15B gives the bound R << M151/294T13/84
for 134/105 5 t 5128/91 (about 1.2762 to 1.2967), which is better than (3.7.3) for t >_ 922/721 (about 1.2788), a number slightly less than the 9/7 above. A small further improvement can be obtained by taking three pairs of A and B steps, and so on indefinitely.
The saving from each further pair of steps is very small. The optimal reduction step is from t large to t close to zero. Although (3.7.3) is better than the iteration for t near 1, stopping the reduction at t near 1 and using (3.7.3) gives worse estimates. We shall see an analogous situation in the rounding error problem, where the double sum of Chapter 18 gives results that cannot be reached by the van der Corput iteration, and yet are not strong enough to yield new results in other ranges by way of the iteration steps.
4 The rational points close to a curve 4.1 MAJOR AND MINOR SIDES OF THE POLYGON We modify the problem of the previous chapter, and consider rational points
(a/q, b/q) close to the curve
y=f(x)=TF(x/M) satisfying
q
-f(q)I
(4.1.1)
Q)
with denominators in a range Q< q < 2Q. We do not require that the highest common factor of a, b and q be one, but that each rational point that can be written as (a/q, b/q) with Q 5 q < 2Q is counted once only, however many triples of integers a, b, q give the required ratios. We can consider the triple a, b, q as a point of the projective plane, although the curve y = f(x) need not be algebraic. This is the same as counting triples (a, b, q) of integers satisfying (4.1.1) with 15 q < 2Q and highest common factor (a, b, q) = 1. Lemma 4.1.1 (The simple asymptotic sieve) Let R be the number of rational points with highest common factor (a, b, q) = 1, q< 2Q, 05 a/q 5 M, satisfying (4.1.1). Then 16 SMQ2
+ O(MQ(Q + 8)).
(4.1.2)
RS 13 +1M(Q+1)(2Q+1).
(4.1.3)
R=
3t(3)
We also have the inequality 88
Proof We start from the Mobius inversion formula
F, µ(d) = 0, dlr
except for r = 1, when the sum is one. We apply this with r = (a, b, q). Thus Q
R= L
F,
F,
F,
q=1 05a5Mq b dl(a,b,q)
µ(d),
The rational points close to a curve
64
where the sum is over b satisfying (4.1.1). We write a = da', b = db', q = dq',
noting that the point (a'/q', b'/q') is the same as (a/q, b/q), so that (4.1.1) still holds. Thus
R = F, µ(d)R(d), d52Q
with 2Q/d
L
R(d) _ E
E
q'=1 05a':5 Mq' b' 2Q/d
/26q'
E (Mq' + O(1))I q'=1
2Q
Q
+ 0(1)
(3(2Q)3+o(Q2
166MQ2
+ 0 Q(MQ d2
3d
+OdQ)+o
+ 6)
`
11 '
and
R=
µ(d)
166MQ2 3
d52Q
d
s
1
a. d52Q d
+0 Q(MQ+S) F
Both infinite series converge to values of the Riemann zeta function. The first sum over d is
1/x(3) +O(1/Q2), which gives the first result. The second result follows from 2Q
R5R(1)5 E
+1
q=1
2Q 26M q=1
5
8SM
Q
2Q
q(q+1)+ >. M(q+1) q=1
(Q + 1)(2Q + 1) +M(Q + 1)(2Q + 1).
0
This lemma gives an asymptotic formula when 6 has order of magnitude greater than one, with no restriction on the function F(x). Now we suppose that F(x) is twice continuously differentiable for 05 x 51, corresponding to
the range 05 a/q <_ M. We can use the rounding error function p(t) to obtain (4.1.2) with a better error term, giving an asymptotic formula when 6 is not too small. When 6 is small, then we cannot expect a non-zero lower bound. When 6
is small, but the integer Q is small, then we can consider each possible
Major and minor sides of the polygon
65
denominator q separately, and use the results of the previous chapter with M and T replaced by qM and qT. The case of interest is when S is small and Q
is large. We want arguments that involve rational points with different denominators. The differencing method does not work, because the difference of two rational numbers usually has a much larger denominator than either. The convexity method fails because the rational points close to the curve do not form a convex polygon. However, we can still argue using determinants.
We number the solutions P, = (a;/q;, b./q;) in order of b/q increasing, and for b/q fixed, in order of a/q increasing. We suppose that &< 4, and we put
f(aib;+S0. qi
q1
with - 2 < 9i < 2. We define the determinants a, b, q, ai Disk = a1
bj
qj
ak
bk
qk
0,
q1
a;
0,
q1
ak
ek
qk
0ijk =
We also write A = T/M2 for the order of magnitude of f" (x). Despite the notation, we do not assume that A is less than unity. We suppose that, for
05x<1,
1/C
0/C < f" (x) 5 CO.
(4.1.4)
Lemma 4.1.2 (The second difference as a determinant) For i < j < k we have ak a; l0,Jkl <-16Q2
- -- , qk
and for some
(4.1.5)
q,
between a./q; and ak/qk we have a, qj ak qk ak qk
Disk = 60iik +
ai
q,
a,
qj
ai
qi
(4.1.6)
2q,gfgk
Proof We apply Lemma 3.2.2 with a; x=-, q1
a; ak t1=---, t2=---. qj qj qk q, a1
a1
Then, on clearing fractions, we have
b,+0,6
b1+06
(a1qj - aJqi)(a1gk - akg1) + (a1gi - a,qj)(ajgk - akgf)
bk+°k6 + (akq, - aigk)(akgt - ajqk)
F (M1
2gagtqk
The rational points close to a curve
66
which can be thrown into the determinant form (4.1.6). To prove (4.1.5), we note that 10,1 <- 2 for each r, and that ak 9itk = Bi qjqk
a,
-q
- O qi qk
qk
(ak 9k
qr)
qk
aj 9-
a,
ai
' ak
ai
(Bigjgk - OJgigk) +
ai
- qi` qj
+ 0k q1 q1 I
)(okqiqi - OJglgk).
qj
Hence IO,jkI516Q2
-Rk
a-'
qk
q1
+
aa)
a,
qj
qi
0
which is (4.1.5).
Lemma 4.1.3 (Non-collinear points) There are at most 64SMQ2 values of r with 1
r,r+l,r+21 >
1B
(4.1.7)
2 8'
and at most 2Q(2C0)1"3 M
(4.1.8)
values of r with ar+1
qr+ 1
ar+2
qr+2
ar+2
qr+2
ar
qr
ar+1
qr+l
ar
qr
1
>_
2grgr+lqr+2
(4.1.9)
2
for some e between 0 and M. Corollary
The number of rational points on the curve which can be written as
(a/q,b/q) with QSq <2Q is 5 2Q(CD)1 "3M +2.
(4.1.10)
Proof The inequality (4.1.7) and Lemma 4.1.2 imply ar+2
a,
qr+2
qr
1
32SQ2'
and the first bound follows from R-2
1
(4.1.11) r=1
qr+ 2
qr
Major and minor sides of the polygon
67
Similarly the inequality (4.1.9) implies
I1
ar+1
ar
ar+2
ar+1 1 ( ar+2
ar
qr+1
qr
qr+2
qr+1 )I` qr+2
qr
1
Z 8COQ3'
and since ar+2
ar l
qr+2
qr
2
) 4 ar+2
ar+1 l r ar+1
qr+l
qr+2
1
1 I`
qr+l
ar qr
we have ar+2
ar
qr+2
q,
1
1
1/3
Q(2'CA
and the second bound (4.1.8) follows from (4.1.11).
In the Corollary we have S = 0. By Lemma 4.1.2, the determinant Dr, r+ 1, r+2 must be a non-zero integer, so (4.1.9) holds: in fact, with the lower
bound 1 replaced by one, so we can omit the factor 21/3 in (4.1.8) to give a valid upper bound for R - 2. O We join the points Pl, ... , P. to form an open polygon. Unlike the case of integer points close to a curve, this polygon need not be convex unless S is very small. We call P,Pr+1 a minor side of the polygon if neither Pr_ 1 nor Pr+2 lies on the straight line PrF.+1. A major side of the polygon consists of a maximal sequence P,P,+1... Pr+r of collinear consecutive vertices. The determinant D;jk in Lemma 4.1.2 is non-zero unless Pi, F,, and Pk are collinear. When D.Jk is not zero, then one of the two terms on the right of (4.1.6) is numerically at least one-half. Putting i + r, j = r + 1, and k = r + 2 (or i = r - 1, j = r and k = r + 1), we see from Lemma 4.1.3 that the number of minor sides is at most 64SMQ2 +2Q(2CA)1/3M+ 1.
(4.1.12)
The difficulty lies in estimating the number of rational points which lie on major sides. We use (4.1.12) in conjunction with the results of Lemmas 4.1.9, 4.2.2, and 4.2.3.
Lemma 4.1.4 (Major sides) A major side of the polygon Pl ... PR lies along a rational line
lx + my + n = 0,
(4.1.13)
where 1, m, and n are integers with highest common factor (1, m, n) = 1. If P. and P,+ 1 lie on the line (4.1.13), then ar+1
qr+ 1
a,
In
- - Z qr
grgr+ 1
The rational points close to a curve
68
Proof We have
la,+mbi+nqi=0 for i = r, r + 1, so 1, m, and n are in rational ratios, and we may take them to be integers with no common factor. Since (1, m, n) = 1, we have qi = (1, m)si
(l, m) I qi,
for some integer si. Eliminating n, we have
l(arsr+l - ar+1Sr) + m(brsr+l - br+lsr) = 0, and
m
u
ml(argr+1-ar+lqr)
(1,,n)I(arsr+l-ar+lsr
Lemma 4.1.5 (Long major sides) If the vertices Pi, Pj, Pk (in order) are collinear points satisfying (4.1.1), then
a,llak-aj15325C
lay qj and
min t
a, %
-,-µ= -
- ai
ak
ai
qk
qi
qi
(4.1.14)
iQ
qj
gk
qi
S
326C
(4.1.15)
AQ
Proof Since Pi, Pj, and Pk are collinear, the determinant Disk is zero. In (4.1.6) of Lemma 4.1.2 we have aj
( qj
a,'1 ak qi
` qk
a1\
ak
qj ( qk
ai
qi)
giqjgk 2
F M)
60ijk
We take moduli on the left-hand side, substitute the bound (4.1.5) of Lemma 4.1.2 on the right-hand side, cancel the factor (ak/qk - ai/qi), and estimate the denominators q,. and the second derivative F" at their minimum values, to obtain the first inequality (4.1.14) of the lemma. The second inequality (4.1.15) follows immediately.
O
Lemma 4.1.6 (Short major sides) Suppose that the consecutive points Pi, ..., Pi+d lie on a major side, whilst Pi+d+1 does not. Then either ai+d+1
ai
d
qi+d+1
qi
646Q 2
(4.1.16)
or
ai+d+1 qi+d+1
ai qi
aai+d
ai
( qi+d _ qi
d 32 d Q3 '
(4.1.17)
Major and minor sides of the polygon
69
so that
- - > min
a,+d+l
d
a;
1/31
2d
1.
64SQ2 ' 4Q (CO )
q,
ql+d+ 1
1
Proof We write j = i + d, k = i + d + 1. The area of the triangle P, Pj Pk is the sum of the areas of the triangles Pr Pr+ 1 Pk for r = i, ... , i + d - 1. Each of these triangles has area > 1/2grgr+lqk ? 1/8Q2gk Hence the determinant D.Jk has size I D;JkI
= 2g1gjqk(area PiPi Pk) > d/4.
In Lemma 4.1.2 we have either I80iikl z d/8,
so that (4.1.16) holds, or, for some e a, ai
q,
ak
qi
ai
qk
ak
qk
AF"(/M)
qi
a,
q,
2gIgjqk
d
8)
giving
a,
a;
ak
a1
ak
a1
qi
q,
qk
qt
qk
q,
d 32CiiQ3 '
which implies the remaining inequalities.
Lemma 4.1.7 (Separation of major sides)
j-i=dz2,
Let P,P1 be a major side, with aj a;
- - = A,
q1
q1
and equation lx + my + n = 0. Then a,+1 q1+ 1
-a
>_min
q1
F 2m
d
d
i
6 4S Q2' 1 6 AQ 2
(4.1.18)
and also aj+1
ai
q1+1
q1
->min
d
d
192 SQ2 ' 48C Q2
m ( 2S )
(4.1.19)
Proof Since the line contains d + 1 rational points with x values at least m/4Q2 apart, we have the inequalities A Z dm/4Q2,
d :!!g 4Q2A/m.
The rational points close to a curve
70
If the second possibility (4.1.17) of Lemma 4.1.6 holds, then
- a,12
(aj+i qt+i
q,
d
d2m
J Z 32C Q3A
128COQ5A2'
which gives (4.1.18). For the bound (4.1.19) we modify the proof of Lemma 4.1.6. Let e = [d/2] > d/3. By (4.1.15) of Lemma 4.1.5 we have as/q3 - ar/qr 5 AA'
either with r = i, s = i + e or with r = i + e, s = i + d. We argue as in Lemma 4.1.6 with i, j, k replaced by r, s, i + d + 1. Again we have
e 5 4Q2µ/m.
µ >_ em/4Q2,
(4.1.20)
Corresponding to (4.1.17) we have ar
ai+i ( qj+i
e
2
-, Z 32C Q3µ qr
elm
>
elm 512C2Q' '
128C OQSµ2
giving the second case of (4.1.19).
Lemma 4.1.8 (Vertical major sides) most 366MQ2 rational points.
O
The major sides with m = 0 contribute at
Proof In the case m = 0 we have (1, n) = 1, so x = -n/l, y = b/q with 1 1q .
Let q = kl. Then ly lies in an interval of length 281/Q, with denominator k ::g 2Q/l. By Lemma 1.2.3, there are between 3 and 281 4Q2 Q
+1 12
88Q + 1 5 128Q 1
1
choices for b/q for each fixed n/l. Also, n/l lies in an interval of length M, and 15 2Q, so by Lemma 1.2.3 again, vertical major sides contribute at most
Fl128Q 1
:!g 128Q(2QM+ 1) 5 366MQ2.
0
n/[
Lemma 4.1.9 (Good major sides) Each individual major side contributes at most 72
(2 S CQ3
(4.1.21)
m
rational points. For A z 1, the major sides with m0 0 and A 5 4A a
(CQ)
(4.1.22)
Major and minor sides of the polygon
71
contribute in total at most 144
2SCQ3
0
0
) + 192ASMQ2
(4 . 1 . 23)
rational points, and for B S 1/S the major sides with m >- 1/B8
(4.1.24)
588B1/2CSMQ2
(4.1.25)
contribute in total at most rational points.
Corollary The number of rational points on major sides is at most 606B112CSMQ2.
Proof We use the notation of Lemma 4.1.7. Since d >- 2, the number of rational points on a major side satisfies d + 1 5 3d/2, and by (4.1.20) in the proof of Lemma 4.1.7 we have
d < 3e S 12Q2µ/m,
(4.1.26)
which gives the first assertion (4.1.21). For the other bounds we adapt the argument of Lemma 4.1.3. We number the major sides satisfying (4.1.22) from
1 to K, and call their endpoints P,(k), Pj(k), with dk =j(k) - i(k). We put Xk = ai(k)/qi(k) Since A + 1) >- j(k) and i(k + 2) > j(k) + 1, the bound (4.1.18) of Lemma 4.1.7 gives xk+ 2 - xk ?
as(k)+1
a'(k)
dk
- qi(k) - 64A S Q2 qj(k)+1 2
Thus K-2
K-2
E dks64ASQ2 k=1
(xk+2-xk)5128ASMQ2. 1
The bound (4.1.23) follows when we use (4.1.26) for the final values k = K - 1
and K, and the inequality dk + 1 < 3dk/2. Similarly, we number the major sides satisfying (4.1.24), and define Xk likewise. By (4.1.19) of Lemma 4.1.7
xk+2 -xk z dk/96B112C8Q2. and by (4.1.26) 2 SCCQ3
d k 5 48BS
48S
2BCQ3 :!9 - 96B112CSMQ2
for each k, since OM2 Z 1. This gives the bound (4.1.25). The corollary
The rational points close to a curve
72
follows when we give B its maximum value 1/8 in (4.1.25), and add the bound of Lemma 4.1.8 for the number of rational points on vertical major
0
sides.
4.2 DUALITY We need the notion of the function g(y) complementary to f(x). It
is
convenient to have f(x) defined for all x, with f"(x)> 0, although we only
use values of x in the range 0 to M. The function F(x) is defined for 0 < x< 1, where it satisfies (4.1.4). Suppose that (4.1.4) holds on a range a <_ x < /3 with a < 0, /3 z 1. For x < a, either F(x) is not defined, or the inequality (4.1.4) fails. Define (or redefine) F(x) for x 5 a as the sum of the first three terms in the Taylor series about x = a, and similarly for x >- /3. This gives a function F(x) defined for all x, twice continuously differentiable, satisfying the inequality (4.1.4), and agreeing with the original definition for
0<x< 1. We continue to put f(x) = TF(x/M). The inverse function h(y) of f'(x) is defined for all y. We put
g(y) =yh(y) -f(h(y)), so that
g'(y) =h(y) +yh'(y) -f'(h(y))h'(y) =h(y), and
g"(y) =h'(y) = 1/f"(h(y)). We can easily check that f(x) is the function complementary to g(y). Figure 4.1 shows the relation between the complementary functions f(x)
and g(y) in the special case when f(x) vanishes at the origin. They are represented as areas bounded by the curve y =f'(x).
g(y)
fl FIG. 4.1
Duality
73
Lemma 4.2.1 (The dual curve) On a straight line lx + my + n = 0, there are at most two disjoint intervals with the property that every point on them has
l y -f(x)l5 S/Q.
(4.2.1)
If I is such an interval, containing two rational points Pi, Pj satisfying (4.1.1), with
A=ai ---aiqi q1
then min XE!
l
28
m
AQ
- + f'(x)
,
(4.2.2)
and
g(
l
n
m
m
2C3S2
S
Q+AA2Q2
(4.2.3)
Proof Since f"(x) > 0, the points x with
f(x)5 -(lx+n)/m+S/Q form an interval, which may be empty. The points with
f(x) S -(lx+n)/m-S/Q form a subinterval, which again may be empty. The difference of these sets is one or two intervals on the real line.
Let U, (xo, yo), be the point on the curve y = f(x) where the gradient is
-I/m. Then xo=h(-l/m), so that Yo + lxo/m + n/m = n/m -g(-l/m). In the one-interval case, the value xo must lie within the interval, so lxo
Yo+m-
+
n
m
S
S Q.
In the two-interval case, the value xo does not lie in either interval. We use the rational points Pi and P. By subtraction
f
so, for some
aj
ai
-f(gi
ll
+
l
a,
ai
m q1 qi between ai/qi and at/qj, we have, by the mean value qt
theorem, 1
MO +In This completes the proof in the one-interval case.
The rational points close to a curve
74
In the two-interval case, the point xo does not lie in either interval. A second application of the mean value theorem gives 28
A CIA-xols
Q.
lies between a./qi and a,/q1, whilst xo does not, then either ai/qi or a!/q, lies between l; and xo. Thus for P, (a/q, b/q), equal to either P, or Pj, Since
we have
la
-x
IS
2CS
AAQ'
V
and Taylor's theorem about xo gives
b qS6=
x +(q -x)
(4
2
for some q. Since f'(xo) _ -1/m, and la + mb + nq = 0, we see that lxo
I
YO +
nI
2C3S2
S
m + m 5Q+-z-
which completes the proof of the lemma.
Lemma 4.2.2 (Points close to the dual curve) The number of rational points on major sides is O(C2SMQ2 + C7/651/2A1/6MQ11/6 )
Proof For technical reasons we want an upper bound for the length A of a
major side (measured in the x-direction). If A exceeds the bound µ of Lemma 4.1.5, then we argue as in Lemma 4.1.7. Let the points on the major
side be Pi, ..., Pi+d. Then for e = [d/2] either Pi,..., Pite or Pi+el
) Pi+d
span a distance at most µ in the x-direction. Instead of considering the whole
major side, we take the shorter of the two halves, which contains at least (d + 1)/2 rational points, including the endpoints. Hence we can suppose that A 5 p., provided that we double the final estimate for the number of rational points. We use Lemma 4.1.8 for major sides with m = 0, and Lemma 4.1.9 with A = C for major sides with length A satisfying A 5 4S
(---) K
.
(4.2.4)
We use Lemma 4.1.9, with a value of B to be chosen, for major sides with m > 1/BS. Let N be the length of the interval taken by f'(x) for MS x5 2M, so that N< CAM. By (4.2.2) of Lemma 4.2.1, major sides for which (4.2.4) is false have 1/m within a distance AQ 1 ( 28 A
52
I\
J« 1i cM
2Cm AQ)
Duality
75
of a value of f'(x). Thus l/m lies in an interval of length M' << CAM. We divide the major sides into blocks for which A < A < 2A, Q' m < 2Q', where A and Q are powers of two; A < 1 is allowed. By (4.2.3) of Lemma 4.2.1, and the bound X15 µ, the point (-1/m, n/m) is close to the dual curve y = g(x), with n S' C3S2 l-m)-mI SQ, -QA2Q2 1
Ig
By Lemma 4.1.1 with max(S', ), there are
M`
,
(4.2.5)
Q replaced by M', Q', and with S replaced by
a
)cMQP2)
0((6' + 1)M'Q'2) = 0I I 1 +
choices of 1, m, and n in such a block. The number of rational points on the major side is O(AQ2/Q'), so that we have Q' << min(AQ2,1/BS ).
(4.2.6)
The block contributes AQ2
C3S2
O I Q, + QA )COMQ,2 rational points, which sums over Q' to 1
OCOAMQ2
BS
B232A)),
and then over powers of 2, A, in the range 2S
(Q Q)
sums to / C3/2I1/2MQ3/2 O
B61/2
( SQ
/ C1/2S 1/2Q7/2 Q2 &"2Q1"2 +C4S2Mminlf1
I C3/2A1/2MQ3/2
=OI
23 2 -5C
5A5µ=
BS1/2
01/2
C4SMQ2
+
B
)
We balance this with the terms
O B1/2CSMQ2 +
rL
' BS,
B2C1/2S3
) )
The rational points close to a curve
76
from Lemmas 4.1.8 and 4.1.9 by choosing
B = max c2, SI Q
Cf/T
rr
O(C2SMQ2 + C7"6(31/2A'/6MQ11/6 + C1/261/20-1/2Q3/2)
and the third term is smaller than the second term, so it may be omitted. Lemma 4.2.3 ((The iteration bound)
0
For any e > 0, the major sides contribute
O I C19/9SMQ2I CQT I E+ C11/661/2A1/6MQ3/2log 1 J (QT)
= OI C10/36MQ2
E
+ (CA)1/3MQ)
(4.2.8)
rational points. The implied constants depend on e.
Proof We can assume that (4.2.9)
Q >_ 2C/e2,
since the result follows from Lemma 4.2.2 if (4.2.8) is false. As in Lemma 4.2.2, we use Lemmas 4.1.8 and 4.1.9 with A = C for major sides with m = 0, m >_ 1/BS or with (4.2.4) false. These cases give the terms of (4.2.7). Let N be the length of the interval taken by f'(x) for M 5 x 5 2M, as in Lemma 4.2.2. By (4.2.2) of Lemma 4.2.1, major sides for which (4.2.4) is false have 1/m within a distance 2S
eAM
1
AQ2
(2CmQ) < 4C 11
eN 4
of a value of f'(x), since N>_ OM/C> 1/C, and (4.2.9) holds. Thus 1/m lies in an interval of length
M'
We divide the major sides into blocks for which A < A < 2A, Q' S m < 2Q', as in Lemma 4.2.2. They correspond to rational lines lx + my + nz = 0 which
are close to being tangents, and these lines correspond to rational points close to the dual curve y = g(x) according to (4.2.5). To set up the iteration, we note that
C,
Duality
77
with A'= I/A. We suppose that we have an upper bound which is a sum of terms of the form CC
1 PMQT
SQ
)
with y >- 2 a. The number of rational points close to the dual curve has an upper bound which is the sum of the corresponding terms S'
CC
a
S (Q)
(QI) O'PM'Q'y
2a
A1-a-PA-2aMQ,y
Multiplying by the estimate O(AQ2/Q') for the number of rational points on a major side, we have 12a
1-a-PA1-2aMQ2Q,y-1
O(CC+1() l Q
(4.2.10)
rational points on major sides with A and m in these ranges. Using (4.2.6) as
a lower bound for A and an upper bound for Q', we see that when y > 2 a Z 1, then the expression (4.2.10) is at most O(B2a- IC'+ 1S 4a- W- a- PMQ2a ).
(4.2.11)
The number of values of Q' is O(logl/S). Since max Q2
I
Q1 SA<µ«
(QQI
when A = C, then the number of values of A is (SCQ3 1 O logminl QQ, , s
1
= Ollogminl MQ, 5)).
Hence for y > 2 a > 1, the sum over Q' and A running through powers of two gives
O(1/(y- 2a)(2a - 1)) times the maximum summand. The upper bound for the number of rational points close to the dual curve
will contain several terms with different exponents a, G3, y and . Which term gives the largest contribution depends on the sizes of S, A, M, and Q. If all the contributions of these terms are O(CSMQ2), then we choose B = 1.
Otherwise we take the largest term of the form (4.2.11) and equate it to O(B1/2CSMQ2), by choosing
B = (63 e
The rational points close to a curve
78
so that the expressions (4.1.20) and (4.1.11) are both O(C(t+2y-4a+1)/(2y-4a+1) 8y/(2y-4a+1) 1),&(I -a - P)/(2y-4a+ 1)
XMQ(4y- 6a)/(2y- 4a+ 1))
The bound for the number of rational points close to the dual curve will always include the contribution (4.1.12) of the minor sides. The term O(SMQ2)
has a = 1, a = 0, y = 3,
= 0, so the term (4.2.11) becomes O(C8MQ2/B), O(C1/3&''3MQ) which can be absorbed by the first term in (4.2.8). The term has a = 0, /3 = '3, y = 1, = 3, so (4.2.10) becomes O(C4/302/3AMQ2),
which sums over powers of 2, A< µ and Q', to Of C11/6S1/2A1/6MQ3/2log
1
I,
(4.2.12)
which is bounded by the second term in the first estimate of (4.2.8). Since AM > 1, the term (4.2.12) can absorb the term O(1/(CSQ3/0)) in (4.1.23) of Lemma 4.1.9. When we use the corollary to Lemma 4.1.9 to count the rational points on major sides of the polygon of rational points close to the dual curve, then the term O(C6 112MQ2) with a = 1/2, 63 = 0, y = 5/2 and = 1 gives (C201/2MQ O
B3/261/2 log S 1)
rational points when we sum (4.2.10) over A and Q', and 1)1/41 0 CS/4S5/8&1/8MQ7/4l log l
S
I
when we choose B >_ 1. This gives a three-term upper bound
+ C5/4S5/801/8MQ7/41 (log S r 00136W
=O1
)1/4
+ C71/651/201/6MQ3/2 log 1/4
log2 S + CS/4S5/801/8MQ7/4 (log
+ (CV/3MQ) , (4.2.13)
for the number of rational points on major sides; the second estimate follows from the first by the geometric mean inequality. This bound (4.2.13) is the first step of an iteration, in which we always get
Duality
79
the same first and third terms, but the second term changes under iteration according to
y
1-an-Nn
an+1=
2y,,-4a,,+Qn+1 =
fly,,-6a,, yn+1=
Sn+1=
2yn-4an+1'
2yn-4an+1 +2yn-4an+1 2yn-4an+1
We put
a,, =1-0,,,
yn=3--9,,,
,,=3/2-K,,.
Then 40,,-71n
3-2in+ 40n' 0i,-f3n
(4.2.14) 60n - 71n
77n +I -
3-271,, +40,, K,, -71n+20n (4.2.15)
3-271,,+40,, '
so that qn-20,, 71n+1-20n+1=
3-271,, +40n
,
(4.2.16)
2
77n+1 - 30n+1 = 3
2-q,,
6 40,,
(4.2.17)
If, for some n, we have 12 On - 71n I < 1, then by (4.2.16) 2 0,, - 71,, -> 0. In particular, we have 120,, - rinI '11,,, < 2 for all large n, so by (4.2.17) 71n - 30n - 0. We now deduce that 0n, a,, (from (4.2.14)) and K,, (from (4.2.15)) all tend to zero. The initial values 00 = 4-z, )30 = 0, 710 = 2, Ko = 2 already have 1200 - 7101 < 1, but the step from n = 0 to n = 1 is exceptional because ao = 2 introduces an extra logarithm power. In (4.2.13) we have a1 = 5/8, /31 = 1/8,
y1= 19/8, K1=1/4, 01 = 3/8, m = 5/8, so 1201 - 7ii1= 1/8. Each further iteration step reduces the exponents, interchanges M and N with an expansion factor 1 + O(e), and introduces a numerical factor. After O(log 1/E) steps we have 1 0,, 1, 1,% I < e/2, and I /3,, I < E, K,, > 0, with a numerical factor
depending on E. The term is E1
O(C3/2SMQ2 cT)
J.
The rational points close to a curve
80
where the order-of-magnitude constant depends on e, and we have used log 1/8 <<
We can absorb the first term in (4.2.13) if we replace -y" by max(y,, 10/3) and K by min(K,,, - 11/6) in (4.2.15). With this iteration rule we have J" - 19/9.
0 We combine the bound (4.1.12) for the number of minor sides with the estimates of Lemmas 4.1.9, 4.2.2, and 4.2.3 for the number of rational points on major sides. Theorem 4.2.4 Let F(x) be a real function twice continuously differentiable for
0 5 x 51, with
1/CM. We write
0=T/M2. Let f(x) = TF(x/M), and let R be the number of points (x, y) which can be written as (a/q, b/q) with a, b, q integers, 0:5 x 5 M, Q:5 q < 2Q, and with
Iy-f(x)I=
b
-f(a
S
)I < Q 9 Then, as either M, Q, or T tends to infinity, we have the bounds R << C61/2MQ2 + (CA)1/3MQ,
(4.2.18)
R << C2SMQ2 + C1/6S1/201/6MQ11/6 + (CO)1/3MQ,
(4.2.19)
and, for any e > 0, R << C10/36MQ2 (
MT S
E
) + (CD)1/3MQ.
(4.2.20)
The constants implied in (4..20) depend on e.
The order of magnitude of the average number of rational points satisfying the inequality is O(6MQ2), corresponding to the first term in (4.2.20). In the
special case F(x) = (x + 1)2, T = M2, all the points (a/q,(a + 1)2/q2) with q2 < Q in the range 0 5 x < M lie on the curve. The number of these points has order of magnitude O(MQ) = O(Ol13MQ), corresponding to the second term in (4.2.20). Hence the bound (4.2.20) is not far from best possible. However, bounds like (4.1.4) for the third derivative of F(x) would rule out the special case y = (x + 1)2. Perhaps the second term in (4.2.20) could then be reduced.
A linear form often near an integer 4.3
81
A LINEAR FORM OFTEN NEAR AN INTEGER
So far we have considered rational points close to a curve y = f(x) with f"(x) bounded away from zero, of order of magnitude A, with AM > 1. There was
no need to consider separately the case of a subinterval of length L 5 M, provided that AL > 1. At the opposite extreme, for a subinterval of length L so short that AL2 << S/Q, we can approximate f(x) to an accuracy O(S/Q) by the first two terms of its Taylor series. We shall see this in Chapter 15 as the approximation step of a divide-and-conquer argument. Accordingly we consider the linear problem where (4.1.4) is replaced by
f(x) = ax +,6
(4.3.1)
for 0 5 x 5 M (we allow M < 1). This case is degenerate, that is, a bad case, in
which unusual things can happen because some constraint has vanished. There may be many solutions of the inequality (4.1.1). If so, then the line (4.3.1) is close to a rational line. Such situations are common in the theory of Diophantine approximation. Lemma 4.3.1 (A non-trivial polygon) If the polygon of rational points satisfying (4.1.1) with f(x) given by (4.3.1) has more than one side, then
R5128SMQ2+1. Proof We adapt the ideas of Lemma 4.1.6. We number the true vertices of the polygon of rational points Pi(1), ..., Pi(K) with dk = i(k + 1) - i(k). As in the proof of Lemma 4.1.6, the area of the triangle Pi(r)Pi(r+1)Pi(r+2) is at least
max(dr,dr+1)/16Q3 _ (dr+dr+1)/32Q3 The triangles Pi(2r_ 1)Pi(2r)P;(2r+1) for 2r + 1 S K do not overlap, and they fit into the parallelogram of points close to the given line segment, whose area is
2 8M/Q. For K odd we have at once d1 + ... +dK_ 1 5
648MQ2.
For K even we have d1 +
+dK_2 <_ 64SMQ2,
and we must also consider the non-overlapping triangles Pi(2r)Pi(2r+1)Pi(2r+2) for 2r + 2:!g K, to get d2 + ... +dK_ 1 5 64SMQ2. In both cases
Lemma 4.3.2 (Rational linear approximation) Suppose that f(x) is given by (4.3.1), and that there are R solutions of (4.1.1) with Q5 q < 2Q on a rational
The rational points close to a curve
82
line lx + my + n = 0, where the highest common factor (1, m, n) is one, and
m>0.Then
m5 Ima+ll --
4MQ2 (4.3.2)
R-1 86Q
R-1'
(4.3.3)
and, for m o 0,
Imn+nI < Proof
128MQ
(4.3.4)
R-1
In the case m * 0 we have
ax+/3+
1x+n
Q
m
(4.3.5)
for an interval I in x that contains R distinct rational points on the line. By Lemma 4.1.4, the interval I has length at least (R - 1)m/4Q2. Since I is a subinterval of 0 5 x 5 M, we deduce (4.3.2). Since (4.3.5) holds throughout the interval I, we have
(R4 1)m
28
Q
(a+-_)
QZ
which gives (4.3.3). We deduce from (4.3.5) that
Im/3+nI5
+Mlma+11 < Q
1RSMQ
-1
If m = 0, then 10 0, and, without loss of generality, 1 > 0. There are R points of the form (-n/l, b/q) with 1 I q that satisfy (4.1.1), so R
na
b
1
q
Q
and, by the analogue of Lemma 4.1.4 for b/q, an interval in y that contains R distinct rational points on the line has length at least (R -1)1/4Q2, so
(R - 1)l 4Q2
5
23 Q
Thus (4.3.3) still holds.
O
Putting the last two lemmas together, we obtain a result. Theorem 4.3.3
Let a and l3 be real numbers. Let S, M and Q be positive, with
A linear form often near an integer
83
Q an integer. Let R be the number of points (x, y) that can be written as (a/q,b/q) with a, b, q integers, 0 <x <M, 1
Q
q
Then either
R 5 max(88Q,128SMQ2) + 1, or all the solutions lie on a rational line lx + my + n = 0 with 1, m, n integers,
15m<_
4MQ2
R-1'
and
Ima+lL<
88Q
R-1
(m/3+nj-
12 Q R-1
There is a similar result when we count triples of integers a, b, q with
laa+q/3-bl5 6, without the condition (a, b, q) = 1. The natural range for a and q is a rectangle of the type 0 5 a S P, 0 < q 5 Q. We can eliminate one variable by the divide-and-conquer method, and then use Dirichlet's approximation theorem (Lemma 1.5.1).
Part II The Bombieri-Iwaniec method
5 Analytic lemmas 5.1 BOUNDS FOR EXPONENTIAL INTEGRALS In this chapter we state the analytic ideas underlying the proof as lemmas. Most of these lemmas concern the exponential integrals
f g(x)e(f(x)) dx where x is a real variable and f(x) and g(x) are real functions, and the corresponding exponential sums
Eg(m)e(f(m)). m
We call the function f(x) the exponent (function), and g(x) the weight (function). A pure exponential sum or integral is one in which the weight function is unity. The first lemma enables us to insert a weight function g(x) whenever we have a uniform upper bound for a pure sum or integral. The other lemmas in this section give upper bounds for exponential integrals. Lemma 5.1.1 (Weight functions of bounded variation) Let g(x) be a real function of bounded variation V on the closed interval [a, (3 ], and let F(x) be bounded and continuously differentiable on [ a, /3 ]. Then
If Rg(x)F'(x)dx S (V+Ig(a)I)max IF(a) -F(x)I. If a and /3 are integers and f(n) is any sequence, then a
Q
g(n)f(n)
5 (V+Ig(a)I) max yZa
a
L f(n) Y
Proof We write g(x) as the difference of two positive monotone functions,
g(x) = h(x) - k(x), with k(a)=0 if g(a)>-0, and h(a) = 0 if g(a)50. We treat the integral; the sum is simpler. We have
f h(x)F'(x)dydx Ifa 'h(x)F'(x)dxI =Ifp x=a y=0 -I fh(/3) fp y=0
x=h '(y)
F'(x)dxdyl
5 h(13) max IF( /3) - F(x)I. x
Analytic lemmas
88
We get a similar result for k(x). The lemma follows since
V=h( /3) +k(/3) -h(a) -k(a), and
h(a) +k(a) =Ig(a)I. Lemma 5.1.2 (First Derivative Test)
Let f(x) be real and differentiable on the
open interval (a, 0) with f'(x) monotone and f'(x) >_ K> 0 on (a, /3). Let g(x) be real, and let V be the total variation of g(x) on the closed interval [ a, /3 ] plus the maximum modulus of g(x) on [a, /3 ]. Then V
fpg(x)e(f(x)) dx
ITK
Remark If f'(x) < - K < 0, then we obtain the same bound by taking complex conjugates.
Proof Integrating by parts, we have
e(f(x))
e(f(x))dx=[2arif'(x)j fp «
p «
p e(f(x)) 21ri
f«
1
d(f,(x)).
Let K = max f'(x). The modulus of the right-hand side is 1
27TK
1
1
21r
27TK
1
1
I pd(f.(x) ) =
IT
(K +
1 K
+
1
1
K
K)
This gives the case of the pure exponential integral with g(x) = 1, V = 1, and the weighted case follows by Lemma 5.1.1. Lemma 5.1.3 (Second Derivative Test) Let f(x) be real and twice differentiable on the open interval (a, /3) with f" (x) z A > 0 on (a, 13). Let g(x) be real, and let V be the total variation of g(x) on the closed interval [ a, /3 ] plus the maximum modulus of g(x) on [ a, /3 ]. Then
fpg(x)e(f(x))dxl5
4V
a
Proof As in Lemma 5.1.2 we need only consider the pure exponential integral. If f'(x) changes sign at a point y in the interval (a, )3), then f'(x) z 8A for x Z y + 8, and I f'(x)I z SA for x < y - S. We use fY+se(f(x))dxl y-s f." < 1/ iTSA + 28 + 1/irSA,
f pe(f(x)) dx 5
8
e(f(x)) dx
+l f p e(f(x))dx y+s
where we have used Lemma 5.1.2 on the first and third integrals. We take
Partial summation for exponential sums
89
S = 1/ (ara) . If y does not exist, or y 5 a + S, or y>- 6 - S, then there is a similar subdivision of the range of integration which leads to a stronger inequality.
Lemma 5.1.4 (rth derivative test) Let f(x) be real and r times differentiable on the open interval (a, f3) with f (')(x) >- µ > 0 on (a, G). Let g(x) be real, and let V be the total variation of g(x) on the closed interval [ a,13 ] plus the maximum modulus of g(x) on [a, R ]. Then
f Rg(x)e(f(x))dxl <<
V
a
The implied constant depends on r.
Proof This lemma is proved by induction on r, dividing the range of integration into at most three parts, estimating trivially on the range on which f (r- 1)(X) changes sign, and using the (r -1)th derivative test on the other two ranges, as in Lemma 5.1.3.
PARTIAL SUMMATION FOR EXPONENTIAL SUMS
5.2
The lemmas of this section allow us to pass in both directions between a weighted exponential sum of fixed length and a pure exponential sum of variable length. We use the convention that a sum with limits a and f3 means a sum over all integers in the closed interval (a, a ] even when a or G is not an integer. Lemma 5.2.1 (Partial summation) Let h(x) be real and continuously differentiable on the interval [M, N]. Then we have N
N
L G(n)e(h(n)) 5 M
(1 + 2Ir fNIh'(x)I dx) sup fm MSxSN
L G(n) x
Proof We put H(x) = e(h(x)) in the partial integration identity N
G(n)H(n) = H(M) M
G(n) + f NH'(x) E G(n) dx. M
X
If the functions G(x) and h(x) also depend on a variable y, and we integrate the modulus squared of the sum in Lemma 5.2.1 over y, then the supremum over x can be taken outside the integration over y. We do not need this extension of Lemma 5.2.1 here. Lemma 5.2.2 (Partial summation in two dimensions) Let f(x, y) be real and
Analytic lemmas
90
twice continuously differentiable on H1 S x 5 H2, N1 5 y S N2. Then, for any coefficients G(h, n), H2 N2
H2 N2
E F, G(h, n)e(f (h, n)) 5 C H, N,
max
xZH1, yzN1
E L G(h, n) x
y
where
C = (1 +21r(H2 -H1)suplf1(x,Y)I)(1 +21r(N2 -N1)suplf2(x,y)I) +4Tr2(H2 -H1)(N2 -N1)supif12(x,Y)I, the suprema being over H1 5 x 5 H2, N1 < y < N2.
Proof This estimate follows from the identity H2 N2
H2 N2
L L G(h, n)F(h, n) = F(H1, Ml) E E G(h, n) H, N,
H, N, H2 N2
+ fH,H2F1(x, N1) E E G(h, n) dx X
N,
H2 N2
+ fN,N2F2(H1,Y)
G(h,n)dy Ht
Y
H2 N2
+ fN2 1H2 F12(x, Y) E F, G(h, n) dx dy, Y=N1 x=H1
x
y
when we put F(x, y) = e(f(x, y)). The partial derivatives are
F1 = 21rifl(x, y)F(x, y),
F2 = 21rif2(x, y)F(x, y),
F12 = 21Tif12(x,Y) - 41r2f1(x,Y)f2(x, y)F(x, y).
0
Our next two lemmas compare an incomplete sum with a complete sum modified by factors of unit modulus. The first lemma is suitable for `additive' problems, the second for `multiplicative' ones. Lemma 5.2.3 (Partial sums by Fourier series) Let N be a positive integer. Let a(n) be any coefficients, and let A(x) be the exponential sum
M+N
A(x) = F, a(n)e(nx). M+ 1
Partial summation for exponential sums
91
Then for any integers N1 and N2 in the range 0 < N1 < N2 5 N we have M+N2
F,
f1/2
a(n) <
-1/2
M+N1+1
K(x)IA(x)Idx,
where
K(x) = min(N, 1/2x), with I1/2
-1/2
K(x) dx =1 + log N.
Proof Let N2-Nj
S(x) _
e(nx).
Then M+N2
a(n)= f
1/2 1/2
M+N,+1
A(x)S(-x)e(-(M+N1)x)dx,
since e(rx) integrates to 0 if r is a non-zero integer. This gives the result, since K(0) > N2 - N1, and, for x IS(x)I
0,
sinT¶((N2 -N1)x) e((N2 -N3)x) - 1 sin 7Tx e(x) - 1 < min(N2 - N1,1/2x) 5 K(x).
0
Lemma 5.2.4 (Partial sums by Mellin transforms) Let K be an integer, K1 and K2 be real numbers with 2
F, g(k) =
1
fcG(s)
KS - KS 2
K,
s
1 ds + O(max Ig(k)Ilog K),
where C is the line segment 1/log K - iK to 1/log K + iK, and
2K-1 g(k)
G(s) _ k=K
ks
Proof We expand the integral term by term. A typical term has integrand LS/sks, where L is K1 or K2. If I k - LI < 1, then we argue trivially. On C, I(L/k)SI <21/logK<e, which is bounded, and we use fC
ds S
<< logK+1.
92
Analytic lemmas
D3
E
4 C
4 D2
01
K2
1/logK
Re s
D1 -iK
FIG. 5.1
If k > L, then we move to the contour D (Fig. 5.1) made up from:
D1: line segment 1/log K - iK to K2 - iK, D2: line segment K2 - iKto K2 + iK, D3: line segment K2 + iK to 1/log K + iK. On D1 we have s = o + iK and Isl z K, so that
L Sds 1 cC L ° ID,(k) s SKJ, (..) and the same estimate for the complex conjugate integral along D3. On D2 we have Isl Z K2, so the integrand has modulus 5 1/K2. The length of D2 is 2K, and thus
L)ads
-s <- -2
I
K.
k When k < L we take D to be made up instead of D2
D4: line segment 1/log K - iK to -K2 - iK, D5: line segment -K2 - iK to -K2 + iK, D6: line segment -K2 + iK to 1/log K + iK. The bounds are very similar: the estimates for the integrals along D4 and D6 are larger by a factor \L/k)1/log
K < 21/ log K < e.
Rounding error sums
93
By the residue calculus
L S ds
fc(k)
rL
S
ds
fDI k) s
s =
since the pole at s = 0 lies between the contours C and D. To add up the error terms we note that, for k > L, log k/L =log( 1 +
k-L L
k-L )
L
We put k=[L]+r, with r>2, and K
E
1
L+1
5 r=2
r-L1
There is a similar bound for the corresponding sum involving log L/k k < L. Thus the error terms add up to z K(log K + 1)max I g(k)I.
for
0
5.3 ROUNDING ERROR SUMS We use the rounding error or row-of-teeth function
p(t) = [t] - t+ I21 to express the number of integers n with a < n 5 /3 as
/3-a+p(/3)-p(a).
(5.3.1)
It would be very convenient if we could manipulate the function p(t) directly. Van der Corput and Vinogradov gave much thought to this, but they did not find any technique which could not be parallelled using the Fourier series W
e(ht)
2Irlh which converges to zero when t is an integer, and (non-uniformly) to p(t) when t is not an integer. Although the partial sums of this series give the best finite approximations to p(t) in mean square, we often want the mean first power, for which it is better to introduce a tapering to the partial sums. Lemma 5.3.1 (Tapered partial sums) Let H be a positive integer, and let k(h) be the coefficients of the second Fejer kernel: 2H-h+1C3 - 4H-h+1C3
k(h) = 2H-h+1C3
k(-h)
for 0 5 h 5 H- 2,
forH -15 h 5 2H - 2,
fort-2Hsh50,
Analytic lemmas
94
where C, is the binomial coefficient n!/r!(n - r)!. Then we have
P(t)
k(h) e(ht) 1 1 +O(H +minl 1, H3 - hO k(0) 27rih IItIl3 ))
Proof We use the Fejer kernel H
2H-2
4
E
K(x) = E e(hx)
k(h)e(hx).
h=2-2H
1
To evaluate k(h), we note that k(-h) = k(h), and that k(h) is the number of solutions of g1 +g2 =h1 +h2 +h in integers with 1 5 g1, g2, hl, h2 5 H. For h >- 0 we put h3 = H + 1- g1, h4 = H + 1 -92, so that 1< h 3, h4 s H, and
h1 +h2 +h3 +h4 = 2H+2 -h.
(5.3.2)
The number of positive integer solutions of x1 + +x, = n is the binomial coefficient -1C,_ 1, the number of ways of putting r - 1 gates in a fence with
n posts. For h > H - 1, each hi must be at most H. For H5 H - 2, the equation (5.3.2) has solutions with hi > H. There is at most one value of i for which hi > H, so we must subtract four times the number of solutions of
h1+h2+h3+h4=H+2-h. If x is not an integer, then we see by summing a geometric progression that K(x) = sin4 irHx/sin4 ?rx = K( -x). For 0 < a < 1 we have f
f'/2K(x) dx << J 1/2 a
a
1
1
dx << 3 . x a
If no integer separates a and /3, then
f RK(x)dx<< II«113 + 11/3113'
and more generally
f RK(x)dx= ([ 13]
- [a]) f 1122K(x)dx+O(II«113 +
it /3113
= ([/3] - [a])k(0) +0(1/11a113 + 1/II /3113). There is also a trivial estimate
f
13
K(x) dx 5 (,G-a+1) f
1/2
-1/2
K(x) dx « (/3 - a + 1)H3,
)
Rounding error sums
95
1
n
0
1
2
FIG. 5.2
so that, even if a and /3 are integers, we have
f 0K(x)dx= Q,6 ] - [a])k(0) + O(min(H3,1/I1 a113+ 1/II /3II3)). a
Integrating /3 from a to a + 1, we find that f«+1
r,
k(h)(e(h/3)-e(ha)) 27rih
R°« hOO
d/3+
f«+1 (a-a)k(0)d/3
= (a - [a])k(0) +O(min(H3,1/11a113) +H2),
o
which gives the result of the lemma.
The expression inside the order of magnitude brackets in the error term of
Lemma 5.3.1 is a periodic function of t called a row-of-glasses function (stakanchik in Russian) from its graph (Fig. 5.2). Its Fourier series converges uniformly. However, we use a finite Fourier series to estimate the order of magnitude of the row-of-glasses function.
Lemma 5.3.2 (Integer points close to a curve) Let y1,..., yM be real. For S
in0<S
D=2[1/86]+1. Let N(6) be the number of values of m for which Ilym11 <- S. Then
N(6)S
2M 4 D-1 +D Re D
dM (1-D)Fe(dy,,,).
Proof Let D
S(x)= 1
(D + 1- 2d)x
Fe(dx)=e((D+1)x\D
2
(
2
)
Analytic lemmas
96
If ym = n + 9 with 1915 6, then since D is odd
cos7r(D+1-2d)ym=cosir(D+l-2d)6zcosar(D-1)S>-cosh/4, and IS(ym )I2 >- D2/2.
The result follows from
M
D2 2
N(S)
M IS(Ym)l2 5
1
e((dl -d2)ym). di d2 m=1
Lemma 5.3.3 (Explicit Weyl criterion) Let yl,... , yM, and z1,..., zM be two sequences of real numbers, with associated weights w1,. .. , WM. Let H be a positive integer, and let
Hi=2[H/2i+3]+1 forlSiS
log H/8 log2
and let k(h) be the coefficients of Lemma 5.3.1. Then
M F, w.. (p(zm) - P ( ym ))
1M <<
m=1
I Wm
H1
I+
k(h) Why,,,) -e(hzm))
H wm h- =1
1
21rih
k(0)
1
H,-1
1
f;-1
h1M
tai Re F 1--)EIwmle(hym) H h=1
+H
i
1
+H
22i
1
I--
h1M E IwmI e(hzm) Hi ) 1
Re E
h=1
Similarly for E wm p(zm), omitting all terms involving ym on the right-hand side of the inequality.
Corollary (Uniform distribution) Suppose that xm is an infinite sequence of real numbers with
M
M F, e(hx)-+0 1
as M -> co for each h = 1, 2,.... Then for any a and 0 with 05 a < p5 1, the proportion of the numbers xm whose fractional part tm = xm
xm I has
a5tm
1 IWmI
(
1
(
+minl11 H3 IlzmIl3) +mini,
H311ym111
(5.3.3)
Rounding error sums
97
We use partial summation in the spirit of Lemma 5.1.1. Let R; = E I Wm 1. IIY,,,II < 2'/H
Then, for i >- 2 we have
E
IWmI=Ri-Ri-1)
2`-'/HSIIYmII<2'/H and M
1
F, IwmImin(l, l/H3Ilym113)
8
1
(R2 -R1) +
1
82
(R3 -R2) +
1
2K Rj We can put positive weights lwml into the argument of Lemma 5.3.2, so we have 2
R; < H;
,4 Iwml + -Re E, H,
h=1
1--h l M H;
I
;-Iwml e(hym).
1
We see that (5.3.3) gives rise to the first, third, and fourth expressions in the upper bound. Proof of the corollary We put zm = xm - [ xm ] - a, Ym = xm - [ xm ] - l3, and we use (5.3.1). We take wm = 1 for 1 < m < M, wm = 0 for m > M. We find
that the proportion is ,3 - a + 0(1/H) for M sufficiently large, depending on H. We let M tend to infinity first, then H. 0 Lemma 5.3.3 is a form of the Erdos-Turan theorem. It is also a form of the
uncertainty principle in harmonic analysis, that the first H terms in an eigenfunction expansion give accuracy to about one part in H. The usual statement of the corollary is in terms of discrepancy. The discrepancy of t1,.. . , tM on the interval [ a, 13) is
1-(13-a)M,
D(a,/3)= a 51,,, < j3
and the absolute discrepancy is D=
sup ID(a, p )j. Osas13s1
By (5.3.1)
M
D(a,13)=
(P(tm-a)-P(tm-13)),
and 1 M--++
LP(tm)=I D(0,13)d/3, 0
so that M
P(tm)
98
Analytic lemmas
Hence bounds for sums involving the function p(t) at a sequence of values tm are equivalent to bounds for the discrepancy of the sequence. 5.4
POISSON SUMMATION
The exponential e(t) is the violin of the mathematical orchestra, and p(t) is the flute; its themes are developed by the violins. Poisson summation is the tuba: very deep, but ridiculous when used too much. Lemma 5.4.1 (Trapezium rule) Let f(x) be twice continuously differentiable on [a, b], where a and b are integers. Then b-1
b
f(n) = jbf(x)dx+ f o.(x)f"(x)dx,
f(n)+'-2
a
a+1
a
(5.4.1)
a
where
o(x)= j0x p(t)dt, 0:5o(x)<$. If f" (x) has constant sign, then the second term on the right of (5.4.1) is O(max If 'WD, the maximum being over a 5 x < b. Proof We have
jo1(x-z)f'(x)dx='-zf(1)+Zf(0)- f f(x) dx. The left-hand side is
f p(x)f'(x)dx = [o,(x)f'(x)]o1
o
f'o
(x)f" (x) dx.
Putting together b - a consecutive unit intervals gives the result.
Lemma 5.4.2 (Poisson summation formula) Let f(x) be twice continuously differentiable on [a, b]. Then b-1
6
a
a+1
R
- f(n) + . f(n) = lim E jbf(x)e(rx)dx. R-
r= -R a
Proof The function o-(x), the integral of p(x), has a uniformly convergent Fourier series which can be written as W
1 - cos 27T rx
r=1
27T 2r2
Poisson summation
99
Integrating by parts, we have
f
b
b (1-cos27Trx)f"(x) dx f 27T22 r r=1 a
o, (x) fn(x)dx
00
a
fb -sin27rrxf'(x) 7rr
a
1
dx
M
a2cos2lTrxf(x)dx a
W
0
f a(e(rx) +e(-rx))f(x)dx.
_
r=1 a
Lemma 5.4.3 (Truncated Poisson summation formula) Let f(x) be real and twice continuously differentiable on (a, b) with f'(x) monotone, andA 5 f'(x) 5 B on (a, b). Then rx + cr) dx e(f(n)) = 1 f be(f(x)
-
aSnSb
q qA --,' SrSgB+,'-
n=c(mod q)
a
q
q
+ O(log(gB - qA + 2)),
(5.4.2)
and
cn
asnsb
e f(n) + - = q
b
y
ix 1
f e(f(x) - - I dx
q(A--L)SrSq(B+3) a
q 11
r -c(mod q)
+ O(log(B -A + 2)), If If" (x)I x A for a 5 x S b; then
Corollary 1
b
F, e(f(n)) «(b-a+1)V +1/sc. a
If I f" (x)I x A for a 5 x5 b, then
Corollary 2
b
E p(f(n)) << (b - a + 1)11/3 + 1/y. a
Proof First we take q = 1. We write g(x) =f(x) - cx, where c is the nearest integer to (A + B)/2. Let R be a positive integer such that
Ig'(x)I s R - 4 on a:!-. x s b. By the Poisson summation formula (Lemma 5.4.2),
e(f(n)) + 1 1 e(f(n)) _
1
a
a+1
f be(g(x) - rx) dx. r=_ a
Analytic lemmas
100
For I r I >- R we integrate by parts: b
I e(g(x) - rx) dx = I a
L
1b
e(-rx)
- 2Trir e(g(x))J + a
b e(-rx) g'(x)e(g(x)) dx. r a
Since a and b are integers, the integrated terms go out when we take -r with r. For r > R the First Derivative Test (Lemma 5.1.2) gives
Ib e(-rx) g'(x)e(g(x)) dx r
These terms sum for r > R to
<
114
4
4
1
1+5+ +4R-3
<-(6+logR).
The terms with r < -R are complex conjugate. Because we have symmetrized the range for r, the sum in the lemma may contain one less term, corresponding to r = R -1 or 1 - R. This extra term too is bounded by the First Derivative Test. The second assertion of the lemma follows at once from the case q = 1 on
replacing f(x) by f(x) + cx/q. We obtain the first assertion by putting n = c + mq, and applying the result to the sum over m. We obtain Corollary 1
by using the Second Derivative Test (Lemma 5.1.3) on the integrals. The logarithm term is smaller than the geometric mean of the other two terms. Corollary 2 is trivial if A >> 1. For k << 1 we apply the Weyl criterion of Lemma 5.3.3 with H x A113, and estimate the exponential sums by Corollary 1.
The assertions of Lemma 5.4.3 combine Poisson summation with a finite
Fourier transform modulo q. There is rich structure of exponential and character sums modulo q, which can occur together with Poisson summation. We give two examples: the proper Dirichlet characters y(n) mod q and the quadratic Gauss sums ( axe + bx \
G(a, b; q) _
e x mod q
,
(5.4.3)
q
where the sum is over any set of q distinct integers (mod q), for instance
0,1..... a-1.
Poisson summation
101
Lemma 5.4.4 (Poisson summation with a character) Let f(x) be real and twice continuously differentiable on (a, b) with f'(x) monotone, andA 5 f'(x) 5 B on (a, b). Let X(n) be a proper Dirichlet character modulo q. Then
E
E x(n)e(f(n)) = T(x)
asn5b
X(r) f bel f(x)
qA - ; rSgB+',
a
-'x)
dx
qJ
`
+O(/log(gB-qA+2)), where T(X) is the Gauss sum
a
T(X) _ F,
X(a)e
amodq
q
Proof We assume the properties of proper Dirichlet characters (see Davenport 1967, Chapter 9). We take the Poisson summation formula with the change of variable used in (5.4.2): ru
1
e(f(n))_-
.
q r
a5nsb
n a c(mod q)
rx crl f be( f(x)--+dx, a
q
(5.4.4)
q
multiply by X(c), and sum over c mod q. On the right-hand side we have the sum cr
X(c)e cmodq
- = X(r)T( x). q
We now have the coefficients in the formula, and the truncation bound follows as in the last lemma, using
IT(x)I = >q.
0
We give some elementary properties of the quadratic Gauss sum of (5.4.3), concerned with simplifying and casting out common factors.
Lemma 5.4.5 (Relations between quadratic Gauss sums) (1) The sum G(a, b; q) is zero unless (a, q) I (b, q). (2) If q is even, q = 2r, say, then G(a, b, 2r) is zero unless ar + b is even. (3) If (c, q) = 1, then G(a, b; q) = G(ac2, bc; q). (4) If (r, s) = 1, then G(a, b, rs) = G(as, b; r)G(ar, b; s).
Analytic lemmas
102
(5) Let u denote the solution x of ux = 1(mod q), the Kloosterman inverse of u mod q. Then we have
eG(a , b ; q ) = e e
4ab2
G( a, 0 ; q )
for q o dd ,
q
/
-
a(b 2)2
-
a(b2 - 1)/4
for q a n d b b oth even,
G( a, 0 ; q )
q
G( a, 1 ; q )
for q ev en bu t b o dd .
q
(6) For each a, I G(a, b; q)12 averages over b mod q to q. (7) For (a, q) = 1, the sum G(a, b; q) has absolute value if q is even, q = 2r and ar + b is even.
if q is odd,
(2q)
Proof For (1) we put d = (a, q), q = de, x = ey + z with y mod d, z mod e. Then
> ---
e-1e /az2
G(a,b;q)=
+
by
q
ymodd z=o
d
+
bz
q
The sum over y is zero unless d I b.
For (2) we write x = ry + z, with y mod 2, and argue similarly, using y2 = y(mod 2). For
(3) we take the
q
distinct summands to be
0,c,2c,...,(q -1)c. For (4) we use the integers r, s with if + ss = 1. For any integer N we have
e(') =e( '1e('), so that factorizing and using (3) gives
G(a, b; rs) = G(as, bs; r)G(ar, br; s) = G(ass2, bss; r)G(arr2, brr; s).
For (5) we substitute x = y + tab, or y + ab/2 or y + a(b - 1)/2 in the three cases. For (6) we note that, in
E
E
E
xmod q y mod q b mod q
axe+bx-ay2-by l
q
the inner sum is zero unless x = y(mod q). Finally, we deduce (7) from (6). If q is odd, then all the sums G(a, b; q) with b varying have the same modulus
by (5). If q is even, then half the Gauss sums have the same modulus as G(a, 0; q), the others the same as G(a, 1; q). By (2) either G(a, 0; q) or G(a, 1; q) is zero.
Poisson summation
103
Lemma 5.4.6 (Poisson summation with a rational quadratic) Letf(x) be real and twice continuously differentiable on the interval (a, (3) with f'(x) monotone, and A S f'(x) < B on (a, 13). Let a, b, and q be integers with (a, q) = 1. Then
E
ant + bn e
a5n5(3
G(a, b + r; q)
e(f(n)) _
q
q
qA- L5r5gB+I
4
4
rx
x f Re f(x)- - )dx+o(log(qB_qA+2)), q
a
where G(a, c; q) is the quadratic Gauss sum.
Proof We argue from eqn (5.4.4) as in Lemma 5.4.4, multiplying by e((ac2 + bc)/q) and summing over c. We use assertion (7) of Lemma 5.4.5 in 0 the truncation bound. Lemma 5.4.7 (Dirichlet's evaluation) We have
G(1,0; q) = E(q)
E(q)
1+i.
i
Proof By Poisson summation (Lemma 5.4.2) we have I x21
°°
el q I =
G(1,0;q)= xmodq
_
/
J
1
e-
qn2 \ 4
n
E
x2
q (
f0 el nx+
fqe(1q x+
q dx
nq )2) 2
I
E qf'+n/2e(,y2)dy+(-i)qq n even
dx
fi+n/2e(gy2)dy n/2
n/2
= q(1 + (- i)q)I//, by Lemma 5.4.5, where I denotes the integral
I = f e(x2) dx. When q = 1 we have
1=G(1,0;1)=(1-i)I, so that
G(1,0; q) =
1+(-i)q 1- i
=
E(q)r. 0
Analytic lemmas
104
In general G(a, 0; q) is ± G(1, 0; q). The ambiguous sign is determined by quadratic reciprocity theory. For instance, if a = c2, a perfect square, then assertion (3) of Lemma 5.4.5 gives G( C2, 0; q) = G(1, 0; q).
5.5 EVALUATING EXPONENTIAL INTEGRALS To make full use of the Poisson summation formula, we must evaluate the
integrals as accurately as possible. First we give a general result using integration by parts. In special cases more accuracy can often be obtained by using contour integration.
Lemma 5.5.1 (Explicit First Derivative Test) Let f(x) be a real function, three times continuously differentiable for a <x 5 /3. Suppose that there are positive parameters M, T, with M >- /3 - a, and positive constants Cr such that
for aSXS f3 If (r)(X)I<_ CrT/Mr
for r = 2, 3. If f'(x) and f" (x) do not change sign on the interval [ a, have
e(f( f3))
f 0e(f(x))dx= 21rif'(/3) «
then we
IT e(f(a)) 1 l +O 2lrif'(a) MZ minlf'(x)13I.
Proof We have
e(f(x))
P
l'3
f e(f(x))dx= [21rif'(x)J«+
1
,
2,7ri fa
f"(x)e(f(x)) f'(x)2
dx.
The integral on the right is << V/min I f'(x)I,
by the First Derivative Test (Lemma 5.1.2), where V is the modulus at an endpoint plus the total variation on the interval [ a, /3 ] of f" (x)/f'(x)2. The total variation is
f°a
f (3)(x) f'(x)2
2f" (x)2
s T
T
1
f"
x
f M f (x)2 dx +Mz s 4ff 3 dx f'(x)3 dx
T
min f'(x)2
1
M2 I If,( f3)I2 These estimates give the result of the lemma.
1
I f'(a)I2 U
Evaluating exponential integrals
105
The explicit form is useful if there is a sum or integration over a second variable. For instance, if we integrate g(x)e(f(x)), where the weight function g(x) tapers to zero at both ends of the interval, then the method of Lemma 5.1.1 gives integrals from x = a(y) to /3(y), where g(a(y)) = g( /3(y)) = Y.
We apply the first derivative test again to the integration over y in the endpoint terms.
Lemma 5.5.2 (Stationary phase integrals) Let f(x) be a real function, four times continuously differentiable for a 5 x < P. Suppose that there are positive
parameters M, T, with M
a, and positive constants Cr such that, for
a < x < 13 ,
If (r)(x)I < CrT/Mr,
(5.5.1)
f" (x) > T/C2M2.
(5.5.2)
for r = 2, 3, 4, and
Suppose also that f(x) changes sign from negative to positive at a point x = y with a < y < 6. If T is sufficiently large in terms of C21 C3, and CO then we have
Ire(f(x))dx=
e(f(y) + 8) + e(f(/3)) f-7(y) 21rif'(/3) M
M4 1
+ 0 T3/2 + 7.2
-
e(f(a)) 2vrif'(«)
1
(y
a)3
/p + \
1
1
-
y)3 J
Proof The values of f(x) outside the interval [a, /3 ] do not occur. We redefine f(x) for x < a to be the sum of the Taylor series about a up to the term in (x - a)4, and similarly for x >,8. Then inequalities of the form (5.5.1) and (5.5.2), but with larger constants, hold on an interval [ a - cM, /3 + cM ] for some positive constant c. We write
A= j- (-/)/2,
g =f(3)( y)/6,
f(x) -f(y) = Ag2(x- y),
(5.5.3)
where g(x - y) has the sign of x - y. We choose a real number r< cM and define u, v by
g(u - y) _ -r,
g(v - y) = r.
Then
f(u)=f(v)=f(y)+Ar2. By Lemma 5.5.1, for a :!g u we have u
e(f(u))
f e(f(x))dx- 2rrif'(u)
M4 e(f(a)) +0(r3T2). 2lrif'(a)
(5.5.4)
Analytic lemmas
106
For u S a we have
f
«
e(f(x)) dx=
U
e(f(a)) 21rif'(a)
e(f(u)) 21rif'(u)
M°
r
+0 (y-a)3T2
(5.5.5)
We treat the range v to G in the same way. On the range u to v we change variable to y = g(x - y). We write
f e(f(x))dxf y=r -r e(f(y) + Ay2) dx. v
U
The Taylor expansions about x = y give
f(x) =f(y) +A(x - ,)2+µ(x- y)3+0(TIx- y14/M4). f'(x) =2A(x- y) +3µ(x- y)2+O(Tlx- y13/M4), f"(x) =2A+6µ(x- y) +O(TIx - y12/M4), f'3'(x) = 6µ + O(Tlx -
y
IxMy14 )
A
2
i/2
c(x-y)+o(Ix_21
y=(x-y) 1+ 11
3
(xy)+2A(xy)2+OIxM21 Substituting repeatedly in (5.5.6), we find that
f'(x) =2Ay+2µ(x- y)2+0(TIx- y13/M4) = 2Ay + 2µy2 +
O(T1y13/M4),
and similarly
f" (x) = 2A + 6µy + O(Ty2/M4 ), f (3'(x) = 6µ + O(TIyI/M4).
We have
dy dx
dx
f'(x) 2Ay
µ
y2
+ y+O M2 (y2)
=1- -y+Ody
(5.5.6)
Evaluating exponential integrals
107
Thus fu
e(f(x))dx=e(f(y)) f r
e(Ay2)(1 r
- A y+G(y)) dy,
where
G(y) <
f- e(Ay2)dy= -
41riAr
M
+0 (r3T2),
(5.5.8)
with a similar expression for the integral from r to -. From the evaluation of the integral I in Lemma 5.4.7, we see that
f e(Ay2)dy =
= (1 + 0
1
(1 - i)F
-
e($)
f"(y)
2Fk
The integral involving G(y) is estimated. We note that
G'(y) =
d
f'(x) dy 2 Ay
+
µ A
After some calculation, we find that
G'(y) << Iyi/M2, and similarly
G"(y) =
d2 f'(x) Y7 2A
1
<<
M2 .
y
22
Next we choose a real number 8 x 1/ F, for which r/6 is a power of 2. By the Second Derivative Test (Lemma 5.1.3) we have
f
2
SSG(y)e(Ay2)dy
<<
<<
T32
.
We split the ranges 5 5 y <- r, - r5 y 5 - S into intervals of the form t 5 y 5 2t (or - 2t 5 y 5 - t). Using the method of the last lemma, we have 12,
f 2tG(y)e(Ay2)dy = G(y)e(Ay2) t
41riAy
t
- f 2te(Ay2) t
d
G(y)
dy 41riAy
dy.
The integral on the right can be estimated by the First Derivative Test, using our bounds for G(y), G'(y), and G" (y) to estimate the minimum modulus and the total variation of the weight function, as 0(1/AM2 ). The integral is then
<< 1/A2M2t «M2/tT.
Analytic lemmas
108
This bound sums to O(M/T3/2) again over ranges with t = 2kS, k = 0, 1, 2.... The integrated terms cancel except at ± S and ± r, with
G(±S)e(AS2) 41riAS
M
6 <<
AM2 << T3/2
We now have
I e(f(x)) dx = e(f(y))
e($)
+
u
+0
M1
(1 + G(y))e(Ay2) 4iriAy
,
rM4
T3/2)
OI
r
r3 T 2
with
e(f(y
(1 +G(y))e(Ay2) r
µ
[e(f(v) + Aye) 41riAy
41riAy
-[
e(f(x)) dx
(1 -
A
xso
47riAy dy Jxmu
1
r
y+G(y)) -r e(f(x))
-
[2if'(x),V
u'
cancelling the explicit terms in (5.5.4) and (5.5.5). We complete this intricate calculation by choosing r in the range
M/T1/65r5cM, which is non-empty if T is sufficiently large.
O
In Lemma 5.5.2 we assumed that f"(x) was positive, and f(x) had a minimum at y. If f(x) has a maximum at y, then we consider the complex conjugate of the sum. Lemma 5.5.3 (Inversion step) Suppose that f(x) is real and four times continuously differentiable for a <x < P. Suppose that there are positive parameters M and T, with M >- /3 - a, and positive constants Cr, such that, for a< x< /3, If (r)(x)I <- CrT/Mr
for r = 2, 3, 4, and
f" (x) >- T/C2M2,
so that f'(x) is monotone with A = f'(a) < f'(x) < B = f'(/3). Let g(x) be a real function of bounded variation V on the closed interval [ a, l3 ]. Then g(n)e(f(n)) = a_-5n_-5P
ASrSB
g(xr)e(f(xr) - rxr+ ) g fd(xr)
+O(V+Ig(a)D
77
+log(B-A+2)1
Evaluating exponential integrals
109
where xr is the unique value in a <x 5 /3 with
f'(Xr) = r. Proof First we consider the unweighted sum. We apply Lemma 5.5.2 to the integrals J
Re(f(x) -rx)dx
in the truncated Poisson summation formula (Lemma 5.4.3) with y = Xr. First we consider the exceptional cases when Lemma 5.5.2 gives less information than the Second Derivative Test (Lemma 5.1.3). This occurs when Xr is within a distance O(M/ FT) of the endpoints a or (3. Now -xr)f"()
1 =f'(xr+]) -f'(xr) _ (xr+l for some
between xr and xr+ 1. Hence
(5.5.9) xr+1 -xrXM2/T. The number of values of r with xr within O(M/ FT) of either endpoint is
MT
<<1 +7T . <<1+ FT M. By the Second Derivative Test these values of r contribute
M
<< - 1+
F
T
7 M
<<
M
+1
(5 5 10) .
.
to the unweighted sum. The two terms in (5.5.10) correspond to the sizes of individual terms in the transformed sum and in the original sum respectively. We can absorb the terms with r:5 A + 1 or r >- B -1 into the bound (5.5.10). We consider next the values of r with (5.5.11) F TIM, and use Lemma 5.5.2 with f(x) replaced by f(x) - rx. The explicit terms
e(f(a) - ra) 21ri(A-r)
e(f(/3) - r/3) 21Ti(B-r) '
sum to O(log(B -A + 2)) over the range (5.5.11) as in Lemma 5.4.3. In the error term, by (5.5.9),
M4
1
T2 (xr- a)3
-
T
1
M2 (r-A)3
whose sum over the range (5.5.11) is bounded. The error term O(M/T3/2) sums to
M
<<- 3/2 (B-A)4 T
M
T
Analytic lemmas
110
We have put back the terms in e(f(xr)) for
B-1-FT /MSr
Ayr
M T << M-+1, <
h(n)e(f (n)) = a5n5J3
fh(B)
e(f(n)) dY max(a,h-'(y))5n5/3
y
e(f(Xr)-rXr+g)
fh(j9) y=p
fu
ASrSB h(x,)zy
(Xr)
+0 T +log(B-A+2)l Ll.r'
h(Xr)
dy
e(f(Xr) - rxr+ ) n
ASrSB
g
Of (Xr)
+O h(,6) VT= +log(B-A+2)J I, and we deduce the weighted version as in the proof of Lemma 5.1.1.
0
Lemma 5.5.3 involves the complementary function f(y) with
.f(y) =xy - f(x),
(5.5.12),
where
y = f'(x) as in Fig. 4.1. Then
-ax =y +xdy
f'(y)dy
f'(x),
so that
f'(y) =x.
(5.5.13)
Equations (5.5.12) and (5.5.13) show that the relationship between f(x) and f(y) is symmetrical. The inversion step sends a sum over e(f(n)) into a sum over e(-f(r)), and vice versa. When the step is repeated, then we return to the original sum. We usually use the inversion step with M > V to change a long sum into a short sum.
Evaluating exponential integrals
111
Next we obtain the basic approximation to the Riemann zeta function which we shall use in Chapter 21. Lemma 5.5.4 (Approximate functional equation) Let s = o- + it be a complex variable with 0 < o- < 1, ItI > 1, and let N and T be integers with T = [I t11 + 2,
Nx FT. Then
(s) =
1
N 1
ns
E
I'(1 -s)
+ I` i JI
(log Tl
1
1 srsltl/2aN- a
rl-s + OII`` 7 o/z
Proof Step one
For s = v+ it with o-> 1, the Riemann zeta function is given by 1
W
(s)= F S. n=1 n
By Lemma 5.4.1 with a = T, b - oo,
(s)
T-1
1
ns
1
T-
1 1
1
+
1
1
+f
2TS
dx+ f
T xs
Tl-S
1
ns + 2TS +
s-1
00 s(s + 1)u(x) X
T
S+z
( s(s + 1)o (x) dx. xs+z + JT
dx (5.5.14)
The integral on the right converges absolutely for Re s > -1. Hence (5.5.14) gives a meromorphic continuation of C(s), and we may use it as the definition of C(s). From the proof of Lemma 5.4.2 we see that
u As + 1)v(x) IT
xs+2
s(s + 1)(1 - cos2.7rrx)
u
dx = fT
2a2r2xs+2
r=1 s
-
1
12
1
US+1 )
( TS+1
E s(s + 1) r=1
1((
21rzrz I TU
/
dx
t
1
xv+zI\eI`rx-
t
+eI -rx- Z-logx T2
1
« Tv + r=1
dx
1 «Tv, r2 Tv+z r 1
1
by the First Derivative Test (Lemma 5.1.2). We let U tend to infinity.
Step two We use the truncated Poisson summation formula of Lemma 5.4.3,
Analytic lemmas
112
with a weight function g(x) =x-° put in as in Lemma 5.5.3. For ease of writing, we consider the case of t positive. We have T
T
1
Fns=E fNe
1
1-
x°dx+O
log T
N°
where the sum is over t
t
1
1 2arT-45r52i7N+4.
The term r = 0 gives T'-g
N'-$
1-s'
1-s
the first term cancels in (5.5.14). The integral with r = [t/27rN + 1/4] is
divided into blocks with x running from Nk to Nk+l, chosen so that
t
Nk+ I < 2Nk for each k. By the Second Derivative Test (Lemma 5.1.3)
fNk"e (
Nk
rx-
2?r
Nk
logx )I
x dx<<
If 0 < o, < 1, then these blocks sum to N1-
f (5.5.15)
log T.
<<
Step three For 0 < o < 1 we can deform the integral for I'(1 - s) to a contour integral along the imaginary axis: eiy
co e-x
r(1 - s) = fo
dx = fo i,-,ys dy.
xS
Hence 1
f e(rx)-xsdx = I 0
21Tr I
i
1
F(1 - s).
J
By the First Derivative Test, for r5 t/21TN - 3/4 we have (
t
rx-21T log x
1
1
dx <<
rT°. JTe For the integral from 0 to N we integrate by parts first: N/
f e rx 2- log x
)X01
1
X° dx
0
X1
1-s N'-S
1-s
e(rx)
N
+
N
x'-°dx 2Tr s-1 0f 27rire( rx--logx t 1
o
0(
1
r Is -11
N1or
t/21rN - r
Evaluating exponential integrals
113
These terms sum to t
log T
log T
Na Na If there is an integer r with t/2IrN - 3/4S r 5 t/27rN, then S-1
N1-' e(rx)dx <<, f xs M
1
o
as in (5.5.15). Thus T
27rrl) s-1r
1
ns
E/2
(
r5 taN \
1 - s) + o( (
t
logT
N1_Q)
+
``
J
Substituting into (5.5.14) gives the result.
Lemma 5.5.4 is related to one of the proofs of the functional equation C*(s) = C *(1 - s), where
r(s/2)C(s) irs/2
The factor r(s/2)7r s/2 in C*(s) can be regarded as a continuous analogue of the Euler factors of C(s): 1
1
1+
+ PS
7
+
The functional equation can be written asymmetrically as sir 2irs-1 sin C(s) = r(1 -s)C(1 -s). 2
(5.5.16)
The first three factors on the right of (5.5.16) are approximately The first two terms in the lemma are a partial sum for C(s) and a partial sum for C(1 - s) multiplied by an approximation to the factor in the functional equation. The traditional name `approximate functional equation' only approximates the content of this lemma. There are versions of Lemma 5.5.4 with tapering weights on the terms of the two sums; these can be especially useful when s is allowed to vary. The next two lemmas are versions of Lemmas 5.5.1 and 5.5.2 for exponential integrals with a differentiable weight function. The proofs are omitted; the method is the same, but the details are more complicated. (21r/i)s- t
Lemma 5.5.5 (Weighted First Derivative Test) Let f(x) be a real function, three times continuously differentiable for a 5 x 5 /3, and let g(x) be a real function, twice continuously differentiable for a S x 5 P. Suppose that there are positive parameters M, N, T, U, with M a, and positive constants Cr such that, for a 5 x 5 /3, If(r)(X)I 5 CrT/Mr, jg(s)(x)I 5 CU/Ns,
Analytic lemmas
114
for r = 2, 3, and s = 0, 1, 2. If f'(x) and f" (x) do not change sign on the interval [ a, /3 ], then we have
g(a)e(f(a))
g(/3)e(f(f3))
I=f g(x)e(f(x))dx= 27rif'()3) (C
+O
r TU I MZ
21rif'(a) M M2 min I f'(x)I
1+-+ N NZ
T/M
1 3 )minIf'(x)I)
The implied constants are constructed from the constants Cr.
Lemma 5.5.6 (Weighted stationary phase integral) Let f(x) be a real function, four times continuously differentiable for a 5 x 5 /3, and let g(x) be a real function, three times continuously differentiable for a <- x ::g /3. Suppose that there
are positive parameters M, N, T, U, with
M>>-/3-a,
NzM/V,
and positive constants Cr such that, for a <x < 0, If (r)(X)I< CrT/Mr, Ig(S)(x)I s CSU/NS, for r = 2, 3, 4 and s = 0, 1, 2, 3, and
f" (x) >- T/C2M2. Suppose also that f'(x) changes sign from negative to positive at a point x = y with a < y < P. If T is sufficiently large in terms of the constants Cr, then we have
I Rg(x)e(f(x)) dx -g(a)e(f(y)+8) +g(/3)e(f(/3)) f'"(y) 21rif'(9) +O
T4U11
+-
g(a)e(f(a)) 21rif'(a)
1
N/2(
+O
3) The implied constants are constructed from the constants Cr. 5.6
1
U (1
(T
+
N/2)
BILINEAR AND MEAN SQUARE BOUNDS
An analytic number theorist, it is said, is someone who is very good at using
Cauchy's inequality. Often in analytic number theory we come across a double sum MZ
N2(m)
F,
F,
m=M1 n=Ni(m)
g(m)h(n)e(f(m, n)).
(5.6.1)
Bilinear and mean square bounds
115
Following Montgomery and Vaughan, we classify double sums as type I if either sequence of coefficients is smooth. If h(n) has bounded variation, then we can estimate the inner sum in terms of pure sums. If both coefficient sequences are irregular, then we use Cauchy's inequality. We state the first lemma for a sum slightly more general than (5.6.1). Lemma 5.6.1 (Type II sums) M2
N2(m)
F,
F,
2
g(m)h(m, n)e(f(m, n))
m=Mi n=Nl(m) M
Ig(m)I2) M2 n1
h(m,nl)h(m,n2)e(f(m,n1)-f(m,n2))
E
x
m=M1
n2
N(m)S n U112
Zm)zn1,n2
Proof We use Cauchy's inequality on the sum over m; the second factor is M2
2
N2(m)
E h(m, n)e(f(m, n))
E
m=M1 n=NI(m) rearranged.
In our next lemma a sum in one variable is treated as a double sum. The method is `divide and conquer'; it is equivalent to splitting the range into short intervals, using Cauchy, and then averaging over the possible points of dissection.
Lemma 5.6.2 (Differencing step) Let f(x) be a real function, defined for M< x:5 N, and let w(M), w(M + 1), ... , w(N) be any coefficients. Let D 5 (N - M + 1)/2 be a positive integer. Then N
E w(m)e(f(m))I
(N-M+2D-1)
2 <-
D
M
N
X ', Iw(m)I2 + 2Re M
D-1 /
d
11
11- D)
d=1 `
J
N-d
x
F,
w(q+d)w(q-d)e(f(q+d)-f(q-d))
q=M+d (5.6.2)
Analytic lemmas
116
Proof We write N
N-2d
D
E
D > w(m)e(f(m)) _ E
w(m + 2d)e(f(m + 2d))
d=1 m=M-2d
M
N-2
D
F, m=M-2D
w(m + 2d)e(f(m + 2d)).
d=1
Msm+2d5N
By Lemma 5.6.1 with n = d, h(m, n) = w(m + 2n), g(m) = 1, we have 2
N
D2
E w(m)e(f(m)) M
min(N-2d1,N-2d2)
5(N-M+2D-1)E E d,
E
w(m+2d1)w(m+2d2)
d2
X e(f(m + 2d1) -f(m + 2d2)). We put q = m + d1 + d2, d = d1 - d2, so that
max(M+d,M-d) Sq s min(N-d,N+d). Given q and d, then max(1, d + 1) s d1 5 min(D, D + d).
Thus there are D -I dl choices for d1. Also the terms with d < 0 are the complex conjugates of the terms with d > 0. This rearrangement gives the D result of the lemma.
The differencing lemma corresponds to an A 1 step in the iteration of Chapter 3. We can form an analogue of the A2 step by applying Cauchy's inequality to the sum over d in (5.6.2). However, this gives a result that is weaker than we would get by iterating Lemma 5.6.2. There is an elementary analogue of Lemma 5.6.2 for sums involving p(f(m)) (van der Corput and Pisot 1939). On the right-hand side stands a discrepancy sum, over
p(f(q + d) -f(q - d) + t), to be maximized over t. But the same result can be obtained using exponential sums and Cauchy's inequality (Cassels 1953).
Whether higher A, steps exist for the discrepancy or the row-of-teeth function is an open question. Van der Corput's method for estimating exponential sums (see Graham and Kolesnik 1991) consists of using repeated differencing steps alternating with inversion steps (Lemma 5.5.4). After each step we get a similar sum, with
the parameter T replaced by a smaller value T' after a differencing step (`process A'), or the parameter M replaced by a smaller value M' after an inversion step ('process B'). When the conditions for the inversion step are not satisfied, or the sums become very short, then we estimate the last sum
Bilinear and mean square bounds
117
trivially as the number of terms times the maximum term. The theorems of Chapter 17 give non-trivial estimates for sums with parameters T' and M' with T' close to M'2 in order of magnitude. The longer the iteration, the less
sensitive the results are to how we estimate the last sum. We set up the
iteration to reach a sum with T' close to M'2 as soon as possible. In applications of the method there are often different order-of-magnitude ranges for M and T to consider, and further parameters that can be chosen to adjust the orders of magnitude. There are two possibilities:
(1) There is always a range with M close to T, and it dominates the other ranges;
(2) ranges with M less than T seem to dominate. In case (1) we use the inversion step followed by a trivial estimate. In case (2) we use inversion and differencing steps to bring T' close to M'2, and then we
optimize the other parameters; if the right sequence of steps was chosen, then this usually brings T' closer to M'2.
For sums in two or more variables we can apply the inversion and differencing steps to either variable individually. A good simultaneous differencing step is not known for two-variable exponential sums, or for two-dimensional discrepancy. The inversion step is used more. The range of summation may contain exceptional regions where a second derivative vanishes, which have to be treated more trivially. Most applications of the method have f(x) a monomial function in the sense of Lemma 3.3.2.
In this case the exponents in the transformed exponential sums are also monomial functions, whose monomial part is easily calculated.
There is an analytic version of Lemma 5.6.2, more clumsy than the elementary one, but made up of two useful lemmas which we state generally.
Lemma 5.6.3 (Mean to maximum) For f(M), .
. . ,
f(N)
real and
g(M), ... , g(N) any coefficients, r >_ 1 ahd T real we have N
r
E g(m)e(Tf(m)) M
Max S(55)r
Tr
T+2 N
-15Q51 'T-2
M
Proof We put N
F(s) _ E g(m)exp2rrsf(m), M
r
it) f(m)) dt.
118
Analytic lemmas
where s = v + it is a complex variable. For 1
C1(y): line segment 1 + iT- iy to 1 + iT+ iy, C2(y): line segment 1 + iT + iy to - 1 + iT + iy,
C3(y): line segment -1 + iT+ iy to -1 + iT- iy, C4(y): line segment -1 + iT - iy to 1 + iT - iy, a rectangle with centre at iT. By Cauchy's residue theorem
F(s)
F(iT) _21T1 - fy=1fc(y) s - iT ds dy. 1
2
We have
r(j2j
IF(iT)Ir <
1
c(y) Is - iTll r/(r
y= 1
X f2
f
y=1 C(y)
Idsldylr-1
])
/lf
IF(s)Ir Idsldy.
Since Is - iTl Z 1 on the contour, the first factor is 1
r
f2
< (21T)
2
y=1
lOr- 1
7
=
(4 + 4y) dy l 111
(2,r)r
In the second factor, since the integrand is positive and C1(y) is a subinterval of C1(2), we have
f2f 1
c,(y)
I F(s)I r dt dy < f
IF(s)Ir dt,
C,(2)
and similarly on C3(y), whilst
f2fca(y) IF(s)Irdo, dy=fIo- - f2 IF(v+iT-iy)I'dydo, 1 y=1 1
<2 max
IT+2IF(o-
-ISu52 T-2
+iy)lydy,
and similarly on C2.
O
Our next three lemmas use the Fourier transform pair sinc2x, A(s), where sinc x =
sin 7Tx
A(s) = max(1- IsI, 0).
, 7T X
These functions are both non-negative, with a maximum at the origin. Lemma 5.6.4 (Short interval means) =
F(x)
Let sp..., sR be real numbers, and let R
E are(srx), r=1
Bilinear and mean square bounds
119
where a1,..., aR are any coefficients. For positive real numbers any real X, let
I(I) =
6 and 0, and
fx+'IF(x)I2 dx, x-o
and
_
a;aje(s;X-sjX)A
S(6) j
i
=
Si - sj
1
6
2
f I L a/e(siX)I dy.
S
ys,i56
then S(6) is positive, and 1T 2
A
I(D) 5
2
S(
).
1
Proof We take X = 0. Since sin 7rx
sinc x = for IxI <-
Z,
7rx
2
Z 7r-
we have
f QIF(x)IZ dx 5 42 f Ir2o
IF(x)I2 since T.
dx
f_
2
_ E F1a,aje(20(s; - sj)y)sinc2 y dy j
20
( 1
7r
i
1
2 S 2Q /I,
since A(s) is the Fourier transform of sinc2y. For X non-zero, we replace a, 0 by a.e(s,X).
To obtain a result like Lemma 5.6.2, we take out an order-of-magnitude factor T from the exponent f(x) in Lemma 5.6.2, and apply Lemma 5.6.3 with r = 2. The integral on the right of Lemma 5.6.3 becomes the integral on the left of Lemma 5.6.4 with a change of notation. We extend the length of the range of integration from 4 to O(T/D). The final sum involves terms in e(f(m)-f(n))with m - n D. Lemma 5.6.5 (Coincidence detection) Let sl, ... , SR be real numbers, and let R
F(x) _ F, e(s,.x). r=1
Analytic lemmas
120
For positive real numbers 6 and 0 let
N(S)= E E 1,
I(0)=fn
j
i
-A
IF(x)I2dx.
Isi - sjIs s
Then
28
( s1 ). 11
More generally, letx = (x1,..., xk), S = (61,..., 6k), A - (A1,..., Ok) With Di = 1/6i. Given k-dimensional vectors sm,..., s(R), let N(S) be the number of pairs of vectors differing by at most 6i in each component. Let = R nk A F(x) e(s(') x), I(0) = f ... f IF( X)12 dx1... dxk. Ak
O1
Then
2
k
ir2
k
2) 61 ... 6k I(0) SN(S) 5 (2 SI ... Skl(0).
(
Proof We use the Fourier transform pair A(x), sinc2s. We have
si-s
>A
N(S) >_
=2
3/2
j
i
f
S
)
f
2 2
2
e(sx - six)sinc2 j
Sx
dx
2 2
S
I.F'(x)I sinc 2 dx Z 2
Sx
C3
IF(X)12
7r/2
dx,
and in the other direction /
z
N(S)5Ei F(2) since1sS 21) ` / / 2
4
26 f E E e(sjx-six)A(2Sx)dx
'72S
0
= 2 C28 11
2S
j
i
f
S
2
IF(x)I A(26x)dx
SIF(x)IZ dx.
p
The k-dimensional case goes similarly.
Lemma 5.6.6 (Double large sieve)
Suppose xWlW,..., x(m) and yU),
are real vectors in k dimensions with
-Xi/2 5 x;"') S Xi12,
-Y/2 < y, ") 5 Y,/2
, y(N)
Bilinear and mean square bounds
121
fori=l,...,k; m=1,...,M; n=1,...,N. Let Ei=1/Xi, Si=X,/(X;Y+1), and put
A(x) = max(1 - Ixl,O). Then 2
E ambne(x(m) Y(")) m
k
4k
17.
(2) A(S)B(e) [1 (X;Y + 1),
(5.6.3)
n
where k
x(m) - x(r) k
A(s) _ E E IamlIarl [l A
r
r
m
L
k
B(E)= '+
bn b
r
n
1
si
1
t
(
AI
y(rn)
E'
I.
y,(r)
Proof As in the last lemma we drop the suffices and treat the onedimensional case. Since E/2
ify- E/2
e(sx) ds = e(xy)
e(Ex/2) - e(- Ex/2)
= Ee(xy)sinc Ex,
21rix
we have
E E ambne(xmyn) E E m
n
m
n
am bn
fYn + E/2
e slnc Exm y,- E/2
= 1(Y+ 0/2
e(sxm) ds
F,
-EAme(sxm)
(Y+ 0/2 m
IYn-sIs /2
where am
Am -
E sine Exm
I AmI S
2E
laml.
By Cauchy's inequality we have ambne(xmyn)
11: m
SI1I2,
n
where I1 and I2 are the integrals
I
z
(Y+E)/2
2
b
I (Y+E)/2I IYn-SISE/2 "
_
bnbrmax(E - IYn n
I1 = f
ds
yrI,O) = EB(E);
r (Y+ E)/2
2
I F, Ame(Sxm)I ds
-(Y+E)/2 M
b,, dS,
Analytic lemmas
122
with 00
7T2
I1 < f 4 since
2
6sl>Ame(sXm)I ds m
00
7r2
rXk -Xm
AmAk4SAI m
k 7T2
IAmAkI m
ir 4 16
k 1
SEZA(S)
46
S
)
maxi`1-
IXk -Xm)
7r4 (XY+ 1) 16
S
,0
A(S).
E
For the k-dimensional case we use Cauchy's inequality inside a k-fold
0
integral with respect to s1, ... , sk.
To show why Lemma 5.6.6 is called the `large sieve', we deduce from it the simplest inequality of sieve theory, the `Brun-Titchmarsh Theorem'. Hardy and Littlewood (1923) found the first result of this type using Brun's selective version of Eratosthenes' method, rejecting multiples of small primes. Later Selberg changed the treatment from linear to quadratic, finding a signed weight function which is large at primes, and summing its square over all the integers in the interval. Linnik meanwhile found sieve upper bounds using his
dispersion method. The basic idea of the dispersion method is to prove uniform distribution by a mean-to-max method, relating the absolute discrepancy to the mean square of the discrepancy on subintervals. Halberstam and Richert (1974) give a detailed account of sieve methods. We give two proofs of the Brun-Titchmarsh theorem from Lemma 5.6.6, the first one of Linnik's type, the second one of Selberg's type. The two proofs are dual in the sense of bilinear forms (see Lemma 10.6.3).
Lemma 5.6.7 (Primes in an interval) There is a constant C such that, out of any M(Z 4) consecutive integers, at most CM/log M of them have no prime factor less than YrM-.
Corollary If the endpoints of the interval are greater than VM_, then at most CM/log M of the integers in the interval are prime.
First Proof Suppose that the vectors y(°) are E-well spaced, in the sense that, if n r, then for some i we have IyI "> -y(r)l > E;.
Then all terms in the sum B(e) with no r are zero, and B(E) _ , Ib"I2. n
(5.6.4)
Bilinear and mean square bounds
123
We choose bn by
k= E ae(x(m) . y(n)). m
The left-hand side of (5.6.3) is B(E)2, so we have k
4k
B(E) _
ame(x(n') .y(n))I2 S (
I
2)
m
n
A(S) F1 (X;Y + 1).
5.6.5)
Now we specialize k to be one (we drop the suffices i), and the points y(n)
to be the fractions of the Farey sequence AQ) with - i < a/q 5 m - (M + 1)/2, so that
X=M-1,
$=1-1/M.
e=1/(M-1),
Y=1,
x(n') _
Since S < 1, we have 12.
A(S) = E la
The minimum distance between members of AQ) for Q >- 2 is 1/Q(Q - 1), so the spacing condition (5.6.4) holds with Q = [vrM-] (the lemma is trivial for
M 4).
Suppose that the interval goes from K + 1 to K + M. We put am =1 if K + m has no prime factor < Q, zero otherwise. When yn = a/q, then
y )= Eame - m-
ame(x
2
=ea(K+M+1))Ea e(a(K+m) q
m
2
m
q
Let
(a)
Eame
q
a(K+m) q
m
which depends only on a/q modulo one. We can now write (5.6.5) as a
Ir 4
2
E E IS(Iq 5-MEIamI2, q=1 amodq 16
(5.6.6)
m
(a,q)= 1
a form of the large sieve familiar to number theorists.
Let R = E am be the number of non-zero terms in S(a/q). Then using Ramanujan's sum (Lemma 1.4.1) we have
S(a)=Eam a mod q
(a,q)=1
q
m
E e\ a mod q
a(K+ m) q
(a,q)=1
= mLam clqE c/ cl(k+m)
(q)= Fla. µ(q)=p p(q)R, m
Analytic lemmas
124
since the only integer c which is a factor both of q and of K + m is one. By Cauchy's inequality (this corresponds to Linnik's dispersion step) we have
F,
p.2(q)R2 5 cp(q)
a mod q
(a,q)=1
Dividing by cp(q) and substituting in (5.6.6), we have 2
R
µ2(q)
ir4MR
cP(q)
16
F,
q--,Q
15 .6.7)
The last step is to estimate the sum in (5.6.7) from below. The Mobius function µ(q) is 0 unless q is a product of distinct primes, Pi, ... , p, say, and then
µ2(q)
1
1
r
cp(q)
cp(q)
Pi...P,
1
1
(1
Pu
1
)
r( II 1+-+ zR +... _ -,n Pi PI Pr 1 1
1
1
1
taken over all integers n whose prime factors are all taken from the set (Pi,. .. , p,). Hence
µ2(q) q ,Q
1
97(q) -
L
n
taken over all integers n for which the product of the distinct prime factors of
n is an integer < Q. Thus µ2(q) q
Q (q)
Q
1
nz
Q+1
dx x
= log(Q + 1)
iz log M.
(5.6.8)
We deduce the result with C = ir4/8. Second proof Similarly, if the vectors x(m) are 6-well spaced, then 4k
2
bne(x('n) y("))1 "I
n
5
k
(2) B(e) F1 (X;Y + 1).
We take k = 1, x(m) and y(") as in the first proof, and choose b by
µ(q) a M+ 1 9P(q)e(q(K+ 2 ))
(5.6.9)
Bilinear and mean square bounds
125
where y(") is the fraction a/q. The points y(") are e-well spaced, so that
B(e) _
µ2(q) _
':
Q µ2(q) P(q)
q=1 amodq PT(q) (a,q)=1
Thus (5.6.9) gives 2
µ(q)
E F, m q=1 (P(q) amodq
e
7r 2M Q µ2(q)
r a(K+m)) q
5
1
(a,q)=1
16
E 1
cp(q)
(5.6.10)
which is Selberg's sieve inequality (with a worse constant). The sum over a is the Ramanujan sum again. In particular, if K + m has no prime factor 5 Q, then the sum over a is µ(q). The left-hand side of (5.6.10) is µ2(q)
> R 1
2
q'(q)
and we complete the argument using (5.6.8) as in the first proof.
The form of the large sieve in Lemma 5.6.6 is due to Bombieri and Iwaniec (1986a). The application in Lemma 5.6.7 follows Mathews (1973). Lemma 20.2.2, which bounds the number of times that an exponential sum is large, is also a sieve result in the widest sense. The other lemmas were either known
to van der Corput, or are later developments of his ideas. For example, Lemma 5.5.2, from Huxley (1994c), corresponds to the leading terms only of van der Corput's asymptotic formula (1935, 1936), without taking f(x) to be infinitely differentiable. There are many forms of Lemma 5.5.2, with different hypotheses about the function f(x), for example, Lemma 10 of Heath-Brown
(1983), Lemma 2 of Karatsuba (1987), and Lemma 3.4 of Graham and Kolesnik (1991). The more specialized Lemmas 10.6.1, 10.7.1 and 21.4.3 are also of van der Corput type.
6 Mean value results 6.1 THE SIMPLE EXPONENTIAL SUM The idea in this chapter is to investigate a sum S by introducing a real parameter x, and then evaluating E2 = f 1IS(x)I2 dx. 0
Apart from its own interest, the integral has three uses. 1. Limitation results. Since IS(x)I >_ E for some value of x, any uniform upper
bound for IS(x)I must be greater than E. These results are called omega
theorems from the notation F = O(G) for `F does not have order of magnitude strictly smaller than G'. 2. Upper bounds. Sometimes there is a mean-to-max argument, using special properties of S(x), as in the Tauberian theorems of convergence theory. 3. Large values results. If x1, ... , xR is a well-spaced set of points at which IS(x,)I is large, then R itself cannot be too large. The large sieve (Lemma 5.6.6) is often used in this way. Theorem 6.1.1 Let f(x) be a real function with R continuous derivatives for M 5 x 5 2M, for which there is a size parameter T and a constant C >_ 1 such that If (')(x)I 5 CrT/M' (6.1.1) and
if W(x)I z T/CSMS
(6.1.2)
hold for certain values of r and s. Then f2 I
ZM1
2
dt=M+O(CM2
L e(tf(m))
Tg M
M
provided that R Z 1 and (6.1.2) holds with s = 1, and 1 f1
2M-1
2
E e(f(m +x)) dx=M+O
C2M2 log M C9M4 T
+
T2
,
M
provided that R >_ 3 and (6.1.1), (6.1.2) hold with r = 2,3 and s = 2, and f (3)(X)
The simple exponential sum
127
does not change sign. The term O(M4/T 2) in the second result may be improved if (6.1.1) holds for larger values of r.
Proof We have
f
2
12Mf 1
2
f2
dt =
e(t f (m)) M
e(t(f (m) - f (n))) dt m
M+
n 1
m
O n
f(m)f(n)I
m#n
by the First Derivative Test (Lemma 5.1.2). Since
f (m) - f (n) = (m - n)f'(e ) for some
between m and n, the error term must be <<
-CMT L 2 m
n
1
Im - n1
CM2 M 1 T ,..1 r
CM2 log M
T
m*n
For the second assertion we have 2M-1 fro
2
E e(f(m +x))
dx
M
=
f
1
e(f(m +x) - f(n +x))dx m
n
e(f(m+1)-f(n+1)) =M+ F, E 2ai(f'(m + 1) - f'(n + 1)) m n m#n
-Lm
n
e(f(m) -f(n)) 21ri(f'(m) -f'(n))
m#n
EE m
C_TIm-n1
min I f'(m +x) -f'(n +x)13 '
M3
n
1
m#n
by the explicit First Derivative Test (Lemma 5.5.1) applied to f(m +x) f(n +x). Since m and n run from M to 2M - 1, all the integrated terms cancel except those in the first sum with m + 1 = 2M or n + 1 = 2M, and those in the second sum with m = M or n = M. Estimating as above, we get
M+O
I C2M2 2M-1 T
n=M+1
1 (2M-n
1
+ n-4 +O
C9M3 T2
1
(m - n) 2 m#n
Mean value results
128
which gives the result. If f(x) has more derivatives, then we can argue as in Lemma 5.5.5, integrating by parts several times before applying the first derivative test.
6.2 THE LATTICE POINT DISCREPANCY Let C be a simple closed curve in the plane, winding once round the origin, given by x =x(t), y = y(t) with the functions x(t), y(t)R(> 2) times continuously differentiable, and with the radius of curvature bounded away from zero and infinity. Let A be the area of C. For M large and for u, v in 0 5 u < 1, 0 5 v 51, we consider the enlarged and shifted curve CM(u, v), the locus of the point (Mx(t) + u, My(t) + v). Let NM(u, v) be the number of integer points inside CM(u, v), and let DM(u, v) be the signed discrepancy Dm (u, v) =NM(u, v) -AM2. We also consider TM(u), the trapezium rule estimate for the area of CM(u, v):
the sum of the lengths intercepted by CM(u, v) on the integer abscissae x = m. The two-dimensional analogues of Theorem 6.1.1 are the mean square values of TM(u) and DM(u, v) taken over u and v, and the mean square of
DM(0,0) taken over M. We use the Fourier series discovered by Kendall (1948), a remarkable generalization of a formula for the discrepancy of an ellipse as an infinite series of Bessel functions first stated by Voronoi (1904).
The Fourier coefficients correspond to short arcs of the curve where the gradient is close to a rational number. Hlawka (1950) and Herz (1962a) have computed the Fourier coefficients to great accuracy for the n-dimensional lattice point problem.
Lemma 6.2.1 (Fourier series for the discrepancy) If there is no lattice point on CM(u, v), then the discrepancy DM(u, v) is equal to the sum of a Fourier series
F, FlaM(h,k)e(hu+kv). h
k
The trapezium rule estimate is
TM(u) =AM2 + F, aM(h,0)e(hu). h
The Fourier coefficients are given by
am (0,0) =AM2, and, for h and k not both zero, aM(h, k) << M112/(h2 +k2)3/4. If the curve C is R times differentiable with R z 4, then
aM(h,k) = -bM(k, -h) -bM(-k, h) + EM(h,k),
The lattice point discrepancy
129
where bM(v) is defined as follows. Let P be the point of the curve C where the vector v is in the anticlockwise direction of the tangent. Let p be the radius of curvature of C at P, and let L(v) be the moment of v acting at P about the origin. Then
(Mp) bM(v) =
21ri jvj3/2
e(-ML(v) + $).
The residual term satisfies
em (h, k) «M- 1/2(h2 + k2)-5/4. Corollary When M2 is not a sum of two integer squares, then the discrepancy of the circle centred at the origin, radius M, is
r(n) IT
n/4
n=1
r
cosl 2,rMv +
ar
4)
(
1
+ 01` M1/2
where r(n) is the number of ways of writing the integer n as a sum of two squares.
Proof Let a = min x(t), (3 = max x(t). The curve C is given by two Cartesian equations y = F(x), y = G(x) for a S x :!g f3, with F(a) = G(a), F(f3) = G( f3) and F(x) < G(x) for a <x <,O. The area of C is A
f"0 (G(x)-F(x))dx,
and the trapezium rule estimate for the area of the enlarged curve CM(u, v) is
E
TM(u) _
(g(m - u) -f(m -u)),
Ma+u Sm SM/3+u
where we have written
f(x) =MF(x/M),
g(x) =MG(x/M).
The number of lattice points inside the shifted curve is NM (u, v) =
E
([g(m - u) + v] - [ f(m - u) + v]).
Ma+uSmSMt3+u
Since NM(u, v) is a periodic function of u and v, it has a Fourier series
E E aM(h, k)e(hu + kv) h
k
with
TM(u) = f
1
NM(u,v)dv = E aM(h,0)e(hu),
v=O
h
where
aM(0,0) = f
1
u= 0
TM(u)du =AM2.
Mean value results
130
We subtract the constant term to get the Fourier series for the discrepancy DM(u, v). Since DM(u, v) is a monotone step-function, then the Fourier series
converges to the value DM(u, v) for all u and v for which DM(u, v) is continuous (Hobson (1957) II, Art. 465).
Suppose that DM(u, v) is continuous at (0, 0). Then, for some S, ij, Dm (u, v) is constant for Iul 5 S, Iv15 77. We consider the partial sum H
E
K
E aM(h,k)=J
H K
f 1/2 DM(u,v)E Ee(-hu-kv)dudv 1/2
1/2
1/2
-H -K
h= -H k= -K
1/2
r
1/2
- 1/2 !_ 1/2 x
Dm (u, v)
sin ar(2H + 1)u sin 7r(2K+ 1)v du dv. sin irv sin aru
For H >- K we integrate with respect to u first. By the counting squares argument of Lemma 2.1.1, for fixed u there are O(M) values of v at which DM(u, v) is discontinuous, and the total variation of DM(u)/sin 7ru for 85 u < i is O(M/S). By the first derivative test (Lemma 5.1.2) we have
f
1/2
Dm (u, v)
sinir(2H+1)u
1
dv << log K,
I
1/21
M SH
du << -,
sin au and similarly for the integral from - i to - S. Since 1/2 sin ir(2K+ 1)v S
sin it v
we can reduce the range of integration over u to - 6 < u 5 S with an error M log K <<
SH Similarly we can reduce the range of integration for v to -q< v < rt at the cost of an error term
OMlog6H '1K
11
The main term is now DM(0,0)
sin ir(2H+ 1)u sin ar(2K+ 1)v dude sin vrv siniru n _S s
n I
I
(
1
8H
1'
qK
if H and K are so large that SH 1, 77K 1, by a similar argument using the First Derivative Test. Hence the partial sum tends to DM(0,0) if H and K tend to infinity in such a way that (log K)/H - 0. (log H)/K -* 0, l>-
The lattice point discrepancy
131
The same argument works at a general point of continuity of DM(u, V). We have proved that partial sums over rectangles tend to DM(u, u). In the
corollary we need the partial sum over a circle h2 + k2 < R2. We pick an integer Q with Q g (MR)1"4(log R)1,2(min(8, ,q))-1/2,
and approximate the circle by lines of the square lattice, side Q, as in the counting squares method of Lemma 2.1.1. The area between the approximating polygon and the square, side 2R, that contains the circle can be divided into O(R/Q) rectangles. The sum of aM(h, k) over each of these rectangles is
O
Q. M314 (
M log R
J« -R
R min (8, 77)
1/2
log R
,
1
I1
min(8, 77)
R1/4
by the argument above. There are O(QR) lattice points between the circle and the approximating polygon. Assuming the bound for aM(h, k) stated in the lemma, we see that these terms contribute <<
QRM1/2
M3/4 (
R312
log
R
1/2 ,
R114 l
the same bound as the rectangles give. This bound tends to zero as R tends to infinity.
We evaluate the coefficients using the row-of-teeth function p(t) (a notation that unfortunately clashes with that for radius of curvature): E aM(h, k) e(hu + ku) = NM (u, u) - TM(u) k#0
E
(p(g(m - u) + u) - p(f(m - u) + u))
M«+uSmSM/3+u
e(kg(m - u) +ku) - e(kf(m - u) +ku) 2 i ik
M«+u5m5Mp+u k#0
Hence for k 0 0 we have aM(h, k) 1
Y" fu-0 M«+u5m5M/3+u
21rik
mm(M/3,m)
(e(kg(t) +ht) -e(kf(t) +ht))
fmax(M«,m-1)
2?rik
_ LE m
e(kg(m - u) - hu) - e(kf(m - u) - hu)
fyrp e(kg(t) + ht) - e(kf(t) + ht)
M«
21rik
dt
dt =J(h, k) -I(h, k),
du
Mean value results
132
say. The curve CM(0,0) has radius of curvature Mp, and
1/Mp =f" cos3 fr,
1/p = F" cos3 41,
where 41 is the tangent angle. Thus
f" (x) >> 1/M. By the Second Derivative Test (Lemma 5.1.3) we have
I(h, k) «M112/Ik13"2, and similarly for J(h, k). By symmetry we have
(M1/2 M1/2 aM(h, k) << minl
1h13/2'
M112
IkF) « (h2 +k2)
We approximate the integrals in the case I h/kI 51; if IhI > IkI, then we treat y as the independent variable, and argue similarly. We split the range at Mat and M)31, where u =Mat corresponds to the tangent angle +yo it/3, and u = M/31 corresponds to the tangent angle 1 = 7r/3. On the range Mf31 to M/3 we use various forms of the first derivative test. Since we have F" = 1/p cos3 41,
F' = tan 41,
3 sin 4-
F(3)
pcos54
dp
1
p2cos341
dcr'
integrating by parts introduces improper integrals as 41 tends to it/2. We split the range of integration at u = M/32, corresponding to 41 = 4/2 with
cos +lr2 =1/ (kM) . Then, by the First Derivative Test (Lemma 5.1.2),
fMp e(kMF(t/M) + ht) Mpg
21rik
dt <<
1
1
k2F'( P2) = k2 tan
«
1
2
On the middle range we integrate once by parts to get Mpg
IMp1
e(kMF(t/M) + ht) dt 21rik
e(kMF(t/M) + ht)
JM$2
(27ri)2k(kF'(t/M) + h)
Ma,
Mpg
e(kMF(t/M) +ht) (21ri)2k(kF'(t/M) +h)2
- Itirp, xF"(-M) M dt.
The lattice point discrepancy
133
We apply the First Derivative Test to the integral, with 2 cos kF" (t/M)/M k (kF'(t/M)+h)2 - Mpcos3t/i (ksinqi )
1
<<
kMcos fi r'
an d
f
kF" (t/M)/M
d
Mpg
dt
dt (kF'(t/M) +h)2
m"P
F'3>(t/M)
2kF" (t/ M)2
(kF'(t/M) + h)2
(kF'(t/M ) + h) 3
Mpg k Mp,
«
'2
M2
k
1
MZ
TO;3-ql
2
1
kM
cost
k
k2
+ cosh
sin ii cos2 4, d
dt
cos3 qi 1 k3 ) M cos c i d 4,
1
<<
kM cos
4,2
Hence we have
e(kMF( N1) +hM/1) Mpg e(kMF(t/M) + ht) dt = 21rik (27ri)2(kF'( p1) + h) Mp
rcos42 +OI k2
1
)
+O k3Mcos 421
Adding the ranges, we have
1MP e(kMF(t/M) +ht) 21Tik
m"0
dt = -
e(kMF(131) +hMp1)
(21Ti)2(kF'(Pl) + h) 1
+0 ( M 1/2 ( h 2 + k2) 5/4 The range Ma to Mat is treated similarly. In the main range from Mat to M)31, the phase of the exponential is stationary when
F'I
t
M
)= -
h k
l (h , k) ,
'r s 4
1
(h , k)
5 4-
.
If k is positive, then the vector (k, - h) is in the anticlockwise direction of the tangent. For convenience of notation, we suppose that the parameter t in the
Mean value results
134
definition of the curve C as a locus is the tangent angle 4. Since F" is positive, the stationary phase term is e(kMF(x(c&l)) +hMx(4,1) +'-g)
27rik (kF"(x(a/r1))/M) (Mpcos3 01) e(kMY(/i1)+hMx(+li1)+ 11-) 21rik3/2
(Mp) +k2)3/4
e(-ML(k, -h) + $),
2rri(h2 where p is the radius of curvature at the point cross product
4'1,
and L(k, -h) is the vector
L(k, -h) = (x(i1),Y(ii1)) x (k, -h). The formula for the stationary phase integral (Lemma 5.5.2) gives
MO, e(kMF(t/M)+ht)
2aik
fMa1
dt
e(kMF((31)+hMf31)
e(kMF(al)+hMa1)
(21ri)2(kF'(, ) + h)
(2Tri)2(kF'(a1) + h)
(MP) 2 rri(h2 + k2)3/4
e(-ML(k, -h) + 8) + e1(h, k),
where e1(h, k) is the remainder term M 1 (1 1 e1(h, k) = OI k (kM)3/2) - O (h2 + k2)5/4 M112 ) The three ranges fit together to give
Ah, k) _
M3
IMa
e(kMF(t/M) + ht) 2arik
dt
(MP) 3 4 e(-ML(k, -h) +'-g) + e3(h, k), 27ri(h2+k2) where
e3(h, k) = el(h, k) + O(1/M1 /2(h2 +k 2)5/4) Similarly we get
J(h, k) =
V (P) k2)3/a
e(ML(-k,h) - '-'-g) + e4(h, k),
2vri(h2 + where p is the radius of curvature at the point i i2 where the vector (- k, h) is
The mean square discrepancy
135
in the anticlockwise direction of the tangent, and E4(h, k) satisfies the same bounds as E3(h, k).
O
6.3 THE MEAN SQUARE DISCREPANCY We can now prove two results for the lattice point discrepancy which correspond to Theorem 6.1.1 for the simple exponential sum. Fourier series lend themselves naturally to mean square estimates, but less well to 'pointwise' bounds. When pointwise estimates are obtained, it is often by use of a mean-to-max argument. In a direct mean-to-max argument we relate the value of a sum to its mean square over a short interval of a parameter. Extra information may be needed: in Lemma 6.3.2 below the jumps of the sum are always jumps to a greater value. Indirect mean-to-max arguments are a type of `divide and conquer': we relate the value of one sum to the mean square of a shorter sum. The differencing step of the van der Corput iteration (Lemma 5.6.2) has this form. An example outside the scope of this book is Burgess's method for character sums (see Gel'fond and Linnik 1966).
Theorem 6.3.1 (Kendall 1948) is the analogue of the second result of Theorem 6.1.1, where we move the curve with respect to the integer lattice, and Theorem 6.3.4 (Huxley 1995b) is the analogue of the first result, where we enlarge the curve with respect to a fixed centre. Theorem 6.3.4 says that the discrepancy has no memory; it returns to about its root mean square size after a bounded time. Theorem 6.3.4 is a localized version of Nowak's mean value theorem (1985a). Nowak (1985b) also gives a limitation result stronger than the lower bound in Theorem 6.3.1. Theorem 6.3.1
For fixed M we have
f01(TM(u) -AM2)2 du
I
5 f ' f1DM(u, v)dudv << M,
provided that the level of differentiability R of C is at least 2, and f01
fI(TM(u) -AM2)2 du >> M,
f1DM(u, v) du dv >
provided that R z 4. The implied constants depend on the curve C.
Proof We drop the suffix M. By Parseval's formula for Fourier series we deduce that
f111 0
0
(N(u, v) -AM2)2 du dv =
F, F, Ia(h,k)12 h
k
(h,k)#(0,0)
<< E E min h
k
(h,k)#(0,0)
M M Ihl'
,
Ikl3
«M.
Mean value results
136
Also
f'(T(u) -AM2)2 du= E la(h, 0)IZ < h#0
0
EE
l a(h, k)I2.
k
h
(h,k)#(0,0)
This gives the upper bounds. For the lower bound we need only prove that some Fourier coefficient is as large as in order of magnitude. If x = y gives the minimum of f(x), then
b(k,0) =
e(kf(y) + $) 21rik
Similarly, if x = 8 gives the maximum of g(x), then
e(kg(6) -
b(-k, 0)-
21Tik3/2 I9°(8)I
If a(0,1) has smaller order of magnitude than M112, then these terms cancel for k = 1, and, for some integer r,
f(y)-g(6)+'-=r+2+e for some small e. Then
2f(y) - 2g(6) +n = 2r+ a + 2e, and b(-2,0) does not cancel b(2, 0). Hence Ia(0,1)I2 + Ia(0,2)12 >> M.
By symmetry the same lower bound holds for la(1, 0)12 + Ia(2, 0)12, and we have a lower bound for E I a(h, 0)12.
O
For our next theorem we consider the curve CM(0,0) for M varying. We assume that the origin is an interior point of C, and we drop u and v from the notation, writing CM, NM, and DM for the curve CM(0, 0), the number of lattice points NM (0, 0), and the signed discrepancy DM (0, 0).
Lemma 6.3.2 (Mean to maximum) Suppose that the origin is an interior point
of C. Let 8 satisfy 0< 85 M/2. If IDMI>-5A6M, then we have I
f MD` d t M-S
VTL
M,
s
Ttrdt
and
f
M+S D` M-s
dt>
SD2 4M
8IDMI
(6M)
137
The mean square discrepancy
Proof Since C, lies outside Cu for t > u, the number of lattice points caught inside C, is a non-decreasing function of t. For t u N,>N,,, Hence, if DM > 0, then for M:!-, t::5; M + 8
D, DM -A(t2 - M2) >- DM -A(t - M)(t + M) Z DM - 5A 8M/2 > DM/2, and similarly if DM < 0, then for M - S 5 t.< M
D,
O
Our next lemma is geometrical. Lemma 6.3.3 (Reciprocation) The lines ax + ,a y = 1 in the plane which do not go through the origin correspond one-to-one to the points (a, /3) other than the origin, the poles of these lines with respect to the unit circle. If the line ax + /3y = 1 is tangent to a given convex curve C enclosing the origin, given in polar coordinates by r = f (0 ), then the point (a,,6) lies on a convex curve D given parametrically by
+ A) a- sin(0 f(0)sin A '
a-
cos(0 + A)
f(O)sin A '
(6.3.1)
where A is the angle between the radius vector OP and the tangent at the point P with polar coordinates r and 0 on the original curve C, with d
cot A= d0logf(0), 0
Proof If ax + 13y = 1 is the tangent to C at the point P (r cos 0, r sin 0), then (6.3.3)
a cos 0 + /3 sin 0 = 1/r,
and the gradient at P is
(rcos 0+
/dr dr desin0 I decos0-rsin0J
1
= (cos 0 + cot A sin 0)/(cot A cos 0 - sin 0) = tan(0 + A) as asserted. Thus
- a/p = tan(0 + A), a cos(0 + A) +,B sin(0 + A) = 0.
(6.3.4)
Mean value results
138
The simultaneous equations (6.3.3) and (6.3.4) for a and 8 give (6.3.1) of the theorem. Differentiating, we get from (6.3.3)
da
d/3
dr
1
d9cos9+desinO=--
d9+asinO-/3cos0 sin 0 sin(O + A) + cos 0 cos(O + A)
cot A
r
r sin A
cos A
+
rsin A
cos A
rsin A
= 0.
(6.3.5)
Thus
-cot0=tan(0+ 2),
da
and the tangent to the curve D at the point Q is xr cos 0 +yr sin 0 = 1,
corresponding to the point P on C; we have verified that reciprocation is a self-inverse transformation. Differentiation in (6.3.4) gives
da d9
cos(0+A)+
d(3
d9
sin(0+A)
_(asin(6+A)-/3cos(0+A))1+
ddA
)
dA
1
(6.3.6)
rsin A(1+ d8
Solving (6.3.5) and (6.3.6) as simultaneous equations for d a/d 8 and d /3/d O gives
da
sin 6
d/3
dA
cos 0
dA
rsin2A(1+ d0)
d0 rsin2A(1+ d9 Since 0 + it/2 is the tangent angle, the radius of curvature of D is given by dB
1
d
(
r sine A
l+
(6.3.7)
d9
The arc length on C is given by ds (__de)
2
= r2 +
dr 2 (d ) = r2 cosec2 A.
Since the tangent angle for C is 0 + A, we hive (
1+
dA
d9 ) P
d(0+A)
ds
r
d6
d(9+A )
sinA
We deduce (6.3.2) from (6.3.7) and (6.3.8).
(6 . 3 . 8) 11
The mean square discrepancy
139
We can now prove our theorem on the mean value of the discrepancy DM under enlargement of the curve CM. Theorem 6.3.4 Suppose that the curve C encircles the origin, is four times differentiable, and has radius of curvature bounded away from zero and infinity. Then for M large f+1IDt12
dt = O(M log M).
M"
The implied constant depends on the shape and size of C, and on its position and orientation with respect to the coordinate axes.
Corollary Under the same conditions
DM = O(M2'3(log M)1"3).
Proof This is not a Parseval inequality, so we must work to get convergence. From the mean-to-max argument (Lemma 6.3.2), we have
IDuI<SA Su+
(6M)
u
D,
S
Ifu-s
t
dtI +
(6M
+S Dt u
S
Ifu
dt
so that du 5 75A252(M+ 1)2 +
216M
JM+1
2
fu+S D, dt
S
M"
M-S
du.
u
We choose 6 = 1/ OMW, so that the first term is O/(M). Next we write
Dt=
(-2i)Imb,(h,k)+O
F,
1 VFM-
(h,k)*(O,O)
with
fu+SDt u
T
dt
-2i
Im
21ri(h2
(h,k)#(o,0) X
VFp (-k,-
fu+se(-tL(k,
h) +k2)3/4
-h) + 8)dt+0(
u
-2i X
)
E
P(k'
h/4 Im e(-uL(k, -h) + 1)
(h,k)#(0,0) (h + k 2)
1-e(-5L(k, -h))
+O(S )
2IriL(k, -h) Imbu+512(h,k) sin7r6L(k,-h) S -2i E +O irL(k, -h) f M )' (u -+5/2)
Mean value results
140
where p(v) denotes the radius of curvature at the point where the vector v is the anticlockwise tangent. Thus MR
5
1728M M+1 ?r2
M-S
2
bu+s/2(h, k)sinc SL(k, -h)
E
(h,k)#(0,0)
+ O(M).
du
u+S/2 (6.3.9)
We use Lemma 5.6.4 with
X=M+(1-S)/2,
A=(1+S)/2,
and the real numbers s, as -L(v) indexed by the non-zero integer vectors v. The corresponding coefficients are a; =
e($)sinc3L(v).
21r iIvl3/2
The integral in (6.3.9) is
5
Ir2(1+o) 4
E E sinc SL(v)sinc SL(w) v
2 V(P(V)P(W)y
(IvIIwl)
W
Xe1 M+ 1 2 S)(L(w) -L(v))A((1 + 5)(L(v) -L(w))). (6.3.10) We want to estimate this sum in order of magnitude. Non-zero terms have
IL(v) - L(01:5 1.
(6.3.11)
Since the origin is an interior point of C, we have L(v) x IvI. (6.3.12) The inequality (6.3.11) introduces the non-uniformity in the position and orientation of C. We use it to deduce that when the A function is non-zero, then IvI X IwI.
Hence the expression (6.3.10) is sinc 2 SL(v) <<
(6.3.13)
F, 1, V
Ivl3
W
where the second sum is over w satisfying (6.3.11), and we have used Cauchy's inequality in the form
2sinc SL(v)sinc SL(w)
sinc2 SL(v)
5
sinc2 SL(w)
Iw13 + to de-symmetrize, interchanging v and w in the second sum.
(IoIIWI)3"2
Ivl3
The mean square discrepancy
141
To estimate the number of vectors w satisfying (6.3.11), we consider the equation
L(k, -h) = -A. By reciprocation (Lemma 6.3.3), the point (h/A, k/A) lies on a smooth curve
D reciprocal to C. If (6.3.11) holds, then v and w correspond to points (hl, k1) on A1D, (h2, k2) on A2 D, with 'l2XAl XIVI.
A2=Al+O(1),
The lattice point (h2, k2) lies within a distance 0(1) of the curve A1D. This is
the configuration of Chapter 3. For the application it is enough to apply Lemma 2.1.1 twice and subtract, to see that, for fixed 'v, there are O(IvI) solutions of (6.3.11).
We divide the sum in (6.3.13) into blocks of the form R5 Ivi < 2R, where R is a power of 2. Thus the sum in (6.3.13) has size
<
/
2
R3 min 1,
R5F
R2 )R M
1+ R> FM
RZ
<< log M.
The main result follows on substituting this bound into (6.3.9). The corollary follows by the mean-to-max argument of Lemma 6.3.2 with 3 = I DM I /5AM.
7 The simple exponential sum 7.1
MOTIVATION
The central problem discussed in this book is to estimate the size of an exponential sum in one variable M2
m
S= E e(TF(m)). M
(7.1.1)
M=M
Here M and T are parameters with M2 < 2M - 1, and F(x) is a real function with at least four continuous derivatives on an interval [1 - 6, 2 + 81 for some S > 0. The function F(x) need not be a monomial function in the sense of Chapter 3. We will require certain derivatives and determinants of derivatives of F(x) not to vanish; these conditions will be satisfied if F'(x) is a monomial function of degree s with s negative. Only values of x between 1
and 2 appear in the sum S; we are allowed to regard F(x) as given on the interval [1, 2], and defined outside that interval by the Taylor polynomial about the point 1 or 2, whichever is nearer. We need the longer interval of definition in order to approximate F(x) by a Taylor series about a point which may lie just outside the interval [1,2]. We will assume when necessary
that M is sufficiently large in terms of S. The constant factor in the upper bound is constructed from the maxima and minima of certain determinants in the derivatives of F(x) on the longer interval [1 - S, 2 + 6 ]. Estimating the sum S is analogous to lattice point questions for enlargements of a fixed curve. The relevant curve is not y = F(x), but y = F'(x). Put
f(x) = Mx/M).
(7.1.2)
The sum S can be large if f(m + 1) -f(m) is close to an integer for a long interval of values of m, and so the exponentials e(f(m)) are approximately constant. As we saw in Lemma 3.2.2, the differenced function f(x + 1) - f(x)
behaves very much like f'(x), which is (T/M)F'(x/M). This analogy explains why the natural order of magnitude for T is about M2, so that M and T/M have the same order of magnitude. If we think of M as a length, then T is an area. The sum S is unchanged if we add to f(x) a polynomial g(x) that takes integer values at integers. If g(x) has degree d, then Jg(x)I >> Md
Motivation
143
for some x between M and 2M. For T about M2, there are many linear polynomials g(x) which we can add to f(x) without increasing its order of magnitude. If T has greater order of magnitude than M2, then we may add quadratic polynomials, and change f"(x). Within the method we will see a splitting into cases M >> FT, when there is a non-vanishing condition involving F"(x), and no condition involving F(x) or F'(x), and M << FT, when the non-vanishing conditions use only F()(x) for r>_ 3. If f(x) is a quadratic polynomial with rational coefficients, then the sum S
can be evaluated as a linear combination of algebraic exponential sums E e(h(m)/q), where h(x) has integer coefficients and the sum is over all integers m(mod q). Since the third derivative f (3)(x) is numerically small but
non-zero, the best that we can do is to approximate f(x) by a quadratic polynomial on a short interval of values of x. If the coefficients have small height, then we get many copies of the same short sum, so that the total sum is large. If the coefficients have large height, then the algebraic exponential sums are small compared with the common denominator q, but we have to add many of them. If we can prove that there is cancellation, then the sum that we want is small. The situation is analogous to the `Hardy-Littlewood
circle method'. This was first used by Hardy and Ramanujan (1918) to evaluate approximately fg(z)dz around a circle centred at the origin for a function g(z) that had the unit circle as natural boundary, with singularities at all points e(a/q) where a/q is rational. They obtained explicit contribu-
tions when a/q has small height, and upper bounds when a/q has large height. Hardy and Littlewood applied this method to the generating functions W
g(z) = E y(n)z" 1
of various interesting sequences y(n). Vinogradov gave the method its modern form (see Vaughan 1981) by considering finite sums and integrating on the unit circle itself. The range of integration is divided into subintervals, each corresponding to a rational approximation, often called Farey arcs. An arc is called major if the approximation is by rationals of small height, minor if of large height. The integrand is usually evaluated on the major arcs, but
estimated on the minor arcs, either pointwise or in the mean or both. We shall use the Hardy-Littlewood method in Chapter 12. In terms of the Hardy-Littlewood method, the Bombieri-Iwaniec method (1986a) corresponds to the integral I2
N
E e(f([Mx] +n)) n=1
with r >_ 3. The case r = 1 gives the differencing step (Lemma 5.6.2).
The Bombieri-Iwaniec method has a `divide and conquer' structure. We divide the sum from M to M2 into short intervals of length about N (suitably
chosen), and approximate f(x) by a polynomial with a rational quadratic
The simple exponential sum
144
term. If the coefficients have small height, then the short interval is called a major arc, and if large height, then a minor arc. The combinatory step is applied to the minor arcs, which are estimated in the mean using the large sieve in the form of Lemma 5.6.6. Since the root mean square of a sum of length N is expected to be at least N, the bound that we obtain cannot be better than
(M/N)WN = M/i in order of magnitude. We also consider a family of functions Fi and the corresponding sums
Si=
M,(i)
rr
m=MI(i)
1
m
(7.1.3)
Here M< M1(i) <M2(i) 5 2M - 1 for each i. Usually we can write
Fi(x) =F(x, yi)
(7.1.4)
where F(x, y) is sufficiently differentiable in two variables, and y,, ... , y, lie in the interval [0,1] with Yi+i -Yi Z 1/U;
(7.1.5)
thus I:!-. U + 1. The object is to bound
E
ISjIr
for some integer r Z 1.
7.2 MAJOR AND MINOR ARCS We divide the range (1 - 6)M to (2 + 6)M into intervals of equal length N;
the intervals at each end of the range may have to be shorter, unless (1 + 26)M is an integer multiple of N. The approximation step is based on the Taylor series for f(m + x), as far as the term in x3. We choose N in order of magnitude so that the remainder term is negligible. We suppose that F(')(x) is non-zero on the interval [1 - S, 2 + 51 for r = 2,3 and, when T is large, for r = 4. Since f(x) = TF(x/M), then there are constants C21 C3, and C4 for which
T/C,M' S If (')(X)I S C,T/M'
(7.2.1)
for (1 - S)M <x 5 (2 + S)M and r = 2,3,4. The method is most powerful for T about M2 in order of magnitude. We suppose that C3T < M3.
(7.2.2)
In a sense, the condition (7.2.2) is unnecessary, as the bounds which we
Major and minor arcs
145
obtain would become trivial if (7.2.2) does not hold. We take N to be an integer with
1
(7.2.3) (7.2.4)
(7.2.5) N 4 T< M4. The upper bounds for N in (7.2.3) and (7.2.5) are not critical conditions which affect the optimal choice of N, but the condition (7.2.4) can be. The second parameter of the construction is R, the unique positive integer with
R2NT >- 2C3M3 > (R -1)2NT.
(7.2.6)
As we shall see, R gives the usual order of magnitude of the denominator q when we approximate the real number f" (x)/2 by a rational number a/q. From (7.2.4) and (7.2.6) we have R2 >- N. (7.2.7) The Bombieri-Iwaniec method improves the classical bounds for exponential sums when R N. (7.2.8) We assume that M and T are sufficiently large (in terms of the constants Cr)
for R and N to exist and be large themselves. It is convenient to express orders of magnitude in terms of M, N, and R instead of M, N, and T. For
(1-SW Sx5(2+8)M we have N4f (4)(x) << 1
by (7.2.5), and 1 << N3f(3)(x) << N by (7.2.4), (7,2,6) and (7.2.8).
We divide the range for x into intervals U: M + (j - 1)N 5 x < M + jN, and we let J, be the interval of values taken by f"(x)/2 for x on U. By (7.2.1) the length of J, lies between NT/2C3M3 and C3NT/2M3, and thus between 1/R2 and 2C3/R2. Since f(3)(x) has constant sign, different intervals J, are disjoint. We pick a rational number a/q = aj/qj in the interval Ji. The construction that follows takes place on a fixed minor arc J,. We use the suffix j only when different minor arcs are being compared. The basic choice
of a/q is the rational number of smallest denominator (and so smallest height) in the interval J,; in some cases we modify the choice of a/q. We call
the intervals Ji or U Farey arcs corresponding to a/q. Having picked a/q, we pick the integer m = m,, the centre of the Farey arc U corresponding to a/q, to be the integer within U for which f"(m)/2 is closest to a/q, so that
f" (m)/2 = a/q + A,
(7.2.9)
with
Al 5 C3T/2M3
1/NR2 << 1/N2.
(7.2.10)
The simple exponential sum
146
For this integer m we put
p.= p =fl3l(m)/6, and we pick an integer b = bj for which b/q is as close as possible to f'(m), so that
qf'(m)=b+K,
IKI<1.
We now define the approximating polynomial
(b + K)x
g(x) =gi (x) = f(m) +
q
ax2
+
+ µx3.
(7.2.11)
9
If K is close to '-2, then we also regard the coefficient of x as (b' + K')/q,
with b' = b + 1 and K' = K - 1, and similarly if K is close to - z. The ambiguity is unavoidable if K = i . In the next lemma we can adjust the coefficient of x by 0(1/N), so K only matters up to 0(q/N). The ambiguous case extends in practice as far as K = i + O(q/N). We define Ij to be the intersection of the intervals U. and EM,M2]. A technical simplification is to use the polynomial gj(x) to approximate
f(mj +x) for mj +x on Ij +2. The integer mj + n lies in Ii +2 for an interval of values N1,..., N2 with
N2-N15N-1.
N253N,
N
It is very convenient to have the order of magnitude of the variable n bounded below as well as above. Lemma 7.2.1 (Polynomial approximation) In the notation above N2
N2
E e(f(m + n)) << max
N3zN,
n=N1
E e(g(n))
Proof Let
h(x) =f(m +x) -g(x). Then, for some 0 in 0 < 0 < 1, we have
h'(x) = f'(m +x) -g'(x)
=Mm) -
b+K q
+x(f"(m) -
q2a
+
x3 6
ox) <<
since x << N. The result follows from Lemma 5.2.1.
N O
We call .1 or Ij a major arc if the rational a/q in Jj with smallest denominator has q< R2/N. Other arcs are minor arcs. There are two modifications to the choice of rational a/q in Jj on the minor arcs.
Major and minor arcs
147
If R2/N < q < R, then we replace a/q by the rational point e/r in Jj whose denominator is least, subject to r z R. We note that a/q and e/r must be consecutive in the Farey sequence ,fi(r), and thus by Lemma
Modification 1
1.2.1
1/R25Ia/q-e/rl =1/qr and r >_ R2/q Z R. To bound r above, we consider the two neighbours of a/q in the Farey sequence of order r - 1. One of these, c/s, say, has 1
a
c
qs
q
s
since neither neighbour lies in J, and s + q z r. Hence
rSq+s Sq+2R2/q
Modification 2
tor. We take e/r in Lemma 1.6.3 to be a/q, and f/s to be one of the neighbours of e/r in .fi(r), chosen so that at least half of Jj lies between e/r and f/s. Since J, has length at least 1/R2, some subinterval of Jj with length S >- 1/4R2 satisfies the conditions of Lemma 1.6.3. In the second case there is already a rational a/q in Jj with small denominator, but R2/N
qr < q(q + 2R2/q) 5 3R2.
We take f/s to be a/q. Some subinterval of Jj with length S > 1/6R2 satisfies the conditions of Lemma 1.6.3. In each case we take Q to be the next
power of 2 following r. Lemma 1.6.3 gives us at least Q2/16R2 rationals, each with denominator q < 32r < 96N. We can average over the approximations corresponding to these rationals. As a point of notation we should index the minor arc approximations by the rational a/q rather than the index j, as
different choices of a/q give different approximating polynomials for the same minor arc. We use the notation J(a/q) for a minor arc with the choice a/q of rational approximation to f"(x). We use Modification 1 on the minor arcs in this chapter, but not Modification 2. The second modification is required in the next two chapters, where the error terms that increase for large q are more prominent. The changes when we consider a family of sums Si given by (7.1.3) are
notational. The inequality (7.2.1) must hold for each function f;(x) = TF;(x/M). If F,(x) is obtained from a function of two variables by (7.1.4),
The simple exponential sum
148
then the differentiations in (7.2.1) are partial differentiations with respect to x. The Farey arcs J;j depend on the index i, and Iii is the intersection of U with the interval [M1(i), M2(i)].
Lemma 7.2.2 (Counting Farey arcs) For positive Q, let W(Q) be the number of Farey arcs Jj on which qj lies in the range Q 5 q < 2Q (for some choice of qj when Modification 2 operates). For major arcs
2CCM 23
(7.2.12)
W(Q) << (M2 - M)Q2/NR2 + 1.
(7.2.13)
R/2:5 Q< 32R2,
(7.2.14)
W(Q) << (M2 - M)R2/NQ2 + min(R2/Q, M/NQ).
(7.2.15)
and we have
For minor arcs and we have The implied constants are constructed from C2 and C3.
Proof The fraction a/q is a value of f" (x)/2, which runs through an interval of length 0 with
0<
M2-M 2
(M2-M)C3T
max J f
2M3
<<
M2-M NR2
So small a value of q can occur only when 1
1
2Q < q < max
If"(x)I 2
C2C3M
C2T
< ZMZ < N(R
- 1)2
On the major arcs we use the first assertion of Lemma 1.6.1 with S =1/R2. On the minor arcs we use the third assertion, which gives << (M2 - M)R2/NQ2 + R2/Q. We can also consider q/a, which lies in a range of length A' with I f (3)(x)I
' < 2(M2 - M)max I f (x)12
2C2C3(M2 - M)M T
(M2 - M)NR2 <<
M2
Each minor arc Ji corresponds to an interval for 2/f"(x) of length 1
1
N2R2
R2 min I f"--- » M2 If q >- Q, then
azQmin
If xI > > Q M 2
R NR.
Poisson summation
149
We apply the third assertion of Lemma 1.6.1 with 6 = S' >> NR/M, to get the bound W(Q) << (M -M)R2/NQ2 +M/NQ. Modification 1 on the minor arcs ensures that q >- R, and Dirichlet's theorem M
(Lemma 1.5.1) ensures that q
Modification 2 then replaces a/q by
fractions with denominators at most 32q. We deduce the bounds for Q. 7.3
O
POISSON SUMMATION
The sum on the right of Lemma 7.2.1 is an exponential 4um over a polyno-
mial with rational quadratic term. We adapt the inversion step (Lemma 5.5.3). First we apply Lemma 5.4.6 to the function Kx/q + µx3. This gives N2
e(-f(m)) E e(g(n)) N3
G(a, b + r; q)
_ r
q
fN
(r - K)x Ze(µx3-
N3
q
dx+O(VlogN)
=EG(a,b+r; q) fN2/ge(µg3y3_(r-K)y)dy+O(/ log N), r
N3/q
(7.3.1)
where the sum is over the range
3µ^2 +K - a
(7.3.2)
We apply the method of Lemma 5.5.3 to the function h(y) = µg3y3 + Ky. The stationary phase occurs at y = yr, where r - K 1/2
3µg3yr +K=r,
Yr =
( 3µq3 )
so that
h(yr) - ryr = -2µg3Y; _ -2
(r - K)3
1/2
27µq3
h" (Yr) = 6µg3Yr = 2(3µg3(r - K))1/2.
Since yr x N/q, we have h" (Yr) x q2/R2.
(7.3.3)
The terms of the sum are twisted by the Gauss sum G(a, b + r; q), whose order of magnitude is given by Lemma 5.4.5 as
JG(a, b + r; q)j /j
150
The simple exponential sum
in all cases. The estimate for the error term is not affected if we simplify the range of summation by shifting the endpoints by a bounded distance. We now have the modified form of Lemma 5.5.3. Lemma 7.3.1 (The transformed Farey arc sum) In the notation above
G(a,b+r;q)
N2
e(-f(m)) E e(g(n)) _ r
N3
(12µg3(r- K))1/4
(r-K)3
e -2
1/2
27µq3
+OR (I +logN)/), q
where the sum is over
3µgN3 < r < 3µgN2.
We make two comments about this lemma. First, there are really two inversion steps together, the first step modulo q, and the second step by Poisson summation. In the analogues in later chapters, both steps are integral transforms. Secondly, the integral in (7.3.1) is an Airy integral, related to the Bessel functions (Jeffreys and Jeffreys 1962, Chapter 17). Bessel functions occur in the same way in later chapters.
At this stage we can deal with the major arcs. On the major arcs the number of terms in the sum of Lemma 7.3.1 is bounded, so that we might as
well estimate each term as << R/ r. Lemma 7.3.2 (The major arcs contribution)
The major arcs contribute
r M-M R12 M lo N r (N) logN+mm RlogN , NR )
to the sum (7.1.1).
l
1
Proof We divide the major arcs into ranges Q:5 q < 2Q, where Q is a power of 2. In the notation of Lemma 7.2.2, the major arcs contribute W(Q)
<< Q
7Q
log N,
where Q runs through powers of 2 in the range (7.2.12). Summing geometrical progressions gives the lemma.
The contribution of the major arcs is very small. In fact, we can avoid major arcs altogether. We used the polynomial g/_ 2(x) on the Farey arc Ii. However, if I/_2 is a major arc, then I/+2 is not (unless R is close to N in order of magnitude). So we can avoid using g/_2(x) as an approximating
polynomial, at the price of using g/+2(x) on two intervals 1, and I/+4.
Poisson summation
151
Conversely, since we have a good estimate on the major arcs, we can modify the construction so that the approximating polynomial for a major arc is used
on several adjacent arcs (called the flanks of the major arc). The integral involving h'(x) in Lemma 5.2.1 has order of magnitude greater than unity. We require this modification in the next chapter. Lemma 7.3.3 (The trivial minor arcs contribution) The minor arcs contribute
M2-M
(
log N + min I` Nv log N,
MlogNl R3 2
)
to the sum (7.1.1).
Proof We divide the minor arcs into ranges Q:5 q < 2Q, where Q is a power of 2. The sum over r in Lemma 7.3.1 has O(Nq/R2) terms, each of size O(R/ r). In the notation of Lemma 7.2.2, the minor arcs contribute
«EW(Q)I NR +V log N) «EW(Q)NV Q
`
log N,
Q
where Q runs through powers of 2 in the range (7.2.14). Summing geometrical progressions gives the lemma. The bound of Lemma 7.3.3 is larger than that of Lemma 7.3.2, since R 5 N.
The object of the method is to improve on Lemma 7.3.3. However, in any complicated argument, it is important to recognize when to `take out the improvements' and revert to something simpler. Lemma 7.3.4 (The exponent pair 6, 3) If M 5 T, and (7.2.1) holds for r = 2 and 3, and as an upper bound for r = 4, then the sum of S of (7.1.1). satisfies S <<
(M2 -M)
T 1/6 log T + vrM- log T.
The order-of-magnitude constants are constructed from C21 C3, and C4.
Proof For M << T 1/3 the result is trivial. When (7.2.2) holds, then we choose
NxMT-1/3. The result is immediate from Lemmas 7.3.2 and 7.3.3 when we note that R x N, and so min(R, M/NR) 5 M/V 5 YOM-,
min(NV, M/R3/2) << 'M .
The simple exponential sum
152
Apart from the logarithm power, this lemma is a standard result from van der Corput's method (see Graham and Kolesnik 1991), obtained by a differencing step followed by an inversion step. Workers on van der Corput's method usually quote bounds in the form
S «(T/M)aMQ, and the real numbers a and 0 are called the exponent pair. In what follows we try to choose R to have smaller order of magnitude than N. If this choice is not possible, then we fall back on Lemma 7.3.4.
7.4 THE LARGE SIEVE ON THE MINOR ARCS The most satisfying way to proceed would be to show that the sum in Lemma
7.3.1 is uniformly small on the minor arcs, as in the Hardy-Littlewood treatment of Waring's problem (see Vaughan 1981) to count the number of solutions of h; + . +h; = N. As we have no uniform upper bound, we use the large sieve (Lemma 5.6.6) to show that the minor arc sums are small on average. This step is analogous to Vinogradov's mean value theorem in Vaughan (1981). We want to separate the variable of summation from the parameters which
depend on the minor arc. This tidying process may be called gardening, making straight paths in the wilderness (see Littlewood 1986, p. 61). Lemma 7.4.1 (Gardening the minor arc sum) For q in the range Q < q < 2Q, the transformed sum in Lemma 7.3.1 has order
<
1/2
2H-1
K(,,)
E G(a,b+h;q)
h=H
1/2
2 Xe
]/2
(h3/2_ 3K2
dal,
(27µq
(7.4.1)
where K(x) is the weight function of Lemma 5.2.3, and the sum is over integers H which are powers of 2 in the range
3NQ/R2 < H< 108C3NQ/(R -
1)2.
(7.4.2)
Proof We use three partial summation lemmas. The sum on the right of Lemma 7.3.1 is
E
G(a,b+h;q)
heV (12Ag3(h - K)) 1/4
-2
(h - K) 3 27µq 3
over some subinterval V of 3µgN2 < h S 27µgN2.
1/2
(7 . 4 . 3)
(7.4.4)
The large sieve on the minor arcs
153
The weight function involving (h - K)114 is monotone, and the order of magnitude of its square was computed in (7.3.3). By Lemma 5.1.1 the sum is bounded by
R
G(a,b+h;q)e -2
q
(h
1/2
)3
-K
27µq3
h E v,
for some subinterval V1 of V. Next we apply Lemma 5.2.1. We put
H(x) =
2
(27q)
Ox) =
-
(x3/2
Kx1/2
)
(x - K)3/2 - H(x).
2
(27µq3)
Then, for some 0 in 0 < 0 < 1, we have
h'(x) =
(µ
(x - K)1/2 - H'(x)
3
3) K2
3
(27µq3)
8(x- OK)3
/2
«
R2 2
qx
<
1
x
when x lies in the range (7.4.2). We apply Lemma 5.2.1 with
G(n) = G(a, b + n; q)e(-H(n)). Hence the sum (7.4.3) is bounded by R q
G(a, b + h; q)e hEv2
2
(h3/2 (27µq3) `
Kh1/2 1
2
JJ
for some subinterval V2 of V1. The interval V2 is a subinterval of the range (7.4.4), and so a subinterval of (7.4.2) by the conditions (7.2.1) and (7.2.6); for example
3QN2T
h > 3µgN2 > C,3M3 >
3NQ 2C3M3 R2
>
C3M3
6NQ RZ
We subdivide the interval V2 into ranges according to the number of (binary)
digits of the integer h. Thus V2 becomes a finite union of subintervals of ranges H < h < 2H - 1, where max h >- H > (min h)/2 > 3NQ/R2. The result now follows from Lemma 5.2.3.
0
The simple exponential sum
154
We use Lemma 5.4.5 to write the sum over h in Lemma 7.4.1 as 2H-1
E Ae(-Y1h2
-Y2h312
(7.4.5)
-Y3h -Y4h112).
H
Case 1
When q is odd, then we have
A = G(a, 0; q)e
(
- 4abq 2 1,
4a Y1 =
q
IAI = r, 2ab
2 Y2 =
(27µq3) '
2K
Ys = q - 1,
Y4
= - (27µq) (7.4.6)
The rational numbers 4a/q and 2ab/q are only defined modulo one, since the complex exponential e(x) is periodic. Case 0
For q = 0 (mod 4), Y2 and y4 remain as above, but ab2 A = G(a,0;q)e a
Y1 =
4q
4q
IAI=
(2q),
ab ,
Y3 =
2q - 77,
(7.4.7)
and the sum is restricted to values of h with h + b even. Case 2 For q = 2 (mod 4), we have y1 and y3 as in case 0, Y2 and ya as in case 1, a(b 2 - 1) A=G(a,l;q)e 4q A (2q),
and the sum is restricted to values of h with h + b odd.
The exponent in (7.4.5) (Y1,Y2,Y3,Ya) with the vector
is minus the inner product of the vector
x(h) = (h2, h312, h, h1/2). The vector x(h) lies on a twisted quartic curve in four-dimensional space. The large sieve of Lemma 5.6.6 is most powerful when there are so many vectors
that the minimum spacing S is comparable to the average spacing. We increase the number of x vectors by choosing a positive integer r, and writing
(E e(-x(h) ..y)), = F, ... F, h
h,
e(-x(h...... h,) Y),
h,
where x(h)
=x(h'.....h,) =x(h1) + ... +x(hr)
The large sieve on the minor arcs
155
This construction entails a chain of Holder inequalities, which we summarize as a lemma. Lemma 7.4.2 (Holder's inequality) Let r > 2 be a positive integer, S a positive real number. Then
ISI' «M 2-M ( % r I1r RN))r/2 log' N I
Rr+2-s
(W(Q))r-1
1 1ogr- 1 N
+ 5r-1
Qr/2+2-5 Q
4
X E E f K(71) 1=0 H
-CO
E(ne(-x(h).y)I da,
a/q
h
where Q runs through powers of 2 in the range (7.2.14), then H runs through
powers of 2 in the range (7.4.2), the rational a/q runs through the rational numbers with Q < q < 2Q used to construct approximating polynomials on the
minor arcs J(a/q), I refers to cases 0, 1, or 2 for q; in cases 0 and 2 we sum separately over vectors h with all entries even, and over vectors with all entries odd. The entries hi of the vector h satisfy H ::g h, < 2H in all cases, and that hi is even in cases 0 and 2, odd in cases 3 and 4. The vectory depends on a/q and i'
according to (7.4.6) or (7.4.7) in the various cases. The implied constant is constructed from r and the constants C21 C3, and C4.
Proof First we consider the contribution to the sum S of the error term in Lemma 7.3.1, which is <<
W(Q)(Q
+logNJi<<
W(Q)IlogN,
Q Q(7.2.14). This sum is smaller where Q runs through powers of 2 in the range than that in Lemma 7.3.3 by a factor R/N, so its size is ( «N N
FR
log N + min( N% log N,
2
MR3/2
" (M2-M +
C(R
log N.
(7.4.8)
By Lemma 7.3.2, this expression is also an upper bound for the contribution of the major arcs. Either S has order of magnitude (7.4.8), or S is bounded by the sum of the transformed terms in Lemma 7.4.1 in order of magnitude. As usual, we split
the minor arc terms into ranges Q 5 q < 2Q. Substituting the expression
The simple exponential sum
156
(7.4.5) into Lemma 7.4.1, we see that the rth power of the transformed sum has order <<
R F f K(,q)% D(r)e(_x(h).y) dq) Q H
<<
h
-0D
Q/2 logr-'HE L K(1l)\I F(1) e(-x(h) y) I dii.
(7.4.9)
00
Let E(a/q) denote the transformed sum of Lemma 7.4.1. Under Modification 2, each major arc Jj corresponds to x qj2/R2 rationals a/q with q, x q. Thus the contribution to S of the transformed sums on the minor arcs is R2 <<
I E(a/q)I,
Q n2 aE
(7.4.10)
where a/q runs through rationals with Q < q < 2Q which are used to construct approximating polynomials. The number of such rationals is << W(Q)Q2/R2. The rth power of the sum (7.4.10) is R S/(r- 1) '- 1
R2r- S
F, -
2r-8
Q
Q
Q
r-1
1
(00
E
2Sk/(r-1))
( IE(a/q)I)
Q a/q R2r-3 Q2w(Q) r R2
Q Qtr-S
'
1
aE
IE(a/q)I
R2- s
1
<< ,3r- 1
(W(Q))r-
Q2-s
I E(a/q)I r. a/q
Q
The lemma now follows on substituting the expression (7.4.9) for I E(a/q)I'
0 Lemma 7.4.3 (The large sieve) Let V >- 1 be a real number. In the notation of Lemma 7.4.2, we have Y)1)2
E(1) e(-x(h) 1
a/q
ABH5 NR2V(1 <<
Q3
h
+
Q)
N
'
where i refers to the sum Si of (7.1.3). Here A is the number of pairs of vectors x(1i) whose difference has entries bounded in absolute value by 27
0'
(2 QNR)'0,
C27Q R2 )
The large sieve on the minor arcs
157
and B is the number of pairs of vectors y corresponding to indices i and rational numbers a/q whose difference has entries bounded in absolute value by 1
1
1
1
2r(2H)3/2' 2r(2H)' 2r(2H)1/2
2r(2H)2V'
The implied constant depends on r.
Proof For fixed ri we apply the large sieve (Lemma 5.6.6) to
EEI i
a/q
h
The vectors x are indexed by r-tuples of integers rather than by integers, and
the vectors y by indices i and rational numbers a/q, but this is purely a matter of notation; in the one-dimensional large sieve used in Lemma 5.6.7, the x vectors are integers, and the y vectors are rational numbers. In Lemma 5.6.6 we have
Xi = 2r(2HP -0/2.
(7.4.11)
Since y, and y3 are defined modulo one, we can take
-zSYi,Y3<
,
Y1=Y3=1.
We can suppose that I K I < 3, even when we allow J K J to be slightly larger than z in the ambiguous case. Thus we can take Y4 = Y2 = (8NR2/27Q3)
,
since (7.2.1) and (7.2.6) imply
Iµlz2/NR2. However, the bounds Xi, Y,, do not have to be sharp. In Chapter 16 it will be helpful to strengthen one of the spacing conditions, and we will choose V > 1, and replace the value for Xl given by (7.4.11) by
Xl = 2r(2H)2V, with X2, X3, and X4 remaining as in (7.4.11). Then X;Y >> 1 for i = 1, 2, 3, whilst X4Y4 = 2r (2 H)
QN
VQ27
We ought to treat the range Q >> N, the flanks of the major arcs, separately, taking out the term in h1/2 in (7.4.1) by partial summation (Lemma 5.2.1),
and applying the large sieve to three-dimensional vectors. However, the largest contribution comes from ranges with Q x R. We have 4
F1 (X;Y+1)
«l1+
Q1
f 4
1
The simple exponential sum
158
Lemma 5.6.6 allows us two coefficient sequences. We take the coefficients on the vectors x(h) to be 1 in case 1, and 1 or 0 in cases 0 and 2, according to whether the vector x(h) occurs in the sum E('). We take the coefficients on the y vectors to have unit modulus, and argument opposite to that of the sum
r(l) e(-x(h) .y), h
so that we get the sum of moduli. We have
5i<1/Y, with 8l < 1 and S3 < 1. Since xl and x3 are integers, any value less than 1 for Sl and S3 has the same effect as 51 = S3 = 0. The concentration functions
A(S), B(e) in Lemma 5.6.6 are bounded above by the counting numbers A and B in the statement of the lemma. 7.5
U
A PRELIMINARY CALCULATION
The analytic part of the argument has left us two problems: to investigate the
spacing of the x and y vectors, and to estimate the numbers of coincident pairs of vectors of each type. The large sieve, together with the two spacing problems, forms the `combine' stage of the `divide and conquer' method applied to the exponential sum S. The spacing problems themselves are suitable for the `divide and conquer' method. In Chapters 8, 9, and 10 we apply the same process to more complicated exponential sums. We consider the First Spacing Problem in Chapters 11-13, and the Second Spacing Problem in Chapters 14-16. We apply these bounds to the simple exponential sum in Chapter 17, and to the more complicated sums in Chapters 18-20.
To show that the new method does give a better estimate than Lemma 7.3.3, we substitute the simplest non-trivial bounds for the two spacing problems into Lemma 7.4.3. These are the estimates used by Bombieri and Iwaniec (1986a,1986b), although the proofs that they gave were different. We take r = 4 in Lemma 7.4.2 and V = 1 in Lemma 7.4.3. In the First Spacing Problem, Theorem 11.2.5 gives, for any a > 0, 2 H4+E'
A «H4 + SHs+E << Q2
(7.5.1)
where S is the bound from Lemma 7.4.3
VQ3
Q2 HR2.
NR2
In the Second Spacing Problem, Lemma 14.3.5 gives the more complicated bound B <
A preliminary calculation
159
where P is defined in order of magnitude by
P/Q^I f"(x)I xM/NR2, and O1 and 02 are constructed from the bounds for the first and second entries of the differences of the y vectors in Lemma 7.4.3, with R4
N2Q2
Al
«O Z X
R2 .
NZ
Thus Q2 (M IN + M2 N4 B« R2
R4
+ R2 +
M2R4
N2
(7 5 2) .
.
1
We suppose that N and R are chosen so that Q2 M2R4 B « R2 N6
(7 . 5 . 3)
and we substitute (7.5.1) and (7.5.3) into Lemma 7.4.3, so that ABH5NR2V ( Q
1+
Q\
Q H9+EM2R3 (
N «R '
N5
Q ) 1
+N
Q10 M2N4+E
R
R6
io
1
+
Q
(7 . 5 . 4)
where we have used (7.4.2). We substitute the bound (7.5.4) into Lemma 7.4.2
with r = 4, 8 = 1. The bound (7.5.4) is independent of rt, so the integration over rt gives an extra factor O(log N). The total logarithm power is log4 N <<
NE/2.
There are finitely many values for H and for 1, and we can simplify (7.2.15) to W(Q) << MR2/NQ2. Lemma 7.4.2 gives ISI4
M
« (_ + M l
NZ
Q 1/2
R4 M4NE
1l4 R2
log4 N+ Q
Q4 NR
(1-
N)
where Q is summed through powers of 2 with Q >_ R/2 in (7.2.14), and R < N, so that ISI4 «
M4R2 N4
log4 N+
M4NE
NR
R 1/2 << NR (i+)
M4NE .
(7.5.5)
This bound is better than Lemma 7.3.3 when R
«N1-e log4 N.
(7.5.6)
160
The simple exponential sum
We can simplify (7.5.2) to give (7.5.3) when
R4 >> N5/M,
(7.5.7)
R2>> N,
(7.5.8) M>>N2. (7.5.9) The condition (7.5.8) follows from (7.2.7). When we use (7.2.6) to express R in terms of M, N, and T, then (7.5.7) and (7.5.8) become N << MT-2/7, N <<
M3/2/T1/2.
(7.5.10) (7.5.11)
We take the order of magnitude of N to be the maximum allowed by (7.5.9), (7.5.10), and (7.5.11), with the constants of proportionality chosen so that (7.5.9) implies (7.2.2), (7.5.10) implies (7.2.5), and (7.5.11) implies (7.2.4). We substitute in (7.2.6), and find that in (7.5.5) ISI4 «max(M9/4T1/2+E, M2T9/14+E, M7/4T3/4+e), so that
ISI <<M9/16T1/8+e +M1/2T9/56+e +M7/16T3/16+e
(7.5.12)
Unless M2 is very close to M, the bound (7.5.12) improves the result of Lemma 7.3.4 for 1
log M
2
3< -log- <3' which is the range in which we can satisfy (7.5.6). The bound (7.5.12) was the
main ingredient in Bombieri and Iwaniec's estimate for the Riemann zeta function
(z + it) «T9/56+e; see Chapter 21. The implied constants in these upper bounds depend on e.
7.6 LONG MAJOR ARCS In this chapter the major arcs have not had special consideration. We choose the parameters so that Poisson summation gives a trivial sum on the major arcs, which is negligible compared to the minor arc estimate. Whenever an upper bound contains two terms, one larger than the other, then we should try to modify the argument to reduce the larger term at the cost of increasing
the smaller one. The contribution of the major arcs can be increased by making the major arcs longer, dropping the conditions (7.2.4) and (7.2.5) which restrict the size of N. But there is no saving on the minor arcs unless we take the major arcs so long that there are no minor arcs. When M2 - M M, this is a very bad construction. When M2 - M is small, then we may be
able to treat the whole sum as a single major arc; this idea is due to Karatsuba (1981).
Long major arcs
161
Lemma 7.6.1 (Long major arcs) 1
q< -C
Suppose that
M 1,3, o. T
IN21+IN31
M s-,
(7.6.1)
Co
for some sufficiently large constant Co. Let bx
ax2
q
q
h(x) = f(m +x) -f(m) - - - -, in the notation of (7.2.11). Then for f(3)(X) positive, we have N2
E e(f(m + n)) _
G(a, b + r; q)
N3
e(f(m)+h(gzy.)-rzr- $) I q2h" (qz,)1
r
G(a, b + r; q)
+
e(f(m)+h(gyr)-ryr+ (q2h"/
qr))
r
(log(N2-N3)
+0
µ1/3 /---
where the first sum, which may be empty, is over positive integers r for which there is a point Zr with
qh' (qzr) = r, N3/q < zr < 0, and the second sum, which may be empty, is over positive integers r for which there is a point yr with
gh'(gyr)=r, 0
Proof We follow the proof of Lemma 7.3.1 without first approximating by a polynomial as in Lemma 7.2.1. Here h(x) has a Taylor expansion
h(x) =
+ Axe + µx3 + O(IxI4 g
and similarly for the higher derivatives of h(x). For IxI Z 1 we have
h'(x)=
+3µx2(1+0(IM 9
h" (x) = 6µx1 + 0(IM
+ IxI
+ IxI )),
h(3)(x)=6µ(1+O(IM))
)),
(7.6.2)
The simple exponential sum
162
We require Co to be so large that h" (x) has only one zero, and that for r >- 1, the two solutions of
h'(qy) = r/q have opposite signs, with
r-c
Yr=
(1
r
(3µq3) +O q+Oµg3M
ll
(7.6.3)
and similarly for Zr, with a minus sign before the square root. We note that (7.6.1) implies µq3 << 1,
(7.6.4)
so that Yr >> W.
The range for x has the form N3 <x < N2, where N3 5 0. We put
N' = max(N2, -N3). The analogue of (7.3.1) is N2
E e(h(n)) _ N3
G(a, b + r; q) r
NZ/g
JN3/q
e(h(cy) - ry) dy + O(/ log N'), (7.6.5)
where for µ > 0, r runs through all the values with
0< r< max(qh'(N3), gh'(N2)) + so that r <<
qN'2
By the Third Derivative Test (Lemma 5.1.4) and the bounds for Gauss sums (Lemma 5.4.5) we have
rNz/ge(h(.V)-ry)dy<< G(a,b+r;q) JN3/q i 3 1/2 A/q
(7.6.6)
This is the only available estimate when r = 0. For positive r we divide the range of integration into five parts aj
fl'e(h(gy) - ry) dy,
J a;
with
a, = N3/q,
Q, = 2zr,
a2 = max(N3/q, 2zr), a4 =Yr/2,
a2 = zr/2, I33 = min(N2/q,Yr/2), 04 = min(N2/q,2Yr),
a5 = 2Yr,
a5 = N2/q.
a3 = max(N3/q, Zr/2),
Long major arcs
163
By (7.6.2), on 11 and IS we have qh'(qy) - r + K>> qh'(qy) >> µg 3Y2,
and on 13 we have
Igh'(9y)-r+KI>> r. We use the method of the explicit First Derivative Test (Lemma 5.5.1) on 11, 13, and I5 to get
I e(h(gy) - ry) dy = a
e(h(gy) - ry) - r)
[ 21Ti(gh'(,y) 1
e(h(qy) - ry)g2h" (qy)
p
27ri Ia
(gh'(gy) - r)2
dY
Hence
e(h(qy) - ry) 11 - [ 2ari(gh'(gy) - r)
R' I
<<
µq3 I yl
Ip1
a1 (µg3y2)2 dy
a '
« 1/µg3(31 « 1/µg3z: << 1/r,
and similarly for I5, whilst
I3 -
r
ry)
r
(qy) - r) J a3
L 21ri(gh
p3 µq31 yl
Ia3
r2
µg 3
dy 1
max(a3 , 33) <<
r.
These integrals give an error term IG(a, b + r; g)I
«
r
r
« r log N',
which can be absorbed by the error term in (7.6.5), by virtue of (7.6.4). The integrals 12 and 14 are stationary phase integrals as in Lemma 5.5.2, with M and T of Lemma 5.5.2 replaced by
- K' M' ^Yr X
T'
x µg3yr
r (g3
1
(r- K)3
^
µq3
The simple exponential sum
164
Lemma 5.5.2 gives
e(h(gz,) - rzr - g I2
e(h(gy) - ry)
- [ 21Ti(gh'QV) - r)
Igzh (qz,) 1
M'
Mi4
M'4
1
T73/2 + T,z
a:
I
zr
1
T,z
I3
(Zr -N3/q
)3 .
(7.6.7)
The Second Derivative Test (Lemma 5.1.3) gives I2
e(h(gzr)-rzr- 8)
M'
«
Ig2h" (qz,)I
1
7"
X
N/g3ZrI
(7.6.8)
I
We use (7.6.8) when N3/q > 2Zr and
M'
N3 Zr
< < 17T
1
1
X
(7.6.9)
.v . I
I µgN3I
3g µg3zrl
We have
r- K
-h(N3)=(qzr-N3)h"(6)^µgN3zr-
q
N
q
for some 6 between N3 and qzr. Hence (7.6.9) implies
r - K - qh'(N3) << 1 µg2N3I .
(7.6.10)
A IN3I3 << 1,
(7.6.11)
If
then (7.6.10) holds for all values of r which give a point Zr of stationary phase. Using the bound from Lemma 5.4.5, we see that the non-zero values of
r for which we must use (7.6.6) or (7.6.8) in place of (7.6.7) contribute to (7.6.5) <<
(1 +
q
I µg2N31)min
µgzN31 <<
1
,
µl/3
<< 1/µl/3 %,
YY + 1/µh/3 V
(7.6.12)
by (7.6.4).
When (7.6.9) does not hold, then the error terms in (7.6.7) are estimated as 1
1
(µ3g9Zr) W4 <<
( µq3) r2
+
( µq3) rs/2 +
IN313/q3
1
IZr13
+ (N3 /q2 - Zr )3
µ2g31N3I3
r(3µ^2 - r)3
Long major arcs
165
These terms sum over r to <<
(µq3) +1+(log N')/jIN313<< log N'
by (7.6.4), since we are not considering values of r for which (7.6.11) holds. Putting in the estimate G(a, b + r; q) << j from Lemma 5.4.5, we see that these error terms contribute to (7.6.5)
« / log N.
(7.6.13)
We estimate similarly in the integral 14. The integrated terms give the difference of
e(h((y) - ry) 27ri(gh'(qy) -r) evaluated at the endpoints a1 and as; if we have used the Second Derivative Test on I2, then a1 is replaced by a3, and similarly 135 may be replaced by R3. Their contribution to the sum (7.6.5) is another term of size 0(rq log N'). 11
Lemma 7.6.1 gives a new exponential sum involving, apart from constant terms,
h(gyr)-r1',=µg3yr -r1',+... = -2
(r- K)3 27µq3
+...
where the terms omitted are smaller, and a term like a(b + r)2/q, where we can suppose a and ab to be reduced modulo q independently. The condition (7.6.1) with Co sufficiently large implies
(r-K)3 27µq 3
We can divide the sum into blocks H S r < 2H, on which h(gyy) - ryr - a(b +
r)2/q behaves as a monomial function with -2 (r3/27µg3) as its monomial part. We have still not evaluated the major arc sum. When K = 0, then the term with r = 0 is approximately
e(f(m))G(a, b; q) µ1/3q
f
e(ix3) dx.
When K is non-zero, then the integral is smaller, by the First Derivative Test.
166
The simple exponential sum
In Sections 7.2, 7.3, and 7.4 of this chapter we used gardening wherever possible to simplify the exponential sum. The more untidy Lemma 7.6.1 has four possible uses: 1. To treat a short exponential sum as a single major arc; 2. To look for cancellation between major arc sums corresponding to different rational numbers with the same denominator; 3. In a double sum, to look for cancellation on the second summation; 4. To set up an iterative argument. Using Lemma 7.6.1 for an iteration would involve treating all Farey arcs, major and minor, the same way. The large sieve provides a better bound in mean square for the minor arc sums. In the next chapter we shall consider a very simple double sum, for which we are able to carry out gardening as in this chapter.
8
The exponential sum for the lattice point problem 8.1 MAJOR AND MINOR ARCS This chapter is parallel to Chapter 7. We consider the more complicated sum, introduced by Iwaniec and Mozzochi (1988), Hz
M2
hT
S=h=H E m=M E e(M
m F(M)).
(8.1.1)
This sum arises when we estimate the area under the curve y = NF(x/M) by counting squares, and analyse the error using the row-of-teeth function. The
details are given in Chapter 18. We write T/M for N to emphasize the analogy with Chapter 7. The analogy becomes stronger when we make F(x) in this chapter correspond to F'(x) in Chapter 7, and allow for the effect of binomial coefficients multiplying the rational approximations to derivatives.
The reason becomes apparent when we replace F(m) in Chapter 7 by F((m + h)/M) - F(m/M). A corresponding double sum is considered in Chapter 9.
The Farey arcs in this chapter are the same Farey arcs as in the Voronoi-Sierpifiski polygon in Chapter 2. We take fewer arcs here, because the curvature is taken into account. If the parameter M represents a length, then T is an area, and the natural order of magnitude for T is about M2. In (8.1.1) the parameters H, M, H2, and M2 are integers with H2 < 2H - 1, and M2 5 2M - 1. The function F(x) is real with at least three continuous derivatives on the interval [1 - S, 2 + 51 for some S > 0. We require the first three derivatives of F(x) to remain non-zero, so that when we write
f(x)= MF(M),
(8.1.2)
then there are constants C11 C2, and C3 for which (8.1.3) T/CrMr+I 5 If (')(x)I 5 CrT/Mr+l for (1- S )M 5 x:5 (2 + 6)M and r =1, 2, 3. As in Chapter 7, we can begin with a function F(x) defined only for 15 x :!g 2, and extend its definition outside that interval using the Taylor polynomials at the endpoints. The sum S is unchanged if we add to f(x) a polynomial g(x) that takes integer values at integers. If T has order of magnitude at least M2, then we may add linear
The exponential sum for the lattice point problem
168
polynomials, and change f'(x). For M2 << T we can extend the final result trivially to functions whose first derivative may vanish on the interval. We choose the integer N, the length of the Farey arcs, to satisfy
64C2H
(8.1.4)
HN3T << M4. (8.1.5) The upper bounds for N in (8.1.4) and (8.1.5) are not critical conditions which
affect the optimal choice of N. The second parameter R is the unique positive integer with R2NT _ C2M3 > (R -1)2 NT,
(8.1.6)
and R must fall in the range 2C2vrH- + 1
(8.1.7)
The condition (8.1.7) implies various order-of-magnitude relations between H, M, and T. In particular 64H3T >- M3 > 4C2HNT> 64C2H2T.
(8.1.8)
From (8.1.8) we see that H is much smaller than M, and the range in which we must choose N is restricted. Because h is small, we obtain the rational approximation to hf(m +x) by multiplying an approximation to f(m +x) by the integer h. For (1 - SW < x S (2 + SW we have HN3f(4)(x) << 1
by (8.1.5), and 1 << HN2f(3)(x) << N
by (8.1.4), (8.1.6), and (8.1.7).
As in Chapter 7, divide the range for x into intervals U: M + (j - 1)N < x < M + jN, and let Ji be the interval of values taken by f'(x) for x on U. By
(8.1.3) the length of Jj lies between NT/C2M3 and C2NT/M3, and thus between 1/R2 and 2C2/R2. Since f"(x) has constant sign, different intervals J, are disjoint. We pick a rational number a/q = a,/q, in the interval Jj. The construction takes place on a fixed minor arc Jj. We use the suffix j only when different minor arcs are being compared. The basic choice of a/q is the rational number of smallest denominator (and so smallest height) in the
interval Jj, in some cases we modify the choice of a/q as in Chapter 7. Intervals Jj for which the rational point chosen has q < R2/H are major arcs.
If R2/H _ R. As in Chapter 7 we find that Modification 1
R :!g R2/q < r < 3H.
This modification ensures that the factor R/q, which occurs in some error terms on the minor arcs, is less than unity.
Major and minor arcs
169
Modification 2 If q > 16R, then there are many rational numbers in Jj with
denominators near q in order of magnitude. Let Q be the next power of 2 following r. As explained in Chapter 6, Lemma 1.6.3 gives us at least Q2/16R2 rationals, each with denominator q :!g 32r< 96H. We average over the approximations corresponding to these rationals, indexing the minor arc
approximations by the rational a/q rather than the index j. We use the notation J(a/q) for a minor arc with the choice a/q of rational approximation to f'(x). Modification 3
Suppose that all rational points in JJ . have q >- 3H. Let x be
the midpoint of J,. By Dirichiet's approximation theoFem (Lemma 1.5.1) there are integers b and r with
Irx-bI<1/3H,
r<3H-1.
Then
1/2R2 5 Ix - b/rI 51/3Hr. Hence r ::r. 2R2/3H, and the Farey arc J(b/r) is a major arc. We assign the
interval J, to the Farey arc J(b/r). The major arcs are thus unions of consecutive intervals J,, with total length
5 2/3Hr + 1/R2 5 5/3Hr. We write U(a/q) for the union of the arcs U. for which JJ . lies in J(a/q), and
I(a/q) for the intersection of U(a/q) with the interval [M, M2]. We take the centre of the Farey arc U(a/q) to be the integer m = m, within U(a/q) for which f'(m) is closest to a/q, so that
f'(m)=a/q+A,
(8.1.9)
IAI S C3T/2M3 X 1/NR2 << 1/N2.
(8.1.10)
with
For this integer m we put
µ = µj =f' (m)/2. It is convenient to write
A= -2µv. We note that
µx 1/NR2,
IvI51+C2C3/M<2,
since we can suppose that M is sufficiently large. Lemma 8.1.1 (Counting Farey arcs) For positive Q, let W(Q) be the number of Farey arcs J, on which q, lies in the range Q s q < 2Q (for some choice of qj when Modification 2 operates). For major arcs,
N(R - 1)2
R2
4Ci Cz M S Q 5 H'
(8.1.11)
The exponential sum for the lattice point problem
170
and we have
W(Q) << (M2 - M)Q2/NR2 + 1.
(8.1.12)
R/2< Q 5 96H,
(8.1.13)
W(Q) << (M2 - M)R2/NQ2 + min(R2/Q, M/NQ).
(8.1.14)
For minor arcs, and we have The implied constants are constructed from C1 and C2.
Proof This lemma is the analogue of Lemma 7.2.2. The term + 1 in (8.1.12) also covers the case when there is a major arc whose centre is outside the
range MSm5M2.
El
8.2 THE MAJOR ARCS ESTIMATE We treat the major arcs by Poisson summation and direct estimation. We consider the sum H2
e(hf(m + n)) h=H
n
m+ne!
where I = I(a/q) is a major arc, centre m. Writing
f(m +x) = f(m) +ax/q+g(x), we have
g'(x) = f'(m +x) - a/q = f'(m +x) - f'(m) - 2µv =4'"(m + Ox) - 2µv for some 0 in 0 < 9 < 1. Hence for m +x in I 1 +Ixl
1
g'(x) « NR2 «NR2
NR2
Hq
1
« Hq'
g" (x) = f" (m +x) << 1/NR2. These orders of magnitude still hold if we allow n to range wider so that m + n can go up to 2N beyond the endpoints of I(a/q) on either side. This allows us to use the same technical device as in the last chapter: on a minor arc Ik we actually use the Taylor expansion corresponding to Ik t 21 so that n is bounded away from zero. This saves a subdivision into cases according to
the order of magnitude of n. The integer n runs through an interval I consisting of a major arc with one or both endpoints shifted by N or 2N. We can write the sum as H2 h
N2
n=Ni
+g(n)
+ahn q
The major arcs estimate
171
where N1 and N2 are the limits of summation for which m + n is in I for N,
E N2
n=NI
e(ahn l
e(hg(n))
q J Nz
1
e hg(x) - LX J dx + O(log(B -A + 2)),
q(A-1)S15q(B+1) IN1 1
q
- ah (mod q)
where
A = hg'(N2)
B = hg'(N1), with
A,B<< h/Hq<< 1/q. The sum is thus
N2e(hg(x)-q/4+O(1)<1
q q
dx+O(1).
1= - ah (mod q)
Since hg'(x) << 1/q, we can use the First Derivative Test (Lemma 5.1.2) to give N,
)dx4z
fN
q , except for bounded values of 1, where the Second Derivative Test (Lemma 9
5.1.3) gives the bound a
<< sup
1
<<
V (NH )
As h runs from H to H2, then the integer 1 takes each value mod q at most (O(H/q) times, and so HZ
N2
hE
n=NI
F, e(hf(m + n))
«H glogq+ VH! q
«H -q (NRH ) log H. 2
Lemma 8.2.1 (The major arcs contribution)
The major arcs contribute
( g H, « (MZ-M)R log H + min R (HN) log (HN)
to the sum (8.1.1).
`I
NI log H)
The exponential sum for the lattice point problem
172
Proof We divide the major arcs into ranges Q < q < 2Q, where Q is a power of 2. In the notation of Lemma 8.1.1, the major arcs contribute
Q
log H,
W (Q) Q
where Q runs through powers of 2 in the range (8.1.11). Summing geometrical progressions gives the lemma. o As in the last chapter, we can treat all arcs as major arcs. If R >- H/4, then minor arcs have q -< 3H<- 3H(4R/H)2 < 48R 21H. Then (8.2.1) holds with
N2 -N1
H/4<-R
S«
(Mz -M)R log H+ min (HN)
M
(1 M2 l H3T« M2-M M + min
H , HT
J,
H
(_j)logH.
The implied constant is constructed from C1 and C2. 8.3
POISSON SUMMATION ON THE MINOR ARCS
After Lemma 8.2.2 we can assume that R < H/4. If a/q and b/r are two consecutive major arcs, then the rationals (at + bu)/(qt + ru) with (t, u) equal to (1,1), (1, 2), (1.3), (2,1), and (3,1) have
Rz/H < qt + ru < 4R2 /H < R, so they correspond to distinct minor arcs (on which Modification 1 applies). Thus any two major arcs are separated by at least five minor arcs. If Jj is a
minor arc, then J,+z and ii-2 cannot both be major arcs or the flanks of major arcs. If, for example, Ji _ 2 is a minor arc, corresponding to the rational number a/q and centre m, then we use the approximating polynomial about m on the arc I,. The variable n runs through a range N1 5 n 5 N2 with
N2 -N1
N1zN,
N2<3N.
Poisson summation on the minor arcs
173
In the notation above
Lemma 8.3.1 (Polynomial approximation)
H2 N2
H2
N2
F,
F, e(hf(m + n)) << max F, E e(hg(n))
h=H n=N1
,
H3 N3
where the maximum is over H3 >- H and N3 >- N1, and ax
g(x) =f(M) +
+ µx2.
9
Proof We write
k(x) =f(m +x) -g(x). Then, for some 0, cp in 0 < 0, cp < 1,
N
x3
k(x) _ -2µvx + 6 f 3>(m + 9x)
N3
1
<< NRZ + MNR2 << H'
and
k'(x) _ -2µv+
x2
2
f(3)(m + cpx) <<
1
NRZ
N
+
1
<<
MRZ
HN'
by (8.1.5) and (8.1.7). By partial summation in two dimensions (Lemma 5.2.2) we have H2 N3
H3 N2
F, E e(hf(m +n)) SCmax L > e(hg(n)) H N,
H3 N3
where
H)I
HN
HN
C«I1+HII+HN)+HN«1. II
13
llllllway
Taking moduli in this than
means that we cannot expect any estimate better
M
N
HM (HN) <<
(HN)
from the minor arcs. Lemma 8.3.2 (Poisson summation over n) In the notation above H2 N2
E F, e(hg(n)) _ H3 N3
2
F,
F
Vi-,i-
lmodq h=H3 1=- -a/h,D351/hsD2
+OI R (HN)logH), where
D; = 2µgN;, and the square root is taken with positive real part.
el hf(m) `
2
4hµq 2 )
The exponential sum for the lattice point problem
174
Proof For fixed h we write
/3=hµ. By (8.1.3) and (8.1.8) we have
/3NSHµN5C2HNT/M35'and by (8.1.7) we have H << R2. By the truncated Poisson summation formula (Lemma 5.4.3), we have l N E e(agn + pn2) - !a)E a+' JN32e(- q + /3x2) dx q(A
N3
q(
!= -ah(modq)
a)
+ O(log(B -A + 2)), where B = 2 /3N2 = hD2/q.
A = 2 /3N3 = hD3/q,
We note that the sum over I has at most one term, since /3N < 4, and that the error term is O(log(2 + /3 N)), which is bounded. We write xh1
1
1
2/3q
2hµq
Then
/3x2-lx/q=/3(x-xhr)2-/3x21 There are now two cases, depending on the sign of µ. We treat the case when p. (and /3 also) is positive. If
N3+1/,5x,i,SN2-11F then we argue as in the proof of Lemma 5.5.2, writing f Nee( /3(x -x21)2 N3 2 e(-/31hl
- Gx21) dx 1
a
(JN+
f ')e(/3(x-xh1)2-dx
(1 N2
_
V-i-
e(-/3xh1)+0I
Nx21
+ xh1
(8.3.1)
- Ns
by the First Derivative Test. When µ is negative, then we still have (21131 In the ranges
N35xh1
N2-1//5xh1SN2,
we have
fNZe(/3(x -x21)2 - ax21) dx N3
2/3i
e(-Rx21) «
1
(8.3.2)
Poisson summation on the minor arcs
175
by the Second Derivative Test. In the ranges
N2 sxhl
N3 - 1/ xhl
j
e( /3(x -xhl)2
N3
- /3x21) dx «
(8.3.3)
by the Second Derivative Test, and in the ranges xhr
xh,>N2+1/vP
we have 1
13x',) dx <<
fN32e(a(x
1
1
(8.3.4)
a INZ -xh1l + lxhr -Nil
by the First Derivative Test.
In the error terms we note that for 1 fixed, h lies in some fixed residue class mod q. If g = h(mod q), then
I(g - h) Ixgi -xh11
1
>> H2µ
Nq
xH.
(8..3.5) 3.5)
Since H << N and R << q, we find that Ixgl - xhIl >> R
I
N
-
When we sum h over a residue class modulo q, then we use (8.3.2) and (8.3.3) a bounded number of times, and (8.3.1) and (8.3.4) at a sequence of values xhl separated by a distance (8.3.5). The error terms sum to <<
E t
1+ VN
1
t«H/q t
I
g=h
-
Ixg, -xh,I
«E log H << gNjlogH<< Rq I
i
(HN)logH.
Lemma 8.3.3 (Poisson summation over h) In the notation above HZ
N2
E E e(hg(n)) h=H3 n=N3 L2
=E E 1=L,
x2(r)
1
(4µq(k - K))
+Ol R (HN)logN),
e
k-K Nq
iikl - abl q - q
q
q
The exponential sum for the lat tice point p roblem
176
where
L2 = 2µgH2N2,
L1= 21kgH3 N3 z 2,
(8.3.6)
and 12
K1(1) = µ q max N3 , L2 NN
(8.3.7)
64,
2
P K2(1) = µq min N22,
LZ
(8.3.8)
N3
and the integer b and the real number K satisfy q
1
3
IKISmin(2 +O(H4).
qf(m)=b+K,
Proof In Lemma 8.3.2 we write H4 =H4(l) = max(H3,l/D2),
H5 =H5(I) = min(H2,1/D3). For each 1 the sum over h takes the form (
H45h5H5
h- -al We can take out the factor
I`
i
2h
1
Je
(h(b + K)
-
12
4hµg2
q
which is independent of h. We apply the inversion step (Lemma 5.5.3) with h = qn-al,
f(n =KnalKq--- 4Iaq 2 (qn-al) 12
M replaced by H/q, and T replaced by I2/4µg2H. When µ is positive, then f" is negative, and H5
1
E
(2hµ)
e(
qKh -
h - -a!
e(f(xk)-kxk-$) (2µ(gxk-al)) f"(xk)l 1
A
s
(H./(.12H\ +O
1
H)
q
2J + lo(B -A + 2)
(8.3.10)
Poisson summation on the minor arcs
177
where the endpoints A and B are given by + al
A=f'
12
= K+ max IzqHs2
R 12
D3
12
1
=K+ 4
4AqHzz'4M R )
ll
=K+max Lz µRNz,AgN3K1(l)+K, z
in the notation of this lemma and the preceding lemma. Similarly B = K2(1) + K.
Changing the limits of summation in the inversion step byt a bounded distance
K does not increase the order of magnitude of the error term. We take the sum over k between K,(1) and K2(1). The stationary phase integral in the Poisson summation formula is part of
the contour integral for a Bessel function, Hi_J/2(x) in the notation of Jeffreys and Jeffreys (1962, article 21.02). In the main term Xk is given by 12
K+
4µq ( qxk - al )
2 =k,
(qxk - al)2 =12/4lcq(k - K). Thus alk
12
f(xk )-kx k= (k - K)x -q -- 4µg2(gxk-al) k al
=-(k-K) xk --q =-1 V(LjLq
akl
12
q
4µq2
(4µq(k - K)) 1
akl
K
q
3
The weight function is 1
2 µ( qxk - al )3
211(gxk-al)
12
1
(2µ(gxk-al)If"(xk)I)
1
12
12
4µq(k - K)
1
(4µq(k - K))
The factor Vi is cancelled by e(- '-g) when µ is positive, and the corresponding factor is cancelled by e(g) when µ is negative.
The exponential sum for the lattice point problem
178
In the error term we have
H 9
« Hl
µgzH 1
l2
V
µH) «
qR z
H)
RH <<
1VR2
Nq
<< 1.
The assumptions (8.1.3), (8.1.4), and (8.1.6) give
KIM >
qH qN2
> RZ - 6R 2
C2M3
2qHN
L1>2µgHN>
NR2
> 64,
>2.
The error term in (8.3.10) is
«
log N
V(HM)
We conclude the proof by replacing the factor (8.3.9) and summing over 1. The range of summation for k is non-empty only when Ll < l 5 L2. The error term sums over 1 to <<
pgHNlogN q << R (HN) log N, (Hµ)
0
which absorbs the error term of Lemma 8.3.2.
8.4 THE LARGE SIEVE ON THE MINOR ARCS We work backwards from the large sieve, in the hope of reducing the number of confusing multiple summations. Lemma 8.4.1 (The large sieve) Let a(k) and b(j) be coefficients of modulus at most unity, indexed by the integer vectors k = (k1, k2,11,12) with entries in
some range K5 k. < 2K, L 5 l; < 2L, and by minor arcs Jj = J(a/q) with a/q = aj/qj, with denominators in some range Q:5 q < 2Q. Let x(k) _
(ilk,- l2k2,11
Y
ci)
a
kl - l2
kz
, 11 -12, l1 / kl - l2/ k2 ),
1
= -q,- (µq3) ,-
ab
K
q ,2 (µq3)
where b, K, µ come from the coefficients of the approximating polynomial on
J(a/q) as in Lemmas 8.3.1 and 8.3.3, and the values of K, L, and Q are consistent with the construction above. Let V >_ 1 be a real number. Then 2
a(k)b(j)e(xck> . y(j)) << ABKL4R2NV/Q3, k
j
The large sieve on the minor arcs
179
where A and B are the number of coincident pairs among the vectors x(k) and y(>>, respectively. Two vectors x(k) form a coincident pair if their corresponding entries differ by numerically less that 2
"iI)i
1, R
Two vectors y(>) form a coincident pair if their corresponding entries differ by numerically less than 1
1
1
FK
8KLV' 4L (2K) ' 4L ' 8L The result is still true for a family of sums Si of the form (8.1.1), taken over subintervals (depending on i) of H< h < 2H, and of M < m < 2M, formed with functions fi(x) satisfying (8.1.3) uniformly in i, when we replace b(j) by Ni, j), y(J) by y('°>), and the sum over j by a sum over i and j. The sequence of rational numbers aj/qj corresponding to the minor arcs may also depend on i.
proof This is Lemma 5.6.6 with
X= (8KLV, 4L (2K) , 8L,16L/V), and 1
Y= 1,2max
(µq3)
1
,1,max
.
(l-+-q3)
we can take Y = Y3 = 1 since the rational numbers a/q and ab/q are only defined modulo one. By (8.1.3) and (8.1.4) we have
2) 3) <
3T ( Q3T
Q
3
We can suppose that K, L, and Q have the orders of magnitude of the sums in Lemma 8.3.3, so that R<< Q<< H,
Kx RQ,
Lx
2
After verifying that
L V11
r NR2) I` Q3
»
R4 NQ Q3 >> Q » H (H2Q2 R2 NR2
we deduce that X;Y + 1 << X;Y
for each i =1, ... , 4.
(8.4.1)
The exponential sum for the lattice point problem
180
The numbers k,, 1, in Lemma 8.4.1 lie in fixed intervals; we actually require the range for l to vary with the minor arc Jj, and that for k to vary both with j and with 1. Moreover, the exponent x(k) y(j) is not quite the same as in Lemma 8.3.3. We tackle these difficulties one by one.
As in the last chapter we use Lemma 5.2.3 to relate the sum over an arbitrary subinterval to sums over the whole interval. There is an alternative treatment using Rademacher's idea of decomposing the interval `in the 2-adic integers', that is, in terms of the binary digits of the endpoints. The combinatorics are intricate. This idea explains the logarithm factor that appears when we use Lemmas 5.2.3 or 5.2.4; it is the number of binary digits in the numbers involved. Rademacher's idea was combined with the large sieve by Uchiyama (1972).
Lemma 8.4.2 (Large sieve over subintervals) Let d(k, 1) be any coefficients of modulus at most unity, and let
g(j, k, l) = d(k, l)e(kly( j) + ly(j) + ly(j)F + ly(j)/F 2 3 4 1
Let L3(j) and L4(j) be sequences indexed by the minor arcs Jj with Q < qj < 2Q, satisfying
L
i
2K-1 L4(j)
F, j
K
2
« ABKL4NR2V Q3 log4 N.
E g(j,k,l) L3(j)
Proof We apply Lemma 5.2.3 to the sum over 1, so that L4(j)
2K-1
2L-1 2K-1
1/2
E g(1, k, l) < f- 1/2 K(x)
L
1=L3(j) k=K
E L g(j,k,l)e(lx) L
K
with
K(x) = min(L, 1/2x). By repeated use of Cauchy's inequality we have 2 2
2K-1 L4(j)
F, K
5
E g(J,k,l)
L3(j)
I
f 1/2 K(x) dx) f 1/2 K(x)
1
1/2
1/2
2K-1 2L-1 L g(j,k,l)e(lx) L K
2
2K-1 2L-1 1/2
K(x)dx1 3f 1/2 K(x) 1/2
L g(j,k,l)e(lx) j
K
L
2
dx 2 2
dx.
The large sieve on the minor arcs
181
Multiplying g(j, k, l) by e(lx) corresponds to changing y3 to y3 +x in Lemma 8.4.1. Since all the y vectors are changed in the same way, this does not affect B, the number of coincident pairs. The result follows from Lemma 8.4.1 and the estimate from Lemma 5.2.3: fl/2
-1/2
K(x) dx<-1 + log L << log N.
Lemma 8.4.3 (Partial sums by Mellin transforms) L2(j)
E
K2(j,l)
E g(k,l) «WL log K+
l=L1(j) k=K1(j,l)
(
In the notation above 1/4
ABLNR2VW2
loge N,
Q3
(
V
)
where the sum is over approximating polynomials used on minor arcs with qj in the range Q:5 qj < 2Q, and W is the number of such polynomials: W= W(Q) when Modification 2 is not used, W << Q2W(Q)/R2 when Modification 2 is
used. The limits of summation for k and 1 are those of Lemma 8.3.3; the argument j has been added to indicate the dependence on the approximating polynomial. Here K and L are powers of 2 whose order of magnitude is given by (8.4.1).
Proof The formula (8.3.7) for K1(j,1) changes at
1 =L3(j) =L2(j)N3(j)/N2(j) = 2µjgjH2(j)N3(j), and the formula (8.3.8) for K201 1) changes at
I =L4(j) = 2µjgjH3(j)N2(j). Both L3(j) and L4(j) lie between L1(j) and L2(j), separating the range of summation for 1 into three subintervals of the form L5(j) < 1 < L6(j). On each subinterval, the range for k is K5(j,1) to K60, 1), where K5(j,1) is one of the two terms in (8.3.7), and K6(j,1) is one of the two terms in (8.3.8). Since L1(j), L2(j), K1(j,1), and K2(j,1) have order of magnitude given by (8.4.1), we can cover the ranges for k and I by a bounded number of intervals of the form K s k < 2K and L < 1 < 2L, where K and L are powers of 2. A typical section of the sum has I running from L7(j) to L8(j), with
L7(j) = max(L, L5(j)),
L8(j) = min(2L - 1, L6(j)),
and K running from K7(j,1) to K8(j,1) with limits given by K7(j,1) = max(K, K501 1)),
K8(j,1) = min(2K- 1, K6(j, W. We write L8(j)
Gj = E
K,Q,1)
F,
1=L7(j) k=K7(j,l)
g(k, l).
The exponential sum for the lattice point problem
182
In Lemma 5.2.4 we have L8(i)
g(k, 1) 1 K6(j,l)s-K5(j,l)s Gi=E 2K-1 E ks )ds+o(LlogK). 12ari f l-L7(i) s k=K 1
Il
C
Let c(j) be coefficients of unit modulus for which c(j)G1 is real and positive. Then either
IGiIc(j)G1<< WL log K,
(8.4.2)
1
1
or, by Cauchy's inequality, 2
E c(j)G1 i
<
(f Ids C
L
fc
S
i
2K-1 L8(j) g(k, 1) s c(j) E F, ks K6(j,l) K
L7(i)
2
dsl S
(8.4.3)
or the corresponding inequality with K6(j, l) replaced by K5(j, 1). We have fc
Ids/sl << log K << log N.
In the second integral on the right of (8.4.3), the modulus squared within the integrand can be estimated as
2K-1 2L-1 g(k,l)
EE K
K6(j,l)
ks
L
2 s
When µig1N2(j)2
K6(j,l) =
in (8.3.8), then we take a factor (K6(j, l)/K)s of bounded modulus outside the sum over k and 1. When K6
)2 j,l = tLj-gil2N3(j L1(, )z =
12
4µ1g1H3(j)z
then we take out a factor (L2/µig1H3(j)2K)s of bounded modulus. The sum is now in the right form for Lemma 8.4.2 with d(k, 1) replaced by
d(k, l)(K/k)s, or d(k,
l)(Kl2/4kL2)s.
Thus 2
c(j)GG
<<
112
ABKL4NR2V 1
Q3
WlogeN.
(8.4.4)
I
Since either (8.4.2) or (8.4.4) must hold, we have the result.
o
The large sieve on the minor arcs
183
Now we have the ranges of summation correct, but not the exponent. There is no point in the argument at which we can use Lemma 5.2.2 without destroying the special form of K(j,1) which is used in the last lemma. The method of last resort is to expand in power series. Lemma 8.4.4 (Large sieve with gardening) and 8.4.3, H2
N2
F,
E e(hg1(n))
QW <<
In the notation of Lemmas 8.3.3
(HN) log N
h=H3 n=N3
ABKL4NR2V`W211/4
R2
+ Q(
loge N.
Q3
The corresponding result for a family of sums Si is H2
E
N2(j)
E e(hgij(n))
h=H3 n=N3(j)
HNQ2VW R2
<<
r R2 ABKL4NR1°VW212
loge N + Q
(
QZ
logo N,
where all the minor arcs with Q< q < 2Q for all the sums S; are counted in B and W.
Proof The exponential in Lemma 8.3.3 has the form e(kly1 + ly3 + lye (k - K) ).
Since k Z 32, we can expand (1 - K/k) 1/2 by the binomial series. The first two terms in the expansion give the required 1y2V +ly4/VK,
which we retain inside the exponential. There
is a
1/ (1 --K/k) in the weight function. We expand ( 11
K
1/2
k)
1 K2
1k1/2
e
(µq3) (8
_...)
k2
as a double power series
P(x,y) =
F, r r
brsxrys
S
in the two variables
x = K/k,
Y = K2l/ (µg3k3) .
similar factor
The exponential sum for the lattice point problem
184
We note that in Lemma 8.3.3 we have P
12
k_- min
16µ9H2,
(
qN2),
1
so that IK12
<_
IYI
,
I ( 16µgH2k) 5 4H
V
4H
4HR2
1
k9 5µ92N2 5 NQ2 S 16
pg3k3
The double series converges geometrically, and xrys
qKr+2s
(4µgk)
is
2( µg3)(s+1)/2 kr+(3s+1)/2
which consists of powers of k and I times a factor depending on the minor arc.
The size of the leading term is 1
2V(µ9k)
NR` KQ
R` Q
After taking out an order-of-magnitude factor, we use Lemma 8.4.3 on each
term in the power series, and add up to get the second term in the upper bound of the lemma. The first term in Lemma 8.4.3 contributes QW R2LW (HN) log N, Q log K << HW log N << R
the last expression being also the contribution of the error terms in Lemmas 8.4.1 and 8.4.2, which come from edge effects in the two-dimensional regions 11 of summation.
9 Exponential sums with a difference 9.1 MAJOR AND MINOR ARCS In this chapter we consider the exponential sum formed with a central difference, introduced by Heath-Brown and Huxle (190),
HMZ
hH
e(TFI
m=M1
111
1
mMmM h )-TF( h))
(9.1.1)
where H, H2, M, M1, and M2 are integers with H2 < 2H, M1 z M + 2H - 1,
M252M-2H, so that
M<m±h <2M. An analogue of the method of Chapter 7 for a general two-dimensional sum does not seem possible, because of the scaling laws. In one dimension, an interval can be divided into N parts, with linear dimensions smaller by a factor 1/N. In two dimensions, the linear scaling factor is only 11N. This increases the accuracy required from the rational approximations to deriva-
tives of the exponent. But a function of two variables has r + 1 partial derivatives of each order r, and the more derivatives to approximate by rationals with a common denominator, the worse the accuracy. We can approach the sum (9.1.1) because it is a double sum involving a function of one variable, and subdivision is only necessary in the m-direction. The exponent is hf(m, h), where f(x,Y)=Y(F(xMY)-Ft(xMY//=
MFj
MY for some 0 between - 1 and 1. The double sum in the last chapter is a linearized version of (9.1.1), with T there corresponding to 2T here, F there
corresponding to F here. Again, the natural order of magnitude for T is about M2. Exponential sums like (9.1.1) appear in the differencing step (Lemma 5.6.2) and in the treatment of short interval means (Lemma 5.6.4). The mean-to-max
argument of Lemma 5.6.3 provides another route to estimating simple exponential sums.
We suppose that F(x) satisfies inequalities of the form for 1
IF(')(x)I >-1/C, IF(')(x)I 5 Cr, (9.1.2) x 5 2, the upper bound holding for r = 2, 3, 4, 5, and the lower bound
Exponential sums with a difference
186
for r = 2, 3 always, and for r = 1 if M>> FT. We write d1 for d/dx and f1 for d1 f, etc., so that (9.1.2) gives Idi-1f(x,Y)I< C Mr
Id,-1f(x,Y)I < 2MT
r
for M+ 2H - 1 <x < 2M - 2H and the corresponding values of r. We suppose that M is sufficiently large, and that integers N and R can be chosen with 15 R :!g H:!' N < M to satisfy 2R2NT >-
C3M3
> 2(R -1)2NT,
(9.1.3)
2C3{H + 1
(9.1.4)
64C3 H < N :!g M/8,
(9.1.5)
HN3T «M4,
(9.1.6)
H3 << N2R.
(9.1.7)
The conditions (9.1.3)-(9.1.6) correspond to the conditions (8.1.4)-(8.1.7) of
the last chapter. These four conditions are used to estimate terms in the Taylor expansions of Section 9.3. The extra condition (9.1.7) is used to estimate additional terms that arise in this chapter, and it can be relaxed a little.
The construction of Farey arcs is the same as in the last chapter, with Ji the interval of values taken by 2T f1(x,0) = MF M for x on the interval U. The length of J, lies between 1/R2 and 2C3/R2. If values of f1(x, 0) outside the range M 5 x < 2M are needed, then we con-
tinue F(x) as the sum of the first six terms of the Taylor series at the appropriate endpoint x = 1 or 2. The centre of a Farey arc U(a/q) is the integer m in U(a/q) for which f1(m, 0) is the closest to a/q. We put
fl(m, 0) = a/q - 2µv,
µ =f11(m, 0)/2, so that
µ x 1/NR2,
IV I < 1 + C3C4/M < 2,
(9.1.8)
when we suppose that M is sufficiently large. Lemma 9.1.1 (Counting Farey arcs) For positive Q, let W(Q) be the number of Farey arcs Ji on which q1 lies in the range Q < q < 2Q (for some choice of qj when Modification 2 operates). For major arcs,
N(R -1)2 4C2C3M
R2
H'
and we have
W(Q) << (M2 - M)Q2/NR2 + 1.
The major arcs estimate
187
For minor arcs,
R/25Q596H,
and we have
W(Q) << (M2 - M)R2/NQ2 + min(R2/Q, M/NQ). The implied constants are constructed from C2 and C3.
9.2 THE MAJOR ARCS ESTIMATE We treat the major arcs as in the last chapter. We consider the sum H
E
E e(hf(m+n,h))I ,
I
h=H
(9.2.1)
n
m+nel
where I = I(a/q) is a major arc centre m. Writing f (m +x, h) = f (m, 0) + ax/q + g(x, h), we have
gl(x, h) = fl(m +x, h) - a/q = fl(m +x, h) + fl(m, 0) - 2µv. Now
T( /m+x+h
f1(m+x,h)= - F'I
M
zP
nl
for some 0 in -1 < 6 < 1. Thus 1+Ix1 +h NR2
gi(x, h) <<
)_F' (
m+x-h)) M
)
1
«NR2
NR2
H9
1
<<
H9,
and
g11(x, h) =f11(m +x, h) << 1/NR2. Let N1 and N2 be the limits of summation for n in (9.2.1), so that N2-N1 << NR2/Hq. The sum in (9.2.1) can be written as H2
N2
F,
hF, n=N1
+g(n,h)
ahn
+ q
We now continue by Poisson summation just as in Section 8.2 of the previous chapter to get the corresponding estimates.
Lemma 9.2.1 (The major arcs) <<
W2 V(HN
(HN )
to the sum (9.1.1).
The major arcs contribute
)RlogH+min R (HN)logH,
M
r I
N) logH
1 13
Exponential sums with a difference
188
Lemma 9.2.2 (The easy estimate)
If there is a choice of N and R with
H/45RSH satisfying (9.1.3)-(9.1.6), then the sum S of (9.1.1) is bounded by
S«
(M2-M)R (HN) (M2-M
M
9.3
g
tog H + min R (HN) to H,
( + min I`
M211
1
) jlog H
N)
log H.
H' HT J 1
POISSON SUMMATION ON THE MINOR ARCS
We continue to follow the previous chapter. In particular, if Jj is a minor arc, then either J,12 or J,-2 (or both) is a minor arc, and we use the approximating polynomial on one of these, with centre m. The minor arc Jj corresponds to x = m + n (or m - n) with n in a range N1 5 n 5 N2 with N2 - N1 < N, N1 z N, N2 5 3N. Lemma 9.3.1 (Polynomial approximation) In the notation above / h3µ
N2
l 3
N3
H2
N2
F,
F, e(hf(m +n,h)) <<max F el -1 E e(hg(n))
H2
h=H n=Ni
H3
where the maximum is over H3 >- H and N3 z N1, and ax g(x) = f(m,0) + + µx2. 4
Proof This lemma is more complicated to prove than its analogue in the last chapter, as the variable h occurs non-trivially. We put
k(x,h) =hf(m +x,h) -hg(x) - µh3/3. In order to estimate k2(x, h), we expand in powers of h first; T
'92hf(m+x, h)M2(F'(
m+x-h
m+x+h M
)-F,(
M
M2F/(mMx)+T F(3)(mMx
+MT
P(4)(m+J 6h)-F(4)(m+M Oh I`
11
J1
Poisson summation on the minor arcs
189
for some 0 in 0 < 0 < 1. Next we expand F and F(3) in powers of x to get 2T
2xT
m
m
a2hf(m+x,h)= M FP(M) + M2 3
+
6M4
(^)() +
T
m
+(x2+h2)M3 T
h2xT OI
M4
I,
for some ,rt in
m+3N+H <,rt< M M m
Thus we have
02hf(m +x, h) = f(m, 0) +xf1(m, 0) + (x2 + h2) f11(m, 0)
(N3T
H2NT H 3 T 1
+ 0+ M4 + M4 11 2
=f(m,0)+xl q -214v) =f(m,0)+
ax
+(x2+h2)µ+O(H),
by (9.1.6), (9.1.4), and (9.1.8). Hence
k2 (x, h) << 1/H. Similar calculations give
kl(x, h) << 1/N,
k12(x,h) «1/HN. We can now apply partial summation (Lemma 5.2.2) as in Lemma 8.3.1.
Lemma 9.3.2 (Poisson summation over n) In the notation above H2 N2
1
e(hg(n)) _ H3 N3
Imodq h=H3
(2hµ)
1= -ah,D351/h5D2
+0 (R (HN) log H) where
D1= 2µgN1 and, the square root is taken with positive real part.
e hf(m,0) -
1
4hµg 2
0
Exponential sums with a difference
190
Lemma 9.3.3 (Poisson summation over h) In the notation above h31 N2 H2
E e N-1 E e(hg(n))
h=H3
n=N3
3
K2(1)
L2
1=L1 k=K¢l)
xe-1
c(l2/(k - K)2) (\4µq(k- K))
J k(3012
a(b+K)l
(k-K)
µq
2
q
/
+OI R (HN) log N), where 9(z) and cp(z) are algebraic functions, one at the origin, and regular inside the unit circle, and L1= 2µgH3N3 > 2,
L2 = 2pqH2N2,
(9.3.1)
a H3 >KZ(1)=µq min a N3+H3,NZ z
Kl(1) = µq maxi`
12
12
N2 + H2 , N3 +
L1
12F
12
12
L1
L2
64,
(9.3.2) (9.3.3)
HZ
and the integer b and the real number K satisfy
gf(m,0)=b+K,
/1 Kminl 2 +OI
If the parameters satisfy
`
q
H
4
`
H3 << N2 R,
(9.3.4)
then we can also take 12
K1(l)=µgmax L2 N2,N3J
(9.3.5)
64,
z
K2(1)=µq min N2
12
LZ
(9.3.6)
N3
Proof In Lemma 9.3.2 we write H4 = H4(l) = max(H3, l/D2), H5 = H5(1) = min(H2, 1/D3). For each 1 the sum over h takes the form
H45h5 H5
h= -al
IF)
el+)
(h(b + K) q
h3
3
l2
4hµg2
1
Poisson summation on the minor arcs
191
We can take out the factor el bh
=e1 - 9l)
which is independent of h. We apply the inversion step (Lemma 5.5.3) with h = qn - al,
f(x)= Kx-
a1K
µ(qx-al)3
+
q
-
12
4µg2(gx-al)
3
M replaced by H/q, and T replaced by l2 /4µg2H. When µ is positive, then 12
f"(x)=2µgz(gx-al)-
2µ(gx - al)3
12
4µ2 g2 (gx-al)4 lz
4µ(gx - al)3 Since qx - al is a value of h, we have
2µq(gx-al)2 1
H5 2H 5 N 532' - 2µgH5 D 5N3 1
3
so f" (x) has the opposite sign to A. The inversion step gives 12 1 Kh µh3 h =H4
e
(2hµ)
4hµgz
3
q
h=- -al
I(f"(xk))I
(2µ(gxk-al))
ASkSB
+O
e(f(xk)-kxk- 8)
1
= L
1
H
(Hµ)
R
4µq22 H) 12
+log(B-A+2)
,
1
where the endpoints A and B are given by P H5 + al = K+µgH5z + A=f I
12
= K + max
2
4M RH5
R
k
4µgHz lz
pglz + µRHz D3 + z > 4µq D3 ) 12
= K + maX kAq--H,2 + µgH2 , µgN3 + 4µRN3 / Iz
K+maX
1z
+ Z µRH3 =K1(l)+K, Lzz µgNZ +NRHz,NRN3 L1
Exponential sums with a difference
192
in the notation of this lemma and the preceding lemma. Similarly
B=K2(1)+K. Changing the limits of summation in the inversion step by a bounded distance
K does not increase the order of magnitude of the error term. We take the sum over K between KI(l) and K2(1). In the main term Xk is given by 12
K+µq(gxk-al)2+ We put
4µq(gxk-al)
2
=k.
h=hk=qxk-al.
Then
12/4µgh2 + µgh2 = k - K,
(9.3.7)
and
h4 - (k - K)h2 + 12 =0 4µ2q2 µq
h2=
k-K
12
1-
2µq
I(i_ (k- K)2 -1
12
1- (k
1+
2µq(k - K)
11
12
(9.3.8)
- K)2
we take the unique root for which h x H. By (9.3.7) and (9.3.8) we have akl (k- K)h
f(xk) - kxk +
-= q
+
q
12
3
4µg2h
12
2µh3
2µg2h
3 4µ282h4
12
+
2µg2h
k-K
=-1
2µq
X 1+ = -1
µh3
3
312
2
1+ Vlk - K)
12
()
3(k- K)
µq3
)
2
1+
Al2
(k- K)2J
rI
2
8
(
(k- K)2 )
(9.3.9)
The large sieve on the minor arcs
193
Similarly we have
I2µhf"(xk)I =
2 h2
1-
4µ g2h4 12
)
2µq(k-K) 1+ I11
r
412
(k- K)2
12 Z (k - tc)
1+
(k-K)2
12
= 4µq(k - K)/cp2(12/(k - K)2).
(9.3.10)
The error term in the inversion step has the same size as in Lemma 8.3.3.
Changing the range of summation for k by O(µgH2) corresponds to changing the range of summation H4 S h < H5 by O(µgH3/K), up to an error term of the same size as the error term already present in Lemma 9.3.2. This corresponds to changing the sum on the left by
<<µH3 K
L
(µH)
<<
H3 IVZR
(HN) ,
which provides the condition (9.3.4). Thus when (9.3.4) holds, then we can take the limits of the sum over k to be given by (9.3.5) and (9.3.6), not (9.3.2)
o
and (9.3.3).
9.4 THE LARGE SIEVE ON THE MINOR ARCS We follow the exposition of Section 8.4 of the previous chapter. The first three lemmas differ only in the notation. Lemma 9.4.1 (The large sieve) Let a(k) and b(j) be coefficients of modulus at most unity, indexed by the integer vectors k = (k1, k2, 11, 12) with entries in
some range KS k, < 2K, LS l; < 2L, and by minor arcs Jj =J(a/q) with a/q = aj/qj, with denominators in some range Q < q < 2Q. Let xik) = Ilk1 - 12k2,
x2k)=l1 kl0(li/ki) -12 k29(122/k2) x3k) 11
11
x4k> =
kl
e
2
(k1
- kl411 5,2 e
=11-12) 11
122
4123
1
( 122
- --k2e 121 + k25,2 e ` k2 1 k2 12
2
(k12
Exponential sums with a difference
194
where 0(z) is the function in Lemma 9.3.3, and let
y(j) = -
a
-
q
- ab
1
q
( µq3)
K
2V-( 1.tq3)
where b, K, µ, come from the coefficients of the approximating polynomial on
J(a/q) as in Lemmas 9.3.1 and 9.3.3, and the values of K, L, and Q are consistent with the construction above, so that K >- 4L. Let V >- 1 be a real number. Then 2
a(k)b(j)e(x(k).y(j))I << ABKL4R2NV/Q3 j where A and B are the number of coincident pairs among the vectors x(k) and k
y(j), respectively. Two vectors x(k) form a coincident pair if their corresponding entries differ by numerically less than 1,
R `N / '1' R `N Two vectors y(j) form a coincident pair if their corresponding entries differ by numerically less than 1
1
1
8KLV' 4L (2K) max0(x)' 4L' 8LmaxIO(x) - 4x0'(x)I where the maxima are over the range 0 S x S 1/4. The result is still true for a family of sums Si of the form (9.1.1), taken over
subintervals (depending on i) of H < h5 2H, and of M + 2H - 1 < m <2M - 2H, formed with functions f;(x) satisfying (9.1.3) uniformly in i, when we replace b(j) by b(i, j), y()) by y(', j), and the sum over j by a sum over i and j. The sequence of rational numbers aj/qj may also depend on i. O
Lemma 9.4.2 (Large sieve over subintervals). of modulus at most unity, and let
Let d(k, 1) be any coefficients
4
xrk,')yri)
g(j, k, l) = d(k, l )e r= 1
Let L3(j) and L4(j) be sequences indexed by the minor arcs Jj with Q5 qj < 2Q, satisfying
L5L3(j)SL4(j)<2L for some fixed powers of 2: L and Q. Then we have 2K-1 L4(j)
j
E E g(j,k,l) K
2 2
ABKL4NR2V <<
Q3
log4N,
L3(j)
in the notation of Lemma 9.4.1.
0
The large sieve on the minor arcs
195
Lemma 9.4.3 (Partial sums by Mellin transforms) L2(j)
K2(j,l)
E
]E
In the notation above
g(k,1) << WLlogN
j l=L1(j) k=Kj(j,1) ABKL°NR2 VW 2 + Q3
1/4
log2N,
where the sum is over approximating polynomials used on minor arcs with qj in
the range QS qj < 2Q, and W is the number of such polynomials: W= W(Q) when Modification 2 is not used, W << Q2W(Q)/R2 when Modification 2 is used. The limits of summation for k and 1 are (9.3.5) and (9.3.6) of Lemma
9.3.3; the argument j has been added to indicate the dependence on the approximating polynomial. Here K and L are powers of 2 whose order of magnitude is given by (8.4.1) of the previous chapter.
In the notation of Lemmas 9.3.3
Lemma 9.4.4 (Large sieve with gardening) and 9.4.3 H2
( µh3 1
N2
el - )e(hj(n)) h=H3
`
W
«
R
n=N3
3
(HN) log N
R2 ABKL4NR2VW2
+Q
1/4
log2N.
Q3
The corresponding result for a family of sums Si is HZ
E h=H3
µh3) N2(j) e(-) F, 3
e(hg;j(n))
n=N3(j)
HNQ2VW R2
R2 I ABKL4NR10VW2
log2N+ Q
1
1/2
log4N,
Q7 )
where all the minor arcs with Q < q < 2Q for all the sums Si are counted in B and W.
Proof The exponential in Lemma 9.3.3 has the form
e(klyl +1y3 +ly2 (k- K) 0(12/(k- K)2)). Since k > 64, we can expand
(1- K/k) by the binomial series, and since
(9.3.1), (9.3.5), and (9.1.5) give 1
k<
2 µgH2 N2
µgN3
5
12H
3
N 516'
Exponential sums with a difference
196
and 6(z) has no singularities inside the unit circle, we can expand 0(12/(k K)2) as a convergent power series in K with coefficients involving k and 1. The constant term and the term in K are kept inside the exponential. For the remaining terms we expand the exponential, and we also expand
/(i-)
z `P
(k - K) z
These terms form a triple power series in the variables K/k, I2/k2, and K21/ (µg3k3) . As in Lemma 8.4.4 we see that each of these variables has modulus at most ,'-6) and that the triple power series has radius of convergence at least i in each variable. For the first result we use Lemma 9.4.3 separately on each term in the triple power series and add up. For the second result we use Cauchy's inequality in the form r=0 s=0 t=0
Erst
Er F, Ft s
(E E E2' +'+IlErst l2)
2r+s+t 1
r
s
t
where the expression E,st contains the sums over i, j, k, and 1. The factor 2r+s+t does not destroy the convergence of the triple power series. We complete the proof as in Lemma 8.4.4.
0
10 Exponential sums with modular form coefficients 10.1 MODULAR FORMS Modular forms, and their generalization to automorphic functions, lie at the centre of mathematics, in the sense that they appear in combinatorics, group theory, number theory, algebraic geometry, geometric topology, functional analysis, and complex analysis. Swinnerton-Dyer once declared that the unsolved problems in mathematics remain unsolved because we have not yet connected them to modular forms. A homogeneous modular form is a function of two complex variables z1 and z2 which is homogeneous: f(Az1, Az2) = A-kf(z1, z2) (10.1.1) (the integer k is called the weight), and depends only on the lattice in the plane generated by z1 and z2, so that
f(z1,z2)=f(w1,w2)
if
(c
)(:2),
(10.1.2)
where the matrix has integer entries and determinant one. It is not defined if the vectors corresponding to z1 and z2 lie in the same straight line. Modular
forms first appeared as constants of ihtegration, independent of z, in the construction of the doubly periodic elliptic functions, which have g(z + w1) = g(z), g(z + w2) =g(z) for two complex numbers w1 and w2. What is important is the additive group of complex numbers w with g(z + (o) =g(z) for all z, rather than any particular pair of generators for the group. There are generalizations in many- directions. One that is essential to the
theory is to allow f(z1, z2) to take a finite number of different values as (z1, z2) runs through all pairs of generators of a given lattice in the plane. For
a matrix M = (c d ), we write f I M (`f twisted by M', or `f acted upon M') for the function f(az1 + bz2, cz1 + dz2). The integer matrices of determinant one for which f I M =f form a subgroup of the group SL(2, 77) of finite index. There is a wealth of relevant algebra. The finitely many different functions f I M give a finite-dimensional representation of SL(2, 7L), and certain linear combinations of them give the irreducible components of the representation.
Since f(-z1, -z2) = (-1)kf(z1, z2), if the integer k is to be odd, then we must permit f I M to take finitely many different values. It turns out that k can be a half-integer without our losing all the algebraic structure.
Exponential sums with modular form coefficients
198
A modular form and the subgroup of SL(2, Z) which fixes it are said to have congruence level N if f(zl, z2) = f(wl, w2) whenever (z1 - wl/N and (z2 - w2)/N are both lattice vectors. This property is equivalent to f I M = f whenever IMI = 1 and M is congruent to I, the 2 x 2 identity matrix, modulo N. The fundamental notion is commensurability. If f is a modular form with congruence level N, and M is an integer matrix with positive determinant D,
then f I M is a modular form with congruence level DN (in general, the subgroups which fix f and f I M are called commensurable with one another when they share a common subgroup of finite index). A suitable average of
the functions f I M with matrices M of the same determinant D gives a modular form of level N again. These averaging procedures are called Hecke operators.
Modular forms occur widely as the generating functions of interesting sequences of numbers. The generating function of a sequence r(1), r(2),... is the function
R(t) = Er(n)t". If the properties of R(t) are known, then we can deduce properties of the coefficients. More generally, R(t) may be related to a known function S(t) by
R(t) = exp S(t),
R(t) = S'(t)/S(t),
R(t) = S(t)/(1- t)2, for example, or the powers t" may be replaced by another sequence of functions. For modular forms the interesting coefficients appear in a Fourier expansion.
An inhomogeneous modular form is constructed from a homogeneous modular form by
F(zl/z2) =zif(zi, z2) =f(Z1/z2,1). The twisted function corresponding to f I M is
(FIM)(z)=(cz+d)-kF
az + b
(cz+d
I.
(10.1.3)
This more complicated formula is the price that we pay for eliminating one variable. Since F
I
1) takes only finitely many values as the integer b
varies, then (whether F has a congruence level or not) there is an integer N
with F o 1) = F, so that f(z + N) = F(z). For each fixed yo for which ( F(z) has` no singularity on the line y =yo, there is a Fourier series F(x+iyo)A"e( nxN)a(nexp 27rin(xN+ iyo) I
with
a(n) =A nexp
27rnyo
N
Modular forms
199
which converges absolutely. We can define G(z) _ E a(n)e2"i"ZIN. n
Then F(z) is an analytic continuation of G(z). The numbers a(n) are called the Fourier coefficients of F(z). When F(z) is invariant over the full group, then these coefficients are canonical. When F I M takes a finite number of different values for M in SL(2, 71), then there are finitely many Fourier expansions. There is a summation formula connecting a sum E a(m)w(m) formed with a smooth weight function w(x) with a sum E b(n)w(n), where w(y) is a transform of w(x), and the numbers b(n) are the Fourier coefficients of F o -1o In the case of functions F(z) of integer weight and congruence level N, we believe that among the linear combinations of twists F I M there is a simplest I
function G(z) whose Fourier coefficients satisfy the multiplicative rule b(mn) = b(m)b(n) when the highest common factor (m, n) is one. The Hecke
operators should take G(z) to a multiple of itself, AG(z) say, and the eigenvalues A for all the Hecke operators should determine the Fourier expansion of G I M for every non-singular integer matrix M. The numbers
b(m)e(am/q) are the Fourier coefficients of G (o
Q ),
with a summation
formula relating them to the Fourier coefficients of
GI
q I
q
Conversely, Andre Weil's Bestimmungssatz (1967, 1971) states that if a sequence of numbers b(m) have summation formulae of the right type for the sums E b(m)w(m)e(am/q), then they are the coefficients of a modular form of a certain weight and congruence level. For modular forms of half-integer weight, the Hecke operator of level D is
connected with b(D2), instead of with b(D). The Fourier coefficients of modular forms without a congruence level await an algebraic interpretation. There are many generalizations of modular forms to higher dimensions.
The algebraic structure does not always allow us to define a sequence of numbers that correspond to the Fourier coefficients in the classical case. A surprising generalization is to eigenfunctions of the Laplacian on twodimensional hyperbolic space, which can be modelled as the upper half-plane H, the complex numbers z = x + iy with y > 0. The Mobius transformations z -* Mz = (az + b)/(cz + d) are translations or rotations. The 'real-analytic modular functions' F(z) have (F I M)(z) = F(Mz) taking finitely many different values for the matrix M in SL(2, 71), and d2F1 y 2(d2F dx2+aY 22JI=AF for some fixed A, which plays a role similar to the weight.
Exponential sums with modular form coefficients
200
10.2 THE WILTON SUMMATION FORMULA Let F(z) and G(z) be modular forms of weight k with Fourier expansions (the old name is `q-series')
G(z) _
F(z) _ F, a(m)exp2irimz, m
b(l)exp
21rilz ,
(10.2.1)
C
1
with
_
r
GI?
I
1
) =z-kGl - z) =F(z).
(10.2.2)
The idea of the Wilton summation formula is to express the coefficients a(m)
as averages of the coefficients b(l). We treat the simplest case when F(z) and G(z) are cusp forms: this means that for any matrix M in the modular group, the Fourier coefficients of F I M are zero for m 5 0, and (F I M)(z) tends to zero as y tends to infinity. To justify convergence, we quote a fundamental result of Rankin (1939): the series corresponds to an integral involving IF(z)I2 over a region in the z-plane.
Lemma 10.2.1 (Rankin's series)
When G(z) is a cusp form, then the series W
Ib(1)I2
I=1
12
R(s) _
converges for Re s > k, and has an analytical continuation with a simple pole at s = k.
Corollary
There is a constant B with L
Ib(1)I2
1=1
l
k
S B2 log2L
(10.2.3)
for any positive integer L Z 1.
Proof of the corollary Since (s - 1)R(s) is continuous, let
A=
max
k5ssk+2
Ks - 1)R(s)I,
where the maximum is taken along the real axis. We put s = k + 1/(log2L). Then 1
Ib(1)I2
Alog2LzR(s)> E 1
which gives the corollary with B2 = eA.
> is
e
E
Ib(1)I2
1
lk
0
The Wilton summation formula
201
Lemma 10.2.2 (The Wilton summation formula) Let F(z) and G(z) be cusp
forms of even weight k satisfying (10.2.1) and (10.2.2). Let g(x) be twice continuously differentiable, with g(x) = 0 for x :!g M or x z M2, where 0 < M < M2. Then M2
F, a(m)g(m) M
(k-1)/2
_ (-1)k/22ir
b(l) fM 2(-)
Jk-1 4"'
V(
g(x) dx.
Proof We write µ = 1/21rM and put h(x) = e2iµxg(x). Let
h(x) = fuM2 h(u)e(-ux) du =M
(10.2.4)
be the Fourier transform of h(x). Then W
f h(x)F(x + iµ) dx = f h(x) E a(m)e(mx)e-2"m dx °°
°°
m
_ E a(m)h(m)e-2,,mµ = F, a(m)g(m). m
m
We use (10.2.2) to obtain
E a(m)g(m) = f h(x)(x + iµ)-kG( -
f Wh(x)(x + 00
=
iµ)-k
1 µ) dx x+i
b(l)e(-
c(x +
iµ)) dx. (10.2.5)
We integrate by parts twice and use the First Derivative Test (Lemma 5.1.2) to get h(x) << 1/IxI3
for large x. Using Lemma 10.2.1 with s = (2k + 3)/4 and Holder's inequality, we find that r
l
µ c(x+iµ)) «(c(x2+'2)\°k3v4
Exponential sums with modular form coefficients
202
Hence the sum and integral in (10.2.5) are absolutely convergent, and we may integrate first. We find that
h(x)(x + iµ)-keI
- c(x + iµ)
I dx
=
l - ux)dxdu fu=M h(u) fx= - (x + iµ)-keI` - C(x+ill)
=
fU=M g(u)fz= -oo+iµ e ( - cz - uz 1 dz du. zk
MZ
I
W+114
MZ
The integrand has an essential singularity at z = 0, and is small in the half-plane Im z < 0. We can deform the inner integral to a clockwise circle around the origin. The substitutions iv
z
(1)
,
)
t = 41T Ku
give
(-i)
k- t Cu (k- t)/2 ( l )
t/
f expl 2I u
v
v-kdv.
(10.2.6)
The coefficient of vk- t in the expansion of the exponential in (10.2.6) is the Bessel coefficient Jk_ 1(t).
Lemma 10.2.3 (Twisted Wilton summation) Let F(z) be a modular form invariant under the full modular group (so that G(z) = F(z) in (10.2.2)), with Fourier coefficients b(m). Let a/q be a rational number and let g(x) be as in Lemma 10.2.2. Then M2
F, b(m)e(
q)g(m)
M
W/2 _ (-1)
21r
q
I (lx) / al) M2(IX) (k-t)/2 g(x)dx, Jk_ll4,7r q b(l)eI` - q fM
where a is defined modulo q by as = 1.
Proof We have
FI z+ q) =FI
(
alq) =FI
( aq a)(0
a1q) =F
aq
l0q)'
The Wilton summation formula
203
where as + qq = 1. The modular form G(z) with
a/q
G
0/=FI`0
(1
1
is given by aq
G=F
-0) =F 1/q -a
1/0gl(0
0
q
'
so
az - -
G(z) = q-'F
q
We now use the Wilton summation formula with c = q2 and a(m), b(l) o replaced by b(m)e(am/q), q-kb(l)e(-al/q). The Wilton summation formula is simplest for analytic cusp forms of even weight. The first cases were Voronoi's formula (1904) for sums of the divisor
function d(m), and Hardy's formula (1915) for sums of the representation function r(m) that counts the number of ways of writing m = a2 + b2 with a > 0, b Z 0. These are cases of Lemma 6.2.1 in which there is a closed form for the Fourier coefficients. The divisor function corresponds to a special non-holomorphic modular form. We quote the twisted formula, using the notation of Jeffreys and Jeffreys (1962, chapter 21) for Bessel functions.
Lemma 10.2.4 (Twisted Voronoi summation) Let g(x) and a/q be as in Lemma 10.2.2. Then M2
M
q
(am)g(m)
d(m)el
1
= q fmM2 log
Xq2
+
2y g(x) dx 1
- 2'r L d(1)eI - al q
+
21T
q
I
fM2Y,0 41r
qJM
r
(lx) q
)xdx
(lx) ( all M2 ( d(l)e - q JI fM K0,°I4rr q
)xdx. 0
To obtain a reflection step we must truncate these series and approximate the Bessel functions. Our next lemma is an exercise in the use of Lemmas 5.5.5 and 5.5.6.
Lemma 10.2.5 (Stokes' approximation) Let I(l, u) be the integral +iµ I(l u) =
1z=--+iµ
I
e (
1 dz - uzJ
CZ
Zk
Exponential sums with modular form coefficients
204
where k z 2 is an even integer. For u Z c/l we have 1/4 Cu (k- 1)/2
C
I(l, u) = 2( 41u)
0
(l)
r lU
Re e (2 1(2k+1)/4 +
1
c ) +8
I
)k-1)
u3/2 (
Proof We move the line of integration from Im z = µ to Im z = ri =1/2lru, and we put z = q(x + 0. Then l cn(x2 +)1) - 7u(x + i) I
I(l, U)
/
1
k-1(x + i) k
2
I
1
'qk-1(x+i)k
e1-a2/(XZ+1) exp(-
x X2A+1
- ix) dx,
(10.2.7)
where we have put A2 = 21rl/c-q = 4ir2lu/c.
This is an exponential integral with x
A2x
2ir(x2 + 1) 1 g(x) = nk-1(x+i)k
21r'
e1-aZ/(X2+1).
Strictly we must consider the real and imaginary parts of g(x) separately, but since ds
Reg(x)
-
I
Re
ds
d3 dxsg(x)I SI
dxsg(x)
the same order-of-magnitude estimates apply formally. We have
A2(x2 -1) )
- -(X2+ f (x)= --I1 2'r ` 1
( x)
/
A2x(x2 - 3) 1r(x2 + 1)2
The equation f'(x) = 0 is a quadratic in x2. For A z 2,7r, the roots are ±x1, ±x2, where 2
x1=1+ 2+0141, x2=A 1-
3 ZAZ
1
+0 (A4))
The Wilton summation formula
205
The subinterval -25 x:!92 of the range of integration
in (10.2.7)
contributes O(uk-1 a-AZ/4)
(10.2.8)
We divide the rest of the range into blocks [H,2H] and [-2H, -H], where H is a power of 2. Let Ho be the nearest power of 2 to x2. We combine two pairs of consecutive blocks as [-2Hp, -Ho/2] and [Ho/2,2Ho]. The endpoint terms cancel between adjacent blocks. At ±2 the endpoint terms are
O
uk21
e A2/4 J, )
A
and they can be absorbed by (10.2.8).
For each block H5 Ixl 5 2H we have If1r)(x)I X A2/Hr, and
uk- 1/Hk+s
for H;-> A,
k 1/H k+3seA2/H' A2su-
for H5 A.
For H< x:5 2H, H < Ho we take A2
TxH
Ux
H3
uk-1e-A2/H2 Hk
MxH,
Nx
12.
The error term of Lemma 5.5.5 is M3U U u k-1 e- A2/H2 << N2T2 << H << Hk+1
(10.2.9)
For Ho/2 5 x :!g 2Ho we take
Uxuk-1/Ho,
TxHO,
MxNxH0.
The error term of Lemma 5.5.6 is MU << T3/2 <<
U HO <<
uk-1
Ak+1/z
(10.2.10)
For HSx52H,HO
Tx H,
uk-1
Ux
Hk
,
MxNxH.
The error term of Lemma 5.5.5 is uk-1
H3U « MU IT « A4 « A4Hk-3
(10.2.11)
Exponential sums with modular form coefficients
206
For H > A2 we use the trivial estimate uk-1 << MU<<
(10.2.12)
WT-1
These terms (10.2.8), (10.2.9), (10.2.10), (10.2.11), and (10.2.12) sum to 1
K
llk- 1
1
k+1/2 + A2k_2
)
The second term is negligible for k > 4.
Since f"(x) is negative, we compute the main term for the range [Ho/2,2Ho] as
g(x2)e(f(x2)
-
(10.2.13)
s)/If" (x2)I
using the complex conjugate form of Lemma 5.5.6. In the exponent (
A
f(x2)= - -+0I
),
A
so that
e(f(x2)-'-'-g)=e(-2
I
lc
I
AJI.
The weight is
g(x2)/
I f"(x2)I
77
=
1+O
k-1xk
2 +0 z
exp 1
2
2
h(rtlA)'rl/2 k- ] k- 1/2
(i+oi_ii11 1
c
-(4lu )
]/4
(l)
(k-1)/2
I
/
1
`
`
II
h(qx2)(1+OI
We also have
h(gx2) = h(,qA) + 0(,q/A).
These approximations give another error term of size (10.2.10). The main term for the range [-2Ho, -Ho/2] is the complex conjugate of (10.2.13). 0 We can now give a truncated form of the Wilton formula. The condition k Z 4 comes from Lemma 10.2.5, and can be removed if we use a different integral for the Bessel function.
207
The Wilton summation formula
Lemma 10.2.6 (The truncated Wilton summation formula) Let F(z) and G(z) be cusp forms of even weight k >- 4 satisfying (10.2.1) and (10.2.2). Let M and M2 be positive integers with M2 < 2M. Let g(x) be s times continuously
differentiable, and zero for X< M and x >- M2. Define the means U, for r = 0,...,s by
f ,Jg(')(x)Idx= U,(M2 -M). M"
Let K + 1 be the power of 2 with
KS
cM(s+1)/(s-1)(Us/U0)2/(s
<2K+ 1.
Then, if M >- c and K z 1, we have M2
F, a(m)g(m) M
K
=2
1/4/
!
(k-1)/2
b(l) fMMZI 41x)
g(x)Re e 2
I
+ $ dx
+O(Bc(k+1)/2Mk/2-1+1/(2s-2)
X (m2 - M)U(2 s-3)/(2s-2)USl/(2s-2)(log K)1/2), where B is the constant of Lemma 10.2.1.
Proof We write Ik(l, u) for I(1, u) in Lemma 10.2.5 to indicate the power of z. Then d
du
Ik(l,u) _ -271iIk-1(I,u).
We integrate by parts to get M2
u) du =
fm
(27ri)s fm
Zg(s)(u)Iks(I, u) du.
Using Stokes' approximation as an upper bound, we have 1(2k+2s- 1)/4 fM 1 cM r fM 2I g(s)(u)I du. Zg(u)Ik(I, ll) du <<
Il
M
J
Lemma 10.2.1 and Holder's inequality now give M
b(l)Ik(1,u) 1=K+1 <<
Bc(2k+2s-1)/4K-(2s-3)/411(2k+2s-3)/4(M2 -M)Us(log
K)1/2.
Exponential sums with modular form coefficients
208
We substitute Stokes' approximation into the terms with 1:!g K in the Wilton formula. The error terms contribute cx (2k+1)/4
< fM Ig(x)I() MM
x-3/2 dx.
(10.2.15) by Lemma 10.2.1 and Holder's inequality again. The choice of K makes both 11 error terms (10.2.14) and (10.2.15) of the size stated in the lemma. Bc(2k+1)/4K1/.4M(2k-s)/4 (M2 -M)U0(log K)1/2,
<<
10.3 FAREY ARCS We present the ideas of Jutila (1987a et seq.) for the sum M2
S = E b(m)e(f(m)),
(10.3.1)
m=M formed with the Fourier coefficients b(m) of a cusp form for the full modular group. Here
f(x) = TF(x/M)
as usual, where M and T are size parameters, and F(x) is a real function with s continuous derivatives, where s Z 4 is an even integer. We suppose that (10.3.2) If`r)(x)I <-CrT/Mr for r = 1, ... , s, and (10.3.3) If (r)(x)I > T/C,M' for r = 2; for T << M we also require (10.3.3) for r = 1. We use Dirichlet's approximation theorem (Lemma 1.5.1) to subdivide the sum. We choose an extended Farey sequence .9(R). When aj/qj and aj+1/q1+1 are consecutive fractions of .A(R), then (aj + a1+ 1)/(q; + qq+1) does not lie in 9(R). Since f"(x) is continuous with constant sign, then f'(x) is monotone, and we can define a function h(y) implicitly by
y =f'(h(y)). Then C2T
h (
(a)+aJ+l\
h
a'Z ai + a+ 1
q1+ql+1(q1)
qj +qi+1
a1 q1
1
=
g1(g1+qq+1)
z
1
2R2
Thus
aj +aj+1
h(qt+qj+1)
-hl
aj
Mz
q,) >N= 2C2R2T
We suppose that
M/8>_NZR, 2
(10.3.4)
Farey arcs
209
the second inequality being the reverse of our previous constructions, and R >- 48C2M/T,
(10.3.5)
NR <M/96C2.
(10.3.6)
which implies
Let a j/qj with j = 1, ..., J be the fractions of 9(R) in the interval
f'(M+2N)
Let No=M+N,NJ=M2-N, and, for j=1,...,J-1 Nj=
h(qj+qj+1 where [[x]] denotes the nearest integer to x. We construct a smoothed characteristic function of the interval Nj_ 1 <x
forx>N,
1
&(x)=
2(1+sins+1
ZN)
0
J
for IxI
for x < -N.
and
wj(x) = w(x-N_1) - w(x-N). Since s is an even positive integer, the function wj(x) has s continuous derivatives, with
wj(h(aj/qj)) =1. Lemma 10.3.1 (Smoothed Farey arcs) We have M2
S=
F, wj(m)b(m)e(f(m)) +O(BMk'2(N log j=1 M=M
M)112).
1
Proof We note that 0
w.(n)= j=1
for x< M,
1
forM+2N<x<M2-2N,
0
forxzM2.
By Lemma 10.2.1 we have
r
S- F, F b(m)wj(m)e(f(m)) j=1 m
M+2N
M2
E +M2-2N E M
I b(m)I << (B2MkN log M)1/'2.
0
Exponential sums with modular form coefficients
210
The bound obtained for the smoothing error using Lemma 10.2.1 is not very good. The bound for the individual Fourier coefficients (Deligne 1974) is
known only in the classical case. We define F(z) to be a primitive modular form or newform (Atkin and Lehner 1970) if F(z) cannot be derived from simpler functions by taking linear combinations of twists. Lemma 10.3.2 (The Ramanujan hypothesis) Let F(z) = E a(m)e(mz) be a newform of even weight k and some congruence level, normalized by a(1) = 1. Then a(n) is an algebraic integer for each n. and l a(n)I 5 d(n)n(k 1)"2.
(10.3.7)
There is a sense in which the Ramanujan hypothesis is the analogue for a certain quadratic polynomial of the Riemann hypothesis for the zeta function: see Ireland and Rosen (1982). For non-holomorphic modular forms, or modular forms without a congruence level, the assertion that a(n) is algebraic
is false, but the upper bound (10.3.7) may still hold. The Jacobi theta functions with weight 'z satisfy Lemma 10.2.1 but not Lemma 10.3.2: most coefficients are zero, but the Rankin series has to diverge at s = k.
Fourth-power analogues of Lemma 10.2.1 have been established by Moreno and Shahidi (1983). Their result is known for more cases than the Ramanujan hypothesis. Lemma 10.3.3 (The fourth-power Rankin series) The series 2
is r
converges for Re s > 2K - 1, and has a pole of order 2 or 3 at s = 2K - 1. The pole has order 2 for forms of even weight and congruence level one.
Corollary
When the pole has order 2, then there is a constant B2 with L
1=1
I b(l)4
lzk- 1 5 BZ (log2L)z
for any positive integer L.
Either Lemma 10.3.2 or 10.3.3 can be used to improve the smoothing error in Lemma 10.3.1.
10.4 WILTON SUMMATION ON THE FAREY ARCS We apply the truncated Wilton summation formula (Lemma 10.2.6), and evaluate the resulting exponential integrals.
Wilton summation on the Farey arcs
211
Lemma 10.4.1 (Explicit Wilton summation) We have (when f" (x) > 0) M2
E wj(m)b(m)e(f(m)) j=1 m=M
_ E 1F, F, b(l)el - 8111 wj(xj + O BMk/2
I MZRZ
N3
)hil
M
(x, )e(gj± (xj ))
11/(2s-2)
+ RZ
)(logM)12),
)
(10.4.1)
where xji and x,-.I are the stationary phase points satisfying
f'(x)=a± 1 (I), qj qj V x
(10.4.2)
the functions g j W, g,-., ( x) are given by
ax
2
qj
qj
(IX)
g (x)=f(x)- '--+-
+8+8,
the weight functions hit (x), h7 (x) are given by
(x/l
)ck-1)/2
(f" (x) ± (l/x3)1/2/2qj)
(4lxgf )1/4
and the length of the reflected sum K1 is the smallest integer with
k1+1 _ 18M/R2. Proof First we fix j, and write a/q for aj/qj. We apply the truncated Wilton summation formula with a(m), b(l) replaced by
b(m)e(am/q), q-kb(l)e(-al/q) as in Lemma 10.2.3. The size of the cut-off K depends on the derivatives ax
dxr
wj(x)e f(x) - q
which are sums of terms involving r,
q
with
ro+r1 +2r2+
+tr1=r.
We have w,(ro)(x) << 1/Nro,
and
f'>(x) << 1/M`-2NR2
Exponential sums with modular form coefficients
212
for i = 2,. .. , s, whilst when wa(x) is non-zero, then a
f'(x) -
NR <<
1
maxi f" (x)I <<
q q 4R By the order-of-magnitude assumption (10.3.4) we have
d w
We l f(x) -
-ax
.
l
1
dx' l q qR for r 5 s. We take the range of integration in Lemma 10.2.6 to be the support of w,(x), of length O(NR/q), so that we may replace M2 - M by O(NR/q) in the error estimate. Then 12/(s- 1)
K
qR)
and the error term is 1/(2s-2)
NVR-
O
(log M)1/2
BMk/2-1
( qR R
Vt
(10.4.3)
The main term is
x
a1
b(1)e -
I (I+(1) +I-(1)),
q
1=1
where
r I ±M = J
(X)(k-1)12
1
ax
2 (lx)
q
q
f(x) - - T
(41xq2 ) 1/4
T
dx.
We treat I+(1); I-(1) is similar. The exponent is stationary at xi given by (10.4.2). For x in the support of w1(x) we have
a.+a ' '+1
+Nmaxl f"(x)I
q1 + qi+ 1
a
1
q
qR
-+
NT
a
1
q
qR
+C2M25-+
1
+2R2.
For l z 18M/R2 we have
f'(x)
Ox I
2q q The weighted First Derivative Test (Lemma 5.5.5) gives (k-1)/2
1
M
(IM q 2)1/4
(1)
<<
I+(1)
q
q
Mk- /2 lk/2
f qs/2
TM M2
(N
M3
1
+ NZT q
(M) 11
!' 1) (-i--)
M 5/4 M +qs/2(l) M 5/4 M
(1)
NRZ Z
q2M
1V2,
312
A large sieve on the Farey arcs
213
By Lemma 10.2.1, K
K,+1
MKT3/2
al )I+ (l) <<
b(1)e(_
`
BM(k-1)/2(R q
(log M)112. (10.4.4)
)3/2
q
For 15 K1 we apply Lemma 5.5.6 with
a=x,, -M/4, 6 =xjl+M/4. If a< M, then we must define the function f(x) suitably in the range 3M/4 <x <M, where all the weights wa(x) are zero, and similarly if /3 > M2. We have
6/R> If'('_ 1) - f'(xx,)I >_ (N_1-xj,)max'I f"(x)I. We now see from (10.3.3) and (10.3.5) of the construction that N,_1 >- xj1-
M/4, and similarly that N.<xj+,+M/4. Hence the interval a <x < /3 includes the support of wj(x), and the endpoint terms in Lemma 5.5.6 are zero. The stationary phase term in Lemma 5.5.6 gives the explicit term in (10.4.1), and the second error term dominates the first. The error terms for I+(l) from Lemma 5.5.6 sum to K,
<< ,= Ib(1)
1
( (lMg2)1/4 l
(k
M
1)/2
I
1
M3
N2T3/2
BMk/2R3/2(log M)112 <<
(Nq)
,
(10.4.5)
where we have used Lemma 10.2.1 again. In each range Q :!g q < 2Q there are O(MQ2/NR2 + 1) Farey arcs by Lemma 1.6.1. We obtain the error term in (10.4.1) by summing (10.4.3), (10.4.4), and (10.4.5) over the Farey arcs; we find that the contribution of (10.4.5) cannot dominate both the other error terms at once. 0 10.5
A LARGE SIEVE ON THE FAREY ARCS
At this point in the argument we have been accustomed to using Lemma 5.6.6 with a bilinear exponent, and symmetry between the `integer' vectors x and the `rational' vectors y. The integers were small, and the number of rationals aj/q1 was large. In this chapter we have L large and J small. We do not want a Holder inequality replacing one variable 1 by two variables E 1; and E 1; ,
where 1, are integers in the range of 1. We treat the sum over 1 and j in Lemma 10.4.1 as a Type II sum in Lemma 5.6.1. Montgomery (1971 Lemmas 1.6 and 1.7) obtained all his large sieve results this way. We simplify the notation of Lemma 10.4.1, writing K for K1,
g;(1) =gjl (x,) - ail/qq,
(10.5.1)
hj(l) = w1(xx, )hj (xJl ).
(10.5.2)
Exponential sums with modular form coefficients
214
Of course similar arguments apply to the sums involving xp. We divide the range for l into blocks L 5 15 2L - 1, and the range for j into blocks with
Q
(10.5.3)
We write E(Q) for a sum over j restricted by (10.5.3), and J(Q) for the number of Farey arcs satisfying (10.5.3).
We cannot separate the variables completely before we apply the large sieve.
Lemma 10.5.1 (The dependence on 1)
Consider l as a continuous variable with 0< 1< K. Then for j fixed with Q 5 qq < 2Q we have in (10.4.2)
dx dl
16C2NR2
(10.5.4)
15Q (IM)
Let u? be the value of x in (10.4.2) when 1 = 0. Then in (10.4.2) we have 16C2NR2Vi
(10.5.5)
15MQ2
We can sharpen (10.5.4) and (10.5.5) to
dx dl
+
1
f" (u; );'-
F
N2R4 O I M2QZ
u
!+ qi
qi
2ui2f"(u;2)
+0
(10.5.6)
I LN2R4
1
M5/2Q3
2qi
(10.5.7)
Proof The stationary phase point x is defined by (10.4.2): q1
1
f (x)= q-+ qj
V
1
(10.5.8)
x/
so
(x)+
1
1
2qI(i)\dxdl
2qt (1x)
(10.5.9)
From (10.3.5) we have
VK 5 /R2) (x3) 2Q 1
2q
MR
5
< - minlf" (x)I. 16CZM2 - 6
Hence
dx
dl S 2qj
16
1
(lx)
15minlf"(x)I'
A large sieve on the Farey arcs
215
which gives the bound (10.5.4). Define z by
x = (uj +z)2. Then we have
-u;l<
1
2V
i dx dl, fo dl
which gives (10.5.5). Expanding f'(x) by Taylor's theorem in (10.5.8) as far as the term in f (3)(x), we find that VT
z
2gjui f" (uf (
1+0
Izl
M
which gives (10.5.7), and (10.5.6) follows similarly from (10.5.9).
Lemma 10.5.2 (A large sieve)
The main terms in Lemma (10.4.1) satisfy 2
min(2L-1, K)
T(Q) b(l)hjU)e(gj(l)) j
1=L
J(Q)z )1/4
Mk-1NR2
Q
5AOB2
+E
(LI -
log M
4VB(O1(V),O2(V),03(V),L4(V))I, (10.5.10)
V
where V is summed through positive powers of 2 satisfying V ::g
LF (LM )1/4
.
(10.5.11)
Here
Q -(L ) LQ2
A2(V) =A2
MV2
NR2
LN2R4
+A2 M3Q2
LQ2 3/4 NR2 +A3 MQ2 ( M VVL 1
A3(V) =A3
04(V) =A4
MQ2 NR-TV--2
LQ 2
0
NR2
0 M) +A4 MQ2 VM
Exponential sums with modular form coefficients
216
and B(O1, 02, 03, 04) denotes the number of pairs i, j with Q < q,, qj < 2Q for which
lizijil < All u;
q ,
u1
qj
(10.5.12)
02 min(02, A3)
if z1
< 04
zii = 0,
0,
if z,. = 0,
(10.5.13)
and
f " (u?)
f (u2)
if
(10.5.14)
where
zit = ai/gi - aj/q>, and
uj =
f'(u12)=ai/q1,
x+-o,
and A0,..., A4 are constants constructed from the constants C1.
Proof Using Lemma 5.6.1, we bound the square on the left of (10.5.10) by 2L-1 lb(l)12
lkh,(l)ht(1)e(gi(l) -ge(l))
1k
1
t
The first factor is O(B2 log L) by Lemma 10.2.1. We estimate the sum over 1 in the second factor using Lemmas 5.1.1 and 5.4.3. The weight function
H(l) = lkhi(l)h1(l) is allowed to depend on i and j. We need estimates for the size and total
variation of H(l) that are uniform in i and j. The bound for dxj,/dl in Lemma 10.5.1 shows that the weight function H(1) is f times a slowly varying function of 1, with
H(1)^
Mk 1NR2 Q
l M1 ,
and the total variation of H(l) has the same order of magnitude. After Lemma 5.1.1, we consider sums e(g,(1) - g,(1))
(10.5.15)
A large sieve on the Farey arcs
217
over subintervals of the range L < I < 2L. We have d ge(l)
dl
_ - gjV(x1l) -
-
and 1
xj
2qj
(13)
d2
dl2
ge(l)
qj
)
dx
lxj, 1
dl
1
1
2qj
,
By Lemma 10.5.1 we have kdl
a, 1 uj gi(1>+- -+-+
d
qj f
dl
32C2 NR
+q
71
2
(10.5.16)
15MQ2
A5N2R4
1
2ql
q1
aj
uj
1
ge(l) + q,
a uif, (ui)
2
MaQs
<_
M V(--)
(10.5.17)
and
d2
dl2
ui
1
g1(l) + 2q/
A6N2R4
5
13/2
(10.5.18)
M2Q3 (LM)
for some constants A5 and A6. In the sum (10.5.15), either u,
u
qj
1 15 qj
32A N2R4L
(10.5.19)
M5123 Q
or else the second derivative of the exponent in (10.5.15) has fixed order of magnitude, which we write as -gj(1))
2
Q
(L) 5
dl2
2
5 Q
(g;(1)
(LM
)
(10.5.20)
corresponding to u;
u!
qr
q11 -
'q
(10.5.21)
Q
for some i with 71 >
2A6LN2R4/M3Q2.
We call pairs i, j for which (10.5.19) holds `bad pairs', and other pairs `good
pairs'. For a bad pair, the first derivative (d/dl)(g;(l) -g,(l)) changes by at most 18A6N2R4 M2Q4
(LQ2 M
which is bounded, since
Q >> min(l, NR2/M).
Exponential sums with modular form coefficients
218
For bad pairs, the truncated sum in Lemma 5.4.3 has a bounded number of terms. For good pairs Corollary 1 to Lemma 5.4.3 gives
(r
e(g,(1) -g1(1)) <<
1/4
(-)
+ (LM)"4
(10.5.22)
We use the Riesz interchange principle, counting how often the sum over I is large. If
(LM)1/4V
e(g;(1) -g1(l))
F
(10.5.23)
with V greater than some absolute constant A7, then either the second term dominates in (10.5.22) with (10.5.24)
-q << LQ2/MV2,
or the pair i, j is bad. Combining both cases, we have LQ
« v2 to
LQZ
N2R4 + MZQ4
(
(10.5.25)
M
which gives u;
u1
LN2R4
LQ2
q1
<<
Mv2 + M3Q2
(10.5.26)
q1
In both cases the truncated sum in Lemma 5.4.3 has a bounded number of terms. The fraction z11= a;/q; - a1/q1 is only defined modulo one, so we can take the largest term to be r = 0, and
e(g,(l) -g1(l)) << I
min(2L-1,K)
L
e(g;(y) -g1(y)) dy
+ 1.
The First Derivative Test (Lemma 5.1.2) gives mint
d
dl
(g1(l) -g'3(1))I
-
(Lm) 11 4
V
There are now two cases. If z;1 is non-zero, then we use (10.5.16) to get
Iz.. <
1/4
(LM) V
+
1
u1
-u1
q,
q1
+
NR2 MQ2
(10.5.27)
If the pair i, j is bad, then the middle term in (10.5.27) is dominated by the third term. If the pair i, j is good, then the middle term has size
Q
I L 1<<
QV
I
M 1
(10.5.28)
A large sieve on the Farey arcs
219
by (10.5.21) and (10.5.24). Since the left-hand side of (10.5.23) is trivially at most L, we deduce the upper bound (10.5.11) for V, which shows that the expression on the right of (10.5.28) dominates the first term on the right of (10.5.27), and we have
z;il «
z
(M)
Q
2
+ MQZ
(10.5.29)
in both good and bad cases.
If zii is zero (so that qi = qi), then we can argue in two ways. We use (10.5.16) to get 1
,r,
I
ui
ui
qr
qi
F I
NR2
« (LM)1/4V + MQ2
so that ui
qi
ui
qi
1 «VFL(
Q2 3/4 NR2 +MQ2 M
LQZ
1
M
(10.5.30) )
We can also substitute (10.5.25) into (10.5.17) to get 1
1
i (su2)
2qZu2"f"(u?)
2qiZu2f
<<
Q
4
(M) +
M2Q3
(M) ,
analogous to (10.5.29). We deduce that
l f" (u?)
MQ2
u? u?
<<
+
NR2V2
NR2 MQ2
L2 LQ2
(-k--).
(10.5.31)
Using (10.5.30), we can replace u?/uj by q?/qj2 = 1 in (10.5.31). We now see that when (10.5.23) holds with V > A7, then the pair i, j is counted in B(A1(V ), ... , 04(V )) when A1,. .. , A4 are chosen to correspond to the implied constants in (10.5.29), (10.5.26), (10.5.30), and (10.5.31). The result (10.5.10) of the lemma follows when we split into cases in which the medulus of the sum (10.5.15) lies in ranges V to 2V.
Lemma 10.5.2 can be applied to a family of sums formed with functions fi(x) satisfying the inequalities (10.3.2)-(10.3.5) uniformly in i. We interpret the sum E(Q) as extending over Farey arcs from all sums of a family. As usual, we suppose that fi(x) =f(x, y.), where y,,..., y, are distinct points in
05y51.
Lemma 10.5.3 (Large sieve for a family of sums) For a family of sums formed with functions x
f(x,Y1)=TF M,Yi),
Exponential sums with modular form coefficients
220
where F12 is continuous and bounded away from zero, let B'(01,..., 04) denote the number of pairs i, j counted in B(i 1, ... , A4), but with
ar/qr $ aj/q,. Then we can modify (10.5.10), replacing the sum over V by 1QV
E v
vQ
B,(z
1(V ), A2(V ), i3(V ), i4(V ))
+J(Q)L +
QJ(Q)log I
(M )
minlyr -Yjl
(10.5.32)
Proof The analogue of (10.5.9) is
fll(xjr,Y) +
xr= -f12(xj,Y),
FX+3
Zqj
aY
so - ayxj+1X
M.
Let g,(l) in (10.5.16) be formed with parameter yr, gj(l) with parameter y, with a,/q; = aj/qj, but r 0 s. Then a
1
a(gt(1)-g;(l))I
=
qj
xjr
1
(l ) - qj V
l yr -Ysl
Q (LM)
I axjr
aYI
x t
1
I Y. -Ysl
Q
M
V1L
and the First Derivative Test gives
IEe(g,(1)-gj(l))I << IYrQYs)
(M)
If y;+ 1 - y; z S for i= 1, ... , I - 1, then I yr - ysl > I r - sl S, and for r fixed 1
s#r IYr-Ysl
2 <-2S 7-1 1-S-(log I+1). n S
0
n=1
10.6 TOWARDS A MEAN SQUARE RESULT We start with the smoothed sum
S(t) _
F, b(m)cej(m)g(m)e(f(m)), j
m
(10.6.1)
Towards a mean square result
221
where f(m) depends also on a parameter t by
f(m) = (T + t)F(m/M), and g(x) is supposed bounded and of bounded variation. We write S(Q) for the sum S(t) with the extra restriction (10.5.3), so that IS(t)12
=IF
2
S(Q)(t) I
s
tlo
R
g2
+1
IS(Q)(t)I2,
where Q is summed through powers of 2 withQ Q5 R. We obtain short interval means for the sums ScQ>(t), but the length of the short interval depends on Q. Changing to uniform Diophantine approximation merely moves the difficulty elsewhere. When we consider a farrtily of sums, then we subdivide the Farey arcs with Q5 qj < 2Q further into transversal sets. We use the same notation E(Q) for a typical transversal set, with IS(t)I2 << log R E IScQ>(t)12, Q
the sum being over all transversal sets for each Q. We write S,(t), Sf Q)(t) to indicate that T takes the value Ti. We sketch how to obtain a short interval mean value for the smoothed sum
S(t), provided that (10.3.3) holds for r = 1 also. We start from a smooth version of Lemma 5.6.4.
Lemma 10.6.1 (Short interval mean with smoothing) Let am be a sequence of real numbers, and let f3(M)...... 3(M2) be any complex coefficients. Let 8 > 0 be real, s >- 4 be an even integer, and let M2
M2
A(t) _ E 13(m)e(amt),
B(x)
F, f3(m)X(x - am),
M
M
where X(x) is the smooth weight with s continuous derivatives X(x)-
?(1+coss+1 S)
forIxl<S,
0
for IxI >- S.
Then
f 1r/3S JA(t)12 dt a/3s
<_
Sf 2
I B(x)12 dx.
Proof We note that X(x) is positive, and rs J X(x) dx = S.
The Fourier transform j(t) of y(x) is given by The
X(t) = f e(-tx)X(x)dx.
Exponential sums with modular form coefficients
222
For Itl < it/35 and btl 5 6, we have I txl 5 it/3, and s
Re X(t)
X(x) s dx >i f
- 26
We now use Parseval's identity for Fourier transforms:
f
I E /3(m)X(x-am)I dx= f 1E,6(m)X(-0e(ant)I dt flr/3S
2
E 13(m)e(a,,t) I $'(-t)12 dt,
-1r/3S
M
0
which gives the lemma.
We obtain a short interval mean
fUIScQ>(t)I2 dt <
S2
f IB(y)12 dy
(10.6.2)
by taking
a. = F(m/M),
6 = it/3U,
ai(m) = wi(m)b(m)g(m)e(arT).
/3(m) = DO /3i(m), i
Since (10.3.3) holds for r = 1, for fixed y, X(y - am) is non-zero for an interval of y of length between 26M/C1 and 2C16M. We take
6M x NR/Q, Lemma 10.6.2 (A short interval mean)
U x MQ/NR.
(10.6.3)
In the notation of Lemmas 10.4.1 and
10.5.2 f U IS(Q)(t)12 dt
-U
/M) k-1/2
« E(Q) j
RI L
`
L
log2 M f
]2
2L-1
2
b(1)e(gi(1)+r11)
d77
1=L
1111
( Q2 M )1/3
+ O B2Mk logz MI
Rz
QR
Q2 M2R2 Q6 M4R4 R6 N6 + R2 N3
where K(x) is the weight function of Lemma 5.2.3 with N replaced by L.
,
Towards a mean square result
223
Proof There are finitely many j for which /3j(m)X(y - am) 0 0. Hence 2
R(m)x(y - a,,,)I
2
« (Q)
aj(m)X(y I
M
am)I
.
(10.6.4)
m
In the notation of Lemma 10.4.1
E wj(m)b(m)g(m)x(y - am)e(TF(m/M)) m
qJ
b(1)e
x
M
wj(xi )g(xj )x
,
+O BMk/2(log M)1/2 r Q 3/2 MR2
+I
R)
N2
+
M
(Q) (QR
R3/2
(NQ)
(10.6.5)
We break the sum over I into blocks L < 15 min(K, 2L - 1) as usual. We write
hj(1) = wj(xj,)g(xj+,)X(y - F(xj+,/M))hj, (x, ), and treat hj(l) as a weight function, with ( M\(k-1)/2 (NR2) hi (1) « L JII
(LMQ2)1/4
We use Lemma 5.1.1 to pass to a subinterval of L 515 2L - 1 with an unweighted sum, and Lemma 5.2.3 to pass from the subinterval to the whole interval, with an integration over a factor e(7 l). Cauchy's inequality gives 2
F, wj(m)b(m)g(m)x(y - am)e(TF(m/M))I M
<< log M B2Mk log M
N 2 R M 1/3 (M2Q (QR
+ F, F,
t
Q3 M2R4 R3 ) + R3 N4 + NQ
f K('q)
L
2L-1
x 1=L
NR2
2
b(I)e(gj(1) +'ql)
L
Q (LM)
(10.6.6)
in the notation (10.5.1). The bound (10.5.6) is uniform in y, but the sum on the left is non-zero only for y within a distance 6 of a value of F(m/M) on
Exponential sums with modular form coefficients
224
the Farey arc, so y lies in an interval of length O(S) by (10.6.3). Integrating (10.6.6) with respect to y, we have fU IS(Q)(t)12 dt
-U
1
<<
M k-1 NR21og2 M
(Q)
F,
i
L E( L)
Q (LM)
2L-1 X f K(77)1
12
b(1)e(gj(1) + 711)
d07
[=L
r N2R M 1/3
+B2Mk log3 M M2Q
QR
Q3 M2R4 R3 R3 N4 + NQ I I
which gives the result of the lemma.
Lemma 10.6.2 has the sum over 1 inside the mean square. We use two lemmas on matrices. Lemma 10.6.3 (Bilinear form duality) Let A be a complex matrix, and A* its complex conjugate transpose. Then the square matrices AA* and A*A have the same non-zero eigenvalues.
Proof If both
w=Av
A*Av=Av, are non-zero vectors, then
AA*w=AAv=Aw. Lemma 10.6.4 (Gershgorin's bound) Each eigenvalue A of a square matrix C satisfies
IA -
E Ic,jI.
i#, Proof Let i be the row in which the corresponding eigenvector v has its largest entry. Consider the ith row of Cv. Lemma 10.6.5 (Dualized large sieve) 2L-1 F(Q) f1/2
i
-1/2
K(n)
In Lemma 10.6.2, for a family of sums, 2
b(1)e(g,(1) + ql)
d,7
!=L
(LM)114
(J(Q+ E VBa(O1(V ), ..., A4(V ))) + L
< < B2Lk loge L
V
+
Qlog I
L
min ly, -y1l
M
10j
Jutila's third method
225
where V is summed overpowers of 2 as in Lemma 10.5.2, and B0(01, . , 04) is the maximum over j of the number of pairs i, j counted in B'(i1, ..., 04). For a single sum we omit the term in log I.
Proof Let A be the J(Q) X L matrix corresponding to the sum in Lemma 10.5.2 or 10.5.3 with e(g,(1)) in row j, column 1. Then F(Q)
2L-1
2
F, b(1)e(gj(l) + r11) [=L
_
a,[,aj1 5A
b(11)e('711)b(l2)e(-7112) [,
Ib(l)12, [
1
12
where k is the largest eigenvalue of AA*. By Lemma 10.6.3, k is the largest eigenvalue of A*A, and by Lemma 10.6.4
A s max L 1
e(g[(l) -g,(1))
[
which we estimate as in Lemma 10.5.2 or 10.5.3. The bound is uniform in q, so we may integrate IK('q)I to O(log L).
10.7 JUTILA'S THIRD METHOD The most powerful form of Jutila's method for the mean square (1990a) uses the large sieve of Lemma 5.6.6. As in Section 2.4.5, we need some heavy gardening. The reward is to make the parameters N and R independent, so that we can choose better ranges. We use partial summations repeatedly.
Lemma 10.7.1 (Partial summation in a modulus squared integral) Let H(x, y) be differentiable for M< x5 N, 0 S y 5 Y. Then for any coefficients G(n, y) we have 2
fY N
F, G(n, y) H(n, y)
0
M
Y
dy5C sup
MSKsN 0
2
K
F, G(n,y) dy, M
where
C=2supIH(n,y)12+2(N-M)sup JMIH1(x,y)12dx. y
Y
N
Proof As in Lemma 5.2.1, we start from a partial integration identity: N
N
M
N
X
g(N, y)H(N, y) - I H1(x, y) F, G(n, y) dx.
G(n, y) H(n, y) _ M
M
M
The lemma follows on squaring and applying Cauchy's inequality.
Exponential sums with modular form coefficients
226
We write (3m = F(m/M), and consider the sum M2
S(t) = E b(m)g(m)X(m)e((3mt), M
where g(m) is a smooth weight function with g'(m) << 1/M, and X(m) is a smoothed characteristic function, zero for m (M or m )M2, rising smoothly to one within a short distance of M and M2, of the type used in Lemma 10.3.1. Lemma 10.7.1 lets us consider sums of the simpler form M2
S(t) _ E c(m)e(/3mt),
(10.7.1)
M
where c(m) = b(m)X(m).
Lemma 10.7.2 (Short interval mean) Given U in 1 < U5 M, there is an integer H in
M/4C1U - 1 < H< C1M/2U + 1,
(10.7.2)
where C1 is the constant of (10.3.2), with T+U IS(t)12
fT-U
U2
m+H
dt << - F,
I
I c(n)e( /3,,T)I
M m n=m+1
2 .
(10.7.3)
Proof Lemma 5.6.4 gives fT+oIS(t)12 dt << AF, E c(n)c(m)e( 13,,T - /3mT)A(2A( Nn -13m)).
T-a
m
n
(10.7.4)
The expression on the right of (10.7.4) is itself a modulus squared integral, by Parseval's theorem for Fourier transforms: 2
-f
c,,e(/33T)I dy,
S
where S = 1/20. We put y = F(x/M). Since F'(x/M) is bounded, the right-hand side of (10.7.4) is 2
2
<< M f I
dx, n
where the sum is over n with
F(x/M) 5 F(n/M) 5 F(x/M) + S, a range of the form x 5 n 5 x+ z with
z = S/F' ( )
Jutila's third method
227
for some . We now average A over U5 A <- 2U, so z is given implicitly by
F(xMz) -F(M)
20'
and for x fixed
x+z
1
MF( M
1
dA
2 dz
)
Thus the integer in (10.7.3) has order U2 <<
M
2 dxd0
2u
ff C/
VU
x5n5x+z
U3
2
<< M2 f f I xSnSx+z
c(n)e( /3nT) I dx dz,
where z is integrated over a subinterval of the range M/4C1U 5 z S C1M/2U.
The integral over z can be bounded by the length of the range, O(M/U), times the supremum over z. For the value of z at which the maximum occurs, the sum over n has either [z] or [z] + 1 terms. Hence the lemma holds with
Hgivenby[z]or[z]+1. We divide the range for m into intervals Ij with length N, where
M/Ux2H5N<< MU/T.
(10.7.5)
In each interval I, we pick a value m1 of m, and we put
a
M
F
I(M
Let Xj(m) be a smooth weight function with 1
X1(x)
0
forx=m+y,m in I105y5H, forIx-mil>2N,
and
Xj(')(x) << 1/N'
for r 5 s. We put ca(n) = c(n)X,(n). Lemma 10.7.3 (linearizing locally) For each a > 1, let K(a, a) be the kernel 2
a
K(a, a) =I
e(ah) h=1
Exponential sums with modular form coefficients
228
Then for each interval Ij, there is a positive integer Hj < H such that T+UIS(t)12
fT- u
2
2
dt << M
ci(n)e(an) I K(Hj, a- aj) d a. (10.7.6)
f1I
Proof We use Lemma 10.7.1 with G(h, m) = c(m + h)e((3mT + ajh), H(h, m) = e( ,6m+hT - (3mT - ajh), corresponding to
+y
H(x,y)=e(TF(x
)-TF(
M
x M)-ajY),
with
IH1(x,y)I=2Tr
T
MF'(x
+y
21rT N+H
5
)-aj
M M by (10.7.5). Hence, for some Hj s H, we have U22
T+U
fT_U
dt <<
M
maxIF"(x)I<< 1/H
2
Hj
L ME j mE
F, c(m + h)e(ajh) h=1
In the inner sum c(m + h) = cj(m + h), so we have H 2
c(m + h)e(ajh) < meld h=1
m h=1
2
c(m + h)e(ajh) Hj
Hj
_ F, F, cj(nl)cj(n2)e(ajnl - ajn2) E E 1 h,=1 h2=1
nl n2
nl-hl=n2-h2 Hj
Hj
cj(nl)cj(n2) E F, e(ajhl - ajh2)
_ nj
h,=1 h2=1
n2
hl-h2=n1-n2
2
= f 1 I L cj(n)e(an) 0
Hj
2
E e(ah - ajh) da, h=1
which gives the lemma.
We summarize the gardening steps. We can take the range of integration
in (10.7.6) to be aj - i to aj + '-2, and replace K(Hj, a - aj) by a bounded multiple of
Kj(a) = min(H2,1/IIa- ajII2).
Jutila's third method
229
Let Do run through the numbers 1/2' with
252'=1/Do
(10.7.7)
Then Kj(a) < 1/Do implies I a - ajI S Do, and we deduce from (10.7.6) that 2
fT
2
n-° II da. (10.7.8)
E fa'
I S(012 dt << M E
j
Ao OZ0
n
o
In thisvform of the method, the parameter R is independent of N. We suppose that
H
(10.7.9)
By Dirichlet's approximation theorem (Lemma 1.5.1) for each a there is some fraction a/q with q
a= q
+(3,
1
Il3I< qR.
We pick a smooth weight function (o(x) with 1
for IxI <- 1/R,
0
forIxIz2/R
and
dr)(x) << 1/Rr for r < s. We have fa,+n°
ct(n)e(an)12 da a,-o° I E n <<
/an
E Ia/q-a1I52G°
f I E ct(n)eI `q n
l2 I w(a) d/3; 1
+ /3n I /
(10.7.10)
q
here we have used the condition H < R in (10.7.8). We apply Lemma 10.2.6 to the sum over n, replacing the interval M :!g n < M2 by the support of Xj(x), whose length is O(N), with g(x) = X,(x)e( /3x),
f Ig(')(x)ldx <
(10.7.11)
Exponential sums with modular form coefficients
230
The condition (10.7.11) is not critical, because Lemma 10.2.6 can be sharpened by taking more terms of the Stokes expansion in Lemma 10.2.5. We have fT+UIS(t)12 dt<
K
1
M Do
0
i
la/q-a Is2Go
f
x(k1)/2 r dl ci(x)Ree --+ <(-l) q
1
b(l)f
(lxg2)1/4
1=1
2
(lx) +(3x+8
dx
q
We use Cauchy's inequality to divide the sum over 1 into ranges L< 15 2L, and we divide the sum over a/q into blocks Q S q5 2Q. We expand out the modulus squared, destroying the symmetry, and consider 2L-1 2L-1 1 xy (k-1)/2
f
1
=L
n
=L
b(1)b(n) f f
M
q (lnxy)
2 (lx)
do
XeI - q + q
2
q
ci(x)cj(y)
(1n)
(/31
(ny) q
do. (10.7.13)
Integrating repeatedly by parts gives (non-uniformly in s)
fe(/3x-E3y)w(g)dR<<
(I
-yl
(10.7.14)
We work to an accuracy Ts. If s >- 2/8, (10.7.15) Ix -yI z QRT5, then the integral in (10.7.14) is O(1/T2) (with constant depending on S), which is quite negligible. Hence terms in (10.7.14) which satisfy (10.7.15) can be absorbed into the error term. We write
2 410 e
q
-2
2V
(ny) e
q
(
q
(f-V )+
2y q
(v - Yy)
When we take x and V as new variables of integration, then we find (with more difficulty) that the terms with I1-nI z
Q
(LM) Ts
can also be absorbed into the error term.
(10.7.16)
Jutila's third method
231
We can now set up a large sieve (Lemma 5.6.6) with
x(1, n) _ (1- n, f - v, F (a
a
(
yq>x,Y)=I- q,
2V 2 q
,q(v -v
and
X=
(min(io(
l
(LM) T8))l,min Y= 1,0(
(2L) O Q
/(jT6)v1(2L))
Q ,O(RTS)
1
The coincidence conditions corresponding to the third entries of the vectors x and y are almost trivial. For HN << M2/T,
H «R2,
(10.7.17)
the coincidence conditions take the forms (10.5.12) and (10.5.13). We take R2 << NT-S,
so that the parameter that corresponds to N in Chapter 7 is our H, not N.
This method leads to a mean square result for S(t) itself, not just the components S(Q)(t).
Part III The First Spacing Problem: `integer' vectors
11
The ruled surface method 11.1
PREPARATION: DIVISOR FUNCTIONS
Let h be the integer vector (hl,..., h,,), and let X(h) =x(h1) + ... +x(h,).
The conditions for coincidence between the vectors x(h) and x(g) can be written as gi+...+gr
=hi+...+hr,
gi12 + ... +gr/2 = hi/2 ..... +h;/2 + O(SH3/2), g1+...
+gr=h,+... +hr,
gi/2 + ... +gi/2 = h1/2 + ... +h;/2 + O(AH1/2). We suppose that all the variables are contained within a range
HSg,,hi52H-1.
(11.1.5)
For the spacing problem in the large sieve of Chapter 6 we have r = 3, 4, 5, or
6 and A = SH. We need to treat S and A as independent in some of the proofs. We write N2r(5, A) for the number of solutions of the equations (11.1.1)-(11.1.5).
The methods of this chapter are mostly elementary number theory. We want to count the number of integer solutions in bounded regions for certain sets of equations and inequalities. The difficulties are partly algebraic, partly analytic. A system of polynomial equations defines an algebraic variety in affine space. The geometry is dominated by two integers, the dimension and the genus of the variety. The size of the set of integer solutions is usually much smaller than the dimension of the variety would suggest. In the case of projective curves, there are only finitely many integer solutions unless the genus is zero or one (Faltings 1983) (an integer point on a projective curve corresponds to a rational point on an affine curve). Dimension and genus are not sufficient to determine the number of integer solutions. The following examples are all parametrizable curves of genus zero.
The ruled surface method
236
We give the number of solutions in the box Ixl
1. straight line y = x: 2. parabola y =x2: 3. parabola 3y + 2 = x2: 4. circle x2 + y2 = N2: 5. hyperbola xy = N: 6. hyperbola x2 -2 y2 = 1:
N, I yl N for large N:
x N solutions;
x V solutions; no solutions; O(NE) solutions for any e > 0; O(NE) solutions;
x log N solutions.
In practice, the nature of the problem seems to change as soon as we add an inequality. The algebraic geometry becomes less relevant. It may be that most solutions of the inequality satisfy not just the given inequality, but a
corresponding equation, and they lie on an algebraic variety of smaller dimension. Many divide-and-conquer arguments go from an inequality to an
equation in this way. For the equations (11.1.1) - (11.1.4), if a and 0 are small enough, then we expect the diagonal solutions to predominate; these are the solutions with g1,. .. , g, equal to hl,..., h, in some order. We start with two variables. Let d(m) denote the divisor function as in Section 2.1 of Chapter 2, the number of ways of writing m as a product of two positive integers. Lemma 11.1.1 (Binary quadratic forms) Let n be a positive integer. Then the number of pairs of integers x, y (of either sign) with x2 - y2 = n is 2d(n) if n is odd, 2d(n/4) if n is a multiple of four, and zero if n is twice an odd integer. The number of pairs of integers x, y with x2 - xy +y2 =n is
6 E X(d), din
where X(d) is the Dirichlet character mod 3 taking the values X(d) = 0, ± 1 according to X(d) = d (mod 3). Corollary If a and b are integers with a2 < 3b, then the number of triples of integers hl, h2, h3 (of either sign) with
hl+h2+h3=a, h; +h2 +h3 =b is
(d
(6-3X(a2))
XI deven,
(11.1.6)
dl(3b-a2)
Proof If x2 - y2 = n, then x + y = e, x - y = d for some integers d and e of the same sign, and either d or e both even, or d and e both odd. Conversely, each such pair of d and e gives a solution. When n is odd, then any pair of integers with de = n gives a solution. When n is even, then if de = n, either d or e must be even. If n is twice an odd number, then the other factor must be
Preparation: divisor functions
237
odd, and we cannot construct a solution. If n is a multiple of four, then any pair of even integers d and e with de = n gives a solution. The equation x2 - xy + y2 = n is treated in the same way in principle, but the factorization is x + py = 6, x + p2y = e, where p is a complex cube root
of one, with p2 =1- p, and 6 and a are algebraic integers of the form a + bp, where a and b are integers (see Hardy and Wright 1960, Chapter 12).
The lemma contains the theorem that every prime number of the form 6m + 1 can be written as x2 - xy +y2. For the corollary we eliminate h3 to get
2h; + 2h2 + 2h1h2 - 2ah1- 2ah2 = b - a2, r and we note that the substitution
x=h1+2h2-a,
y=h2-h1
(11.1.7)
makes
x2 - xy +y2 = 3hi + 3h2 + 3h1h2 - 3ah1 - 3ah2 + a2 = (3b - a2)/2. (11.1.8)
Conversely, if x and y satisfy (11.1.8), and if a is a multiple of three, then (x +y)2 is a multiple of three, and so is x +y, and we get integer values for h1 and h2 from (11.1.7). If a is not a multiple of three, then
2(x +y)2 - -a2 = 2a2(mod3). Half the solutions have x + y = a, and the other half have x + y = -a. Only the solutions with x + y = - a give integer values for h and h2 in (11.1.7). 1
We deduce the formula (11.1.6).
In the next few lemmas we see that the sum over d in Lemma 11.1.1 is not too large, especially if we can average over n. Thus when h4, .... h 91, ... , gr have been chosen in (11.1.1) and (11.1.3), then the last three variables take
care of themselves. We need the more general divisor function d,(n), the number of ways of writing the positive integer n as a product of r positive integers. We give the standard bounds. The averages in Lemma 11.1.3 may be replaced by asymptotic formulae. The leading terms in the asymptotic formu-
lae may be obtained by elementary arguments as in Lemma 11.1.3, or by counting lattice points as in Chapter 2, or by using the Dirichlet series generating function
d,(n) ns
Lemma 11.1.2 (Bounds for divisor functions) We have
d,(mn) 5 d,(m)d,(n),
(11.1.9)
with equality if the highest common factor (m, n) is one, and
d,(n)ds(n) S d,s(n).
(11.1.10)
The ruled surface method
238
For any S > 0, there is a constant B(r, 8) with
dr(n) SB(r, 6)NS.
(11.1.11)
Proof If m = k1k2 ... kr,
and n = 1112 ... lr, then mn = g1g2 ... qr, with q1= k,!1. If (m, n) = 1, then q, determines k. _ (q,, m) and 1, = (q,, n). If (m, n) > 1, then there are several sets of factors k1,.. . , kr and 11, ... , Ir which give the same products q1, ... , qr. This proves (11.1.9). If n = k1k2 ... kr = 1112 ... ls, then we construct rs numbers q1j whose product is n as follows. For any factor e of n, we put ml(e) = (k1, e), and, for i > 2, m1(e) = (k1k2 ... k1, e)/(k1k2 ... k;-1, e). Then q,1 = m,(11), and, for j > 2, q,1= m1(1112 ...11)/m,(1112 ... 11_ 1).
The fraction q,1 has the form
(kk',Ii')(k,1) (k, ll')(kk',1)
with k=klk2
k1_1, k' =k1, 1=1112
1i_1, 1' =1,. Let d=(k,l), k=du,
1= dv. Then (u, v) =1 and
(uk', vl') (uk', vl') q11= (u, vl')(uk', v) (u, l')(k', v) ' which is an integer. It is easy to check that the rs numbers q,, determine
kl,...,kr and When n is expressed as a product of powers of distinct primes
n=pSI...pkk, then
dr(n) = fl dr(p,'). For p a prime, dr(ps) is the same as the number of ways of typing a sentence
of s words on paper with r lines. This is the number of ways of typing r + s - 1 things, each either a word or a carriage return, with s words and r - 1 carriage returns. Thus d"(ps) _
(r+s - 1! <2 r+s-1 (r - 1)!S!
If P'> 2", then, for all s > 1, dr(ps) <
2r+s-1
< 2rs Spsb
There are finitely many primes with ps < 2", and for each of these
(r+s - 1)!
1
(r - 1).s.
pss
239
Preparation: divisor functions
as s - oz. Hence for these primes d,(ps)/psa has a maximum at some integer sp z 0. We have dr(n) d,(ps,) = S), s 5 as, n
pv
p>2'/6
attained when
n= F1 psr.
o
pG2'/b
Lemma 11.1.3 (Averages of divisor functions) we have
For r and N positive integers
N
L dr(n) < N(log N + 1)r-1,
(11.1.12)
1
N d(n)
L r nS(log N+1)r.
(11.1.13)
For r = 2 (the Dirichlet divisor problem) we have
IN\
N
Ed(n)=N logN+(2y-1)N+2 E pl dJ+0(1) d<1N
1
=NlogN+(2y- 1)N+O(iR).
(11.1.14)
where y is Euler's constant.
Proof First we note that N
1
L -51+ fj
e=1 e
Ndx x
1
We have N
N
F, dr(n) = L L...L1 n=1 el
1
e,
el ...e,= n
N
N
E
...
L
E
1
e,-I=1 e,SN/el...e,_I
e1=1
N
N
N
e,_1=1
el...er-1
5 F, .. L e1=1
SN(logN+1)r-1
Similarly N
L 1
d(n) r n
N
= L L...L n=1
(N 1
1
el
e,
el...e,= n
e1e2...er
1
1
e
'
The ruled surface method
240
For the asymptotic formulae we use a divide-and-conquer idea of Dirichlet. We arrange el,... , e,. in order of increasing size (with a subdivision of r! cases) and sum the largest first. For r = 2 the method is quite palatable. We want to count the number of pairs of integers e, f with of < N, which is
E
E
e5FN e
=2er
E 1+ E E 1
1+ L
fyf<eSN/f
e:5 FN f=e
/
1
([1_e+)=2ee P(e)+2 e FI e -eJ.
Let M = [/]. Then, by the trapezium rule (Lemma 5.4.1)
2eE (e -e1 =N-1+M-M+2 fMi x -xdx+4 fM /
N3x)
dx
11
=N+2 j +4N f
x
( N
N
o' (x)
x
1
3
I
/N
dx+Ol
3 x
=NlogN+4IN+O(1), where I is the integral
I=f
o(x) x33
dx.
1
The simplest sum involving I is
e=2-+2M+Mfdx-+2fM v(x) x3 dx
M1
1
1
=logM+2I+1/2+O(1/M). This is the limit which defines Euler's constant of integration y (about 0.5772). Thus I = y/2 - a . 11 In counting arguments we often have to add up divisor functions at the integers in some set. If the set is fairly dense in an interval, then the divisor functions average to logarithm powers. If the set is thin, then it is better to use the number of elements in the set multiplied by the maximum of the divisor functions. This leads to a bound with an exponent e which may be
taken arbitrarily small, at the cost of increasing the order-of-magnitude constant. The aim in this account is to average divisor functions where possible, but not to put much work into minimizing the resulting logarithm power. To show what can be done, however, we quote without proof a result of Hall and Tenenbaum (1986, Theorem 3) on Hooley's divisor concentration
Families of solutions
241
functions. Let ',(n) be the maximum over (real) E1,..., E, of the number of ways of writing
n=e1e2...er with the factors in ranges
E,<e,<2E; for i = 1, ... , r. This corresponds to knowing not only the product of r numbers, but also their individual orders of magnitude. Since there are fewer possibilities for the factors, the logarithm powers in the averages of Or(n) are
smaller than in the corresponding results for dr(n). Lemma 11.1.4 (Average divisor concentration) For t z 1 a real number and N large we have
(ti(n))` <
N)r'-rr+c-1+s
n5N
where, for fixed r and t, log log log N 1/2 <<
log log N The implied constants depend on r, t, and S.
11.2 FAMILIES OF SOLUTIONS The equations (11.1.1) and (11.1.3) correspond to the algebraic variety V with equations +Yr, (11.2.1) +xr =Y1 + x1 + X1
+x; =Yi +
+y
(11.2.2)
in 2r-dimensional affine space, a quadric generalizing the hyperbola in two dimensions. Since the defining equations are homogeneous, the variety V is a cone with vertex at the origin: if (x1, ... , xr, Y1, ... , y) is a point on V, then so is (t1,. .. , ty,) for any t. Thus any integer multiple of an integer point on V is also an integer point on V. More usefully, V is a cylinder with axis along the
vector (1,1,...,1). If (x1,...,xr,y1,...,Yr) is a point on V, then so is (x1 + t, X2 + t, ... , Yr + t) for any t. This construction forms the basis of Watt's counting method (1989a, b).
The tangent hyperplane to the hypersurface (11.2.2) at a general point P (x1,...,Yr) intersects the hypersurface in an r-dimensional plane. The variety V is the intersection of the hypersurface with a hyperplane. There is an (r - 1)-dimensional plane through P lying in the variety V. We have found two lines in this plane, one joining P to the origin, the other in the direction of (1, 1,-, 1). For r >- 4, there are more lines through P in linearly independent directions. We are lucky to have a rational line in the direction of a
The ruled surface method
242
vector with bounded height. There are inequalities of Minkowski (see Cassels 1959) relating the heights of the coefficients in the equations for a plane to the heights of the coordinates of a set of basis vectors for the plane. If one
basis vector is short, then the geometric mean of the heights of the other basis vectors will be large. Our first two lemmas use these vector methods. Lemma 11.2.1 (Three each side) When r = 3, then the number of solutions of (11.1.1), (11.1.3) and (11.1.5) is
< 14H3(log H + 3).
Proof We put k.=g1+h1 Then
(1,1,1) 1=11+12+13=0,
Case 1 The vector 1 is zero. There are H choices each for hl, h2 and h3, which are equal to g1, g2 and g3, so H3 solutions in this case.
Case 2 The vector k is parallel to (1,1,1), so k1 = k2 = k3 = k, say. Since 1= k (mod 2), we have 3k ° 0 (mod 2), and k, 11, 12 and 13 are all even. There
are at most H possibilities each for k, l1, and l2, which determine 13, so at most H3 solutions in this case. Case 3 The vector I is non-zero, and k is not parallel to (1, 1, 1). Then 1 is parallel to the vector cross product
(1,1,1) xk=(k3-k2,k1 -k3,k2-k1). Let d be the highest common factor of 11, 12, and 13, and let 1. = dm1. Then, for some non-zero integer e, we have k3 - k2 = em1, k1 - k3 = em2 k2 - k1 = em3.
There are at most 4H choices each for m1 and m2, which determine m3. When m1, m2, and m3 are chosen, then there are at most H/M choices for d and 2H/M choices for e, where M = max (m11. When k1 has also been chosen, then we know all six original variables.
When M has been chosen, then M= ±m1, ±m2, or ±m3. If M= -m1, say, then m2 + m3 = M. The number of solutions in case 3 is at most 4H
6E
M 2H2
M=1 m2=0
MZ
4H M+1
H= 12H3 E
M2
M=1
< 12H3(log H + 1 + x,2/6) < 12H3(log H + 3).
Families of solutions
243
Lemma 11.2.2 (Equal sums of three squares) The number of integer solutions
of 91 +g2 +g3 = hi +h2 +h3
(11.2.3)
with
0
O(H4).
Proof We put
ki=gi+hi,
li=gi - hi.
There is one solution with gi = hi = ki = 0 for all i. For the other solutions we let d be the highest common factor of k1, k2, and k3, and put ki = dmi. We consider m = (m1, m2, m3), an integer vector of length M_, where
M=mi +mZ +m3. The condition (11.2.3) becomes k111 + k212 + k313 = 0,
so that if I is the vector (11,12,13), then
We consider the vector m to be fixed. Let A be the lattice generated by m and the integer vectors I with I m = 0. If r is a vector of A, then for some integer t. Conversely, if r is an integer vector with M I r in, then r
is in A. Since the highest common factor of ml, m2, and m3 is one, then
there is an integer vector r with r in = 1. We deduce that r in takes all integer values. Thus A has index M in the full integer lattice, and the parallelepiped formed by the basis vectors of A has volume M. One basis vector is m of length VM-. The other two basis vectors, u2 and v3 say, are at right angles to in, and they form two sides of a parallelogram of area V. These vectors v2 and v3 span a two-dimensional lattice A2. The integer vector I lies in the lattice A2, and it has entries at most H in modulus. In order to apply Lemma 2.1.1, we make a change of coordinates in the plane of A2 so that v2 and v3 are the unit vectors. The vectors with IliI
the origin. If v is any non-zero vector of A2, then (H + 1)v has some coordinate numerically greater than H, and the image of (H + 1)v lies outside S. We see that S has diameter at most 2(H + 1), and perimeter O(H). By Lemma 2.1.1 the number of integer points in S is (
Or
Hz
+H
.
These integer points correspond to vectors I in A2 with Ilil 5 H.
The ruled surface method
244
For each power of 2, R say, there are O(R3) integer vectors m with
R2 <M=MI +M2 +M2 < 4R2. The number of choices of d, m and 1 is
0 d5H RSH/d
r H2
l
`
l
R3I R +HI
,
where R is summed through powers of 2. This sum is
2 H2 = O(H4). H2
0
dsH d
Lemma 11.2.3 (Families of solutions) The integer solutions of (11.2.1) and (11.2.2) form families: lines parametrized by
xi= ,+t,
y,=rj,+t,
(11.2.4)
1, ... , , and -q1, .... rt, are fixed, and t is variable. When r = 4, then the number of families which include an integer solution 91, ... 194, hl,..., h4 in the range (11.1.5) is O(H4).
where
Proof We verify that if
1, ... , .... , rt, is a solution, then (11.2.4) also gives a solution. Now suppose that r = 4, and that g1, ... 194, hl,..., h4 is an integer solution. We put t = -(g4 + h4). Then 2h4 - t = -(2g4 - t), and 3
3
(2g, - t)2 = 1
(2h1- t)2, 1
with 12g, - t1, 12h, - tI 5 2H. By Lemma 11.2.2 with H replaced by 2H, there are O(H4) possibilities for the integers 12g1- t1, 12h, - tI with i = 1,2,3. For each of these, there are at most 64 ways of choosing the signs. Since 3
3
(2g;-t)-
2h4-g4= 1
(2h1-t), 1
we know 2g4 - t and 2h4 - t and g1, ... , g4, hl,..., h4 are determined up to the additive constant t.
Our aim is to show that an average family contains a bounded number of integer solutions of (11.1.1), (11.1.2), and (11.1.3). Some families contain many integer solutions, such as the families with g, = hi for each i. We must count how many families give several integer solutions.
-
Lemma 11.2.4 (Families with many solutions) Suppose that 2 < r 4, and that g1 + t, ... , g, + t, h1 + t, ... , h, + t gives a family of solutions to (11.1.1) and
Families of solutions
245
(11.1.3). Suppose that this family contains L solutions to (11.1.2) in the range (11.1.5), where L>3-22r.
Then for each i = 1, ... , r we have r
r
FI (gi - hj),
j=1
H (h, -gj) << nHr,
j=1
(11.2.5)
with
'i1 « SH/L. If the L integer solutions of (11.1.2) also satisfy (11.1.4), then (11.2.5) is true with
ri << min(SH/L, 0).
Proof We write
=xi + ... +x; -y;
DS(x1,...,
-Yr, where the exponent s is not necessarily an integer. The equation
D3/2(g1 +t,...,gr+t,h1 +t,...,h,+t) = a gives rise to a polynomial equation in t of degree at most 3.22x. Hence the solutions of the inequality ID3/2(g1
+t,...,g,+t,h1 +t,...,hr+t)I S SH3/2
form at most 3.22r intervals on the real line. When L > 3.22', then some interval I of t corresponds to T + 1 integer solutions, where T is a positive integer with
TEL/3.22x- 1. We substract a suitable integer from each of g1,. .. , g, hl,..., h, to normalize the family so that I includes the interval 0 5 t S T. This does not affect the differences g, - h j. By the mean value theorem applied to the interval 0 5 t S T, there is a real number T in 0< T< T for which Z TD1/2(g1 + T, ..., gr + 'r, h1 + T, ..., h, + T)I <- 26H1/2 .
We write x1 _
(g, _+T), y; _
(h; _+T). Then, for s = 1,..., r, we have
xi+...+x;=Yi+...+Y:+O()Hs/2), where
4SH
= maxi S, 3T)
4SH
3T '
since T 5 H - 1. If the solutions in I also satisfy (11.1.4), then we can also take
r = max(8, A).
The ruled surface method
246
We denote the elementary symmetric function of r variables by
Pl(xl,...,xr) =x1 + ... +Xr ..., Pr(x1,...,xr) =x1...xr. By Newton's formulae for the sums of powers of roots of equations, we see that Ps(xl,..., xr) =Ps(yl,..., yr) +O(71Hsl'2)
for s = 1, 2, 3, 4. Since r
r
xr+ E
(-1)3Ps(y1,...,yr)xr-s= fl (x-y;),
I=1
s=1
we deduce that r
r )=1
(-1)sPs(xl,...,xr)x;-s+O(-7Hr/2)
(x;-y) =x; + E s=1
1
r
_ F1 (xi -x)) + O(-qHr/2) =
O(-qHrl'2).
)=1
Since x; > Y1H--, we have r )=1
r
($t - h))=
rj (x2 - y?) = O(?1Hr). )=1
Interchanging the roles of x; and y) gives the corresponding inequality with C] g, and h) interchanged.
We can now deal with the case r = 4. This account is based on Watt (1989a). We give the simpler argument which loses a factor H. The proof has a divide-and-conquer structure. We divide the integer solutions into families, and count the number of families which have many solutions. This brings in another important idea, the Riesz interchange, which is the discrete analogue of the transformation
f13If(x)Idx= f/3 a
suplfxI flf(x)Idydx= fy=o
x=a y=o
f
dx dy,
where S(y) is the set of values of x for which I f (x)I ? y. This idea is useful when we can describe the set S(y). Theorem 11.2.5 The number of solutions of (11.1.1), (11.1.2), and (11.1.3) with r = 4 in the range (11.1.5) is
O(H4 + 6Hs+E) for any e > 0. The implied constant depends on e.
Proof By Lemma 11.2.3, the solutions fall into families. There are O(H4) solutions belonging to the families that contain at most 768 solutions.
Families of solutions
247
Families with more than 768 solutions are divided into blocks according into which range L to 2L - 1 the number of solutions falls, where L = 21 is some power of 2. We note that there are at most 24H4 trivial solutions in which 91, ... 194 form a permutation of h1, ... , h4. If (after renumbering) we have g4 = h4 and g3 = h3, then the solution must be trivial, since
gi
+g2 hi + h2, =
g1 + 92 = h1 + h2,
and so g1g2 = h1h2, and
(h1 -g1)(h1 -g2) = h; - (h, +h2)h1 0. Then either h1= g1 (whence h2 =g2), or h1=g2 (whence h2 =g1). If one solution in a family is trivial, then all solutions are trivial. Hence we can refer to trivial or non-trivial families. We estimate the number of non-trivial families with at least L solutions of
(11.1.3). We can normalize the family (renumbering if necessary) so that 0,g,z0,h4=0. By Lemma 11.2.4 with r = 4, 91929394
<<,nH4
(11.2.6)
with 71 << SH/L. Moreover, if one of the gi is zero, then after renumbering we can suppose that g4 is zero, and then Lemma 11.2.4 with r = 3 gives 919293 << 77H3.
(11.2.7)
By Lemma 11.1.3, the number of sets of positive integers g1, ... 194 satisfying (11.2.6) is
0( E d4(n) I = O('qH4 log3 H). I
nc71H4
The number of sets of/ positive integers g1, g2, and g3 satisfying (11.2.7) is T, d3(n)) = O('qH3 log2 H). nanH3 There are also O(H2) triples satisfying (11.2.7) for which one of g1, g2, or g3 is zero. When 911 ... , g4 have been chosen, then
01
h;+h2+h2=9 +2+2+92 h1+h2+h3=g1+g2+g3+g4' Let
F =F(g1,..., 94)
.21 (g1 + ... +g4)2 - i (g1 + ... +g4 )
=Eg2- EF, gigi i
i
i
(11.2.8)
248
The ruled surface method
By Lemma 11.1.2, if F(g1,...,g4) is zero, then there is one choice for hl,..., h4- If F(g1,...,g4)>0, then the number of solutions for hl, h2, and h3 is
< 6 E X(d) < 6d(F) <<
HE/2.
dIF
If F is negative, then there is no solution. The number of families containing at least L solutions is thus O(HE(H2 + 77H 4)) z O(H2+E/2 +
5H5+e/2/L).
Summing L through powers of 2, we find that the number of such non-trivial families is
E LH2+e/2 + 5H5+E/21 L=2'«H f = O(H3+E/2 + 5H5 +e/2 log H).
O(1:
We get the result on adding the number of trivial solutions, and using He12 log H << He.
Finally we give a limitation result, which shows that the last estimate is not too far from best possible. Lemma 11.2.6 (Limitation result) We have SOH2r-3
N2r(S, A) >_ maxr!(H-
r)r,
3(1 + 8)(1 + A)r4
Proof We can choose h1,. .. , hr to be distinct in
H!/(H-r)!> (H-r)r ways, and then have g1, ... , gr a permutation of h1, ... , hr in r! ways; these are diagonal solutions. For the second bound, we divide four-dimensional space into boxes B,,, and we let n; be the number of vectors x(h) (in the notation of Lemma 7.4.2) in
each box. Two vectors (distinct or not) falling into the same box give a solution of (11.1.1)-(11.1.5). Let I be the number of boxes. Then Cauchy's inequality gives (Ln,)z
with
L n, =Hr,
L n? 5 N2i(3, A).
The boxes have length 1 in the first and third coordinates, 6H3/2 in the
A comparison argument
249
second coordinate, and AH1/2 in the fourth coordinate. Since the sth coordinate runs through a range of length (2s/2 - 1) rHs/2, we have r(V - 1)H3/2 1 1)H1/2 I < 3rH2 + 11 rHl + 11 SH3/2 AH1/2 S)`(1
(r(/ -
<
3(y - 1)(1- 1)r4H3(1 +
+ A)
SA
which gives the result. 11.3
A COMPARISON ARGUMENT
We want to compare the numbers N10(S, A) for different S and A. First we
need to bound the number of families in which each gi is close to the corresponding hi. Lemma 11.3.1 (Model families) Let PS(x1,... , xr) denote the elementary symmetric function of degree s, the sum of all products of s different variables x, chosen from x1, ... , xr. For a given family of solutions, define the model family with respect to g5 as the family of solutions (not necessarily real) containing Kl,... , K41 g5, hl,..., h5, where Kl,... , K4 are the unique complex numbers satisfying
(11.3.1) Pr(K1,..., K41 95) =Pr(hl,...,h5) for r = 1,-, 4. The closeness of the model is defined as min I K; - hiI for
i, j = 1, ... , 4. The model families with respect to the other g, and hi are defined
similarly. Then, for d > 0, the number of families containing solutions of (11.1.1), (11.1.3), and (11.1.5) for which g, # hj for any i and j, and
I Pr(g1,...,g5) -Pr(hl,...,h5)1 < SHr (11.3.2) for r = 3, 4, 5, with some model family of closeness at most d, is of order O(SdH5+E) (11.3.3) for any e > 0, with constant depending on e. Proof Equation (11.1.1) implies (11.3.1) with r= 1, and (11.1.1) and (11.1.3) together imply (11.3.1) with r = 2. Since 5
(x - a1)...(a - a5) =x5 + E (-1)rxs-r(a1,..., a5), r= 1
then we have
I(x-h1)...(x-h5) - (x-g1)...(x-g5)I S7SH5 for HS x < 2H. In particular 1(g; -hI)...(g; - hs)I-<7SH5, I(h1-g1)...(hi -95)1:!g 76H5,
The ruled surface method
250
for each i and j. Since we are counting families, not individual solutions, then we can take gs to be fixed. The number of possibilities for hl, ... , hs is
F,
d5(n) << SHs log4 H << SHs+e/2
n573H5
by Lemma 11.1.3. The numbers K11 ... , K4 in the model family are now determined. If g4 is to be within a distance d of one of the numbers Kl,... , K41 then there are at most 4(2d + 1) choices for g4. The conditions (11.1.1) and (11.1.3) fix g1 +g2 +g3 and gi +g2 +g3. By Lemma 11.1.1, the number of possibilities for g1, g2, and g3 is at most 6d(n) for some integer n = O(H2), and so at most O(H/2) by Lemma 11.1.2. This model accounts for O(SdHs+e) families, and the other models differ only in the numbering of g1 and hi. O Lemma 11.3.2 (Close families) The number of families with g. # hj for any i and j, satisfying (11.3.2) of Lemma 11.3.1 for r = 3, 4, and 5 with closeness greater than d is O0 2H8+e/d2) Irrespective of closeness, the total number of families satisfying these conditions is
0(54/3H6+e) The implied constants depend on e.
Proof The elementary symmetric functions in five variables satisfy
Pr(K1,..., K4,gs) =P.(h1,...,hs) =Pr(gl,...,gs) +O(SH') for r =1, ... , 4 (with exact equality for r = 1 and 2), so the symmetric polynomials in four variables satisfy
Pr(K1,..., K4) =Pr(g1,...,g4) +O(SH'). As in Lemma 11.3.1 we deduce that (g1 - K1) ... (g1 - KO << 8H4 so that
d s Ig1 - K,I «S1/4H for some j, and that (Kj
g1)...(Kj -g4) «SH4,
so that (KK -g2)(KI
g3)(Kj -g4) << 3H4/d.
For some i = 2, 3, or 4,
kj -g; << OH 4
14113.
We deduce that g1 -g1 << S1/4H+ (SH4/d)1/3 << (SH4/d)1/3 = D,
A comparison argument
251
say. We renumber if necessary so that g1 - g2 = O(D). Similarly g3 - g, _ O(D) for some i = 1, 2, or 4. If i = 1 or 2, then g1, g2, and g3 are all within O(D) of one another, and then g4 is within O(D) of some g, with i = 1, 2, or 3, and all four are within O(D) of one another. In both cases g3 -g4 = O(D). Next we consider the model family with respect to g1. We see that g5 - gi = O(D) for some i, and either all five are within O(D) of one another, or g5 is within O(D) of one of the pairs g1, g2 or g3, g4. We renumber if necessary so that g2 - g5 = O(D). Finally we consider the model family with respect to g3. We have g1, g2, and g5 all within O(D) of one another, and now g4 must be within O(D) of one of these three. Thus, all the g, must be within O(D) of one another. By symmetry, all the hi are within O(D) of one another, and g1, . . . , g5 have the same average as hl,... , h5, so that all ten integers lie within some interval of length O(D). We can take g5 to be arbitrary, pick hl,..., h5 and g4 in O(D6) ways, and then g1 +g2 +g3 and gi +g2 +g3 are known. As in the previous lemma, there are O(HE) choices for g1, g2, and g3. This gives the first result of the lemma. This bound may be refined: after picking hl,..., h5 in O(D5) ways, we know the
K4 of the model family with respect to g5, so that g4 lies
in one of four intervals whose length can be bounded above. The second result of the lemma follows on combining the first result with (11.3.3) of the previous lemma, and choosing
d=[S1'3H].
0
Our next lemma involves a technical condition which will be removed in the next chapter. Lemma 11.3.3 (Comparison of families)
4/H2 < 5:!g 1/8H,
Suppose that
1/H:!-, 0 < SH.
Let NN0 (6, 0) be the number of solutions counted in N10 (S, 0) for which
max(gl,...,gs,h1,...,h5) <min(g1,...,g5,h1,...,h5)+H/2. Let T be a real number with
2
N10(6, A) << 7 N10(ST 2,2 AT) + N100, B 6T), where B is a constant independent of H and T. The term N10 (6, B ST) may be replaced by
O(HN8(B ST) +D4/3T1/3H7+E) for any e > 0, where
D = min(ST, 0). The constant implied in the order-of-magnitude symbol depends on e.
252
The ruled surface method
Proof Let gl,... , g5, hl, ... , h5 be a set of ten integers which contributes to Nio (S, 0). We write 5
5
(hi + t)S -
D5(t) _ 1
(gi + t)5.
(11.3.4)
1
Then D1(t) and D2(t) are identically zero, and ID112(0)I < AH1'2.
ID312(0)I < 6H3"2,
(11.3.5)
We consider only values of t for which H
and H- 1. For some 0, p in 0 < 0, cp < 1, the mean value theorem gives D3/2(t) = D3/2(0) + 2 tD112(0) + 8 t2 D- 1/2( 001
D112(t) =D1/2(0) +
t 2
D-1/2((pt).
For t in I we have
<5/V. A little calculation shows that if I tI < T/2, then ID312(t)I 5 ST2H312,
(11.3.6)
ID1,,2(t)I < 2 iTH1/2.
(11.3.7)
The inequalities (11.3.6) and (11.3.7) are weaker that those in the definition of Nf0(5, A), and we try to show that they have more integer solutions. The real line can be divided into a finite number of subintervals on which D3/2(t) and D112(t) are both monotone. If none of these intervals contains more than one integer solution with t in I, then this family contributes a bounded number of solutions to Nf 0(S, 0), but at least T/2 solutions to N1O(ST 2,2 AT). The difficult case is when a subinterval of t on which D312(t) and D112(t)
are monotone contains l + 1 > 2 integer solutions with t in I. We consider the subinterval for which l is maximal. The mean value theorem tells us that if f(t) is a twice differentiable function with I f(t)I 5 E on an interval (a, a + I], then there are points /3 in the interval (a, a + 1/3) and y in the interval (a + 21/3, a + 1) with 1
3
1
If'( PA, 3 If'(y)I s2E.
Repeating the process, we have
(y- i3)If"(e)I s 12E/l,
for some that
If"(e)I s 36E/12, between 0 and y. We use this with f(t) = D3/2(t), E = 6H312, so f'(t) = i D1
f"(t) = aD1/2(t),
/2(t).
A comparison argument
253
We can now rerun the argument of Lemma 11.3.1 with r = 5 to deduce that such symmetric function of /j,..., gs is equal to the same function of hl ,..., h5 up to a factor 1 + OOH 2/12).
In particular SH2 I D_s/2(l; A 5 CS
H-S/2.
(11.3.8)
12
The constant C., is difficult to compute. However, the numbers D_s/2 satisfy an eleven-term linear recurrence relation in s. Since linear recurrences grow (at worst) exponentially, we have <
CS
for some positive A, uniformly in S. The Taylor expansions of D3/2(6 + t) and Dl/2(6 + t) converge for
Iti 5 H/4A2, and
SltiH3/2
D3/2(6+t) «SH312+ SH3/2
Dl/2(e+t) <<
I
1
+
+
St2H3/2 12
(11.3.9)
5ItIH3/2 (11.3.10)
I2
If (11.3.11)
Iti 5 AlT
with A sufficiently small, then (11.3.6) holds with argument 6 + t in place of t, whilst (11.3.7) also holds at 6 + t provided that (11.3.12)
Al >_ SH.
When (11.3.12) is false, then by the mean value theorem there is some -1 in the interval with
D1/2(,q) «0vrH-,
D_ 1/2(,q) << pVrH /1,
(11.3.13)
and similarly, for s Z 1, D-,/2(77) s CS
OH
H-s/2
a result sharper than (11.3.8). We find that (11.3.11) again implies that (11.3.6) holds with argument 77+t in place of t, and that (11.3.10) may be replaced by D1/2(v1 + t) << 0VH + O1t1VH /11
The ruled surface method
254
and (11.3.7) holds with 77 + t in place of t provided that A is sufficiently small in (11.3.11). If (11.3.14) I
then at least half the values of t satisfying (11.3.11) lie in the interval I, and this family contributes 0(1) solutions to N10(S, 0), but some number >> IT of solutions to N10(ST2, 20T). The comparison fails if (11.3.14) is false. In this case (11.3.10) gives
D1/2(t) << STU for I tl <_ 1, so the family contributes a number >> I of solutions to N10(S, BST)
for some absolute constant B. We note that BST < B/4H, so BST is smaller
that 0 unless both S and 0 are close to 1/H in order of magnitude. Considering in turn each family that contributes to N10(S, 0), we get the first result of the lemma. We can also estimate directly the contribution to N10(S, A) from families for which (11.3.13) is false. If some g. = hi, then, after renumbering, g5 = h5,
and g1,...,g4,h1,...,h4 gives a solution counted in N8(5,BST). These families contribute O(HN8(S, BST)) solutions; in this argument these fami-
lies are trivial families. We apply Lemma 11.3.2 to non-trivial families. Consider the families for whom L < 1 < 2L, where L is a power of 2 with L > H/2T. The elementary symmetric functions of g1,.. . , g5 and of hl,..., h5 satisfy (11.3.2) with S taking some value
S1 = OOH 2 /V).
Similarly, when (11.3.13) holds for t in the interval I, then we can deduce bounds for the symmetric polynomials with S = 62, where
S2 = O(OH/L). By Lemma 11.3.2, these families contribute 0(L504/3H6+e) to N10(S, A), where 60 = min (S1, S2), which sums over L to O(min(64/3T5/3H7+ e, Q4/3T1/3H7+,)).
0
12 The Hardy-Littlewood method 12.1
INTEGRALS THAT COUNT
We treat the First Spacing Problem by exponential si}ms, following Huxley and Kolesnik (1991). Let ( h 113/2
2H-1
S(a,/3,u,v)= E e(ah2+ah+ul Hl
(h 11/21
+v HJ
J.
(12.1.1)
h=H
By coincidence detection (Lemma 5.6.5) in four dimensions, the counting number N2r0, A) of the previous chapter satisfies
N2r(S,i)_I
1/2 1/2
1/2
I I 1/2
1/2S 1/28
f
1/20
IS(a,13,u,v)12rdvdud/3da.
1/20 (12.1.2)
The Hardy-Littlewood method is simplest when there is only one condition defining the counting function on the left, and only the integration over a on the right. It has a divide-and-conquer structure.
D: Divide the range of integration into short intervals (Farey arcs), each labelled by a rational number a/q approximating the variable of integration, a. A: Approximate; there are two cases. If the rational a/q has small height, then the sum S can be approximated by a simpler function. These arcs
are the major arcs. If the rational a/q has large height, then the approximation used on the major arcs disappears into the noise, but the modulus of S cannot be too large. These arcs are the minor arcs.
C: Combine; again there are two cases. The approximate values of the integrals on the major arcs are carefully added; there may be cancellation between the contributions of different major arcs with the same denomi-
nator q (this will not happen in (12.1.2), because the integrand is a modulus squared, real and positive). On the minor arcs we have an upper bound for ISI, so that we can estimate trivially. If possible, it is better to take some factors ISI out at their maximum, and to leave IS121, for some
small integer t, inside the integral sign. The integral of IS12t over the minor arcs cannot be greater than the integral of IS12t over the whole
range - i to 2, and this is the counting function for some simpler problem, which may have an elementary treatment.
The Hardy-Littlewood method
256
The Hardy-Littlewood method first appeared in Hardy and Ramanujan's work (1918) on the partition function, where they use an integral like that in Section 10.2 of Chapter 10 to find the Fourier coefficients of a modular form with poles at the rational points a/q on the real axis. The contour could not cross the line of poles, which formed a natural boundary, but the part of the line of integration close to a rational number of small height still depended on the residue at the pole. Hardy and Littlewood explored the method in a long series of papers, for example (1923). Vinogradov made it his own. For an account by a modern tactician see Vaughan (1981). First we use (12.1.2) to remedy a technical defect in Lemma 11.3.3.
Lemma 12.1.1 (Restricted ranges) Let N2r(3, 0) denote the number of 2r-tuples of integers g1,. .. , g, hl,..., hr counted in N2i(8, 0) for which M < (H - 1)/2, where
M=max(g1,...,g2,hl,...,hr),
m=min(g1,...,gr,h1,...,hr).
Then
N2r(5,1) << N2r(6, A).
Proof We write S(a, f3, u, v) as S1 + S21 where S1 is the sum in which h
runs from H to H + [(H - 1)/2], S2 is the sum in which h runs from H + [(H + 1)/2] to 2H - 1. Then IS(a, f3, u, v)12i <-
22r-1(IS1(a,13,
u,
V)12r + IS2(a, 13,
u, v)12r
Hence the integral (12.1.2) is bounded in terms of the corresponding integrals with S replaced by Sl or S2. Each of these is bounded by a counting number
like N2i(3, 0), but with the numbers gi and h, lying in only half of the original interval [H, 2H). All solutions of these types are counted in N2r( 6, A).
0 We can now replace NN0(6, 0) by N10(6, 0) in the conclusion of Lemma 11.3.3, which now says that a bound for N10(6, 0) for larger values of 6 and 0
implies a weaker bound for N10(6, 0) for smaller values of S and A. The condition involving 0 is always satisfied if 0 >- 5. Thus we have N10(6, 0) < limit N10(8, A) Q
a
=6: 1/2 i/2
f
1/2 1/2
f 1/28 IS(a,/3,u,v)) dud/da. (12.1.3) 1/28
10
We use (12.1.3) for 6> 1/2H. The integral over u in (12.1.3) corresponds to
an inequality. For inequalities we often have a very simple form of the Hardy-Littlewood method, in which 0/1 gives a major arc, and all other fractions give minor arcs. We only want an upper bound for N10(6, A), not an asymptotic formula. We can use Lemma 5.6.5 in the reverse direction.
Integrals that count
257
Lemma 12.1.2 The major arc in the u-integration) We have
f f f 1/2
1/22
1/2
15
/2
1/
15/2
fl/2
fl/z
5
f IS(a,(3,u,v)I10dvdud/3da 5
f1s/z
1/2 -1/2 -15/2
IS(a, /3, u, 0)110 du d f3 d a x H7
Proof By (12.1.2) these integrals are the same order of magnitude as N10(15/2,5) = N10(co, c): in other words, no inequality conditions. By the H7. Since limitation result (Lemma 11.3.2), the order of magnitude is at least I S(a, /3, u, v)I cannot exceed the length of the sum, these, integrals are at most H2 times the corresponding eighth-power integrals, which have the order of magnitude of N$(co,co), which is H5 by Lemmas 11.2.4 and 11.2.6. C1
On the minor arcs we have to bound an exponential sum. Let f(x) be the exponent in (12.1.1):
f (x) = axe + (3x + u(x/H)312 + v(x/H)1/2,
(12.1.4)
so that
f'(x) = 2 ax + /3 + 3Ux1/2/2H3/2 + v/2H1/2x1/2, f" (x) = 2a + 3u/4H3/2x1/2 - v/4H1/2x3/2, f 131(x) = 3(vH -
ux)/8H3/2x5/2.
Lemma 12.1.3 (Minor arcs in the u-integration) Let T = Jul. Then
H2/FT IS(a, (3, u, 0)12 <<
H3/2 HT113
for 1 5 T:5 H, for H< T < H3/2, for H3/2 < T 5 H3/8.
Proof We use the differencing step (Lemma 5.6.2) to compare S(a, /3, u, 0) with the sums 2H-1-d
E e(f(h + d) - f(h - d)).
h=H+d
We have for some 0 in -1 < 0<1 and H < x < 2 H, 2
d
2(f(x+d)-f(x-d))=2df(3)(x+6d)x H3
By Corollary 1 to Lemma 5.4.3 (the truncated Poisson summation formula), we have
e(f(h+d)-f(h-d))« (HI +H
(j ).
The Hardy-Littlewood method
258
Lemma 5.6.2 gives
IS(«, (3 ,
T)
D-1
U, 0)12 «
H+
(H)
+H
(
)
2
<<
D
+ (DHT)+H2
(T).
The bounds follow on choosing D to be [H/2], [H2/2T], or [H/2T1/3] in the three cases. Theorem 12.1.4 For S>_ 1/2H3/2 and any A we have N1O(8, A) << SH7+E
The implied constant depends on e.
Proof First we note that, for D >_ 1, as in (12.1.3), we have
1 f1/2 D
(D 1 ) << H4 +H5+E/D by Theorem 11.2.5. For a function w(u) which is a decreasing function of Jul, the Riesz interchange principle gives 1/2
1/2 f D/2 f 1/2
8
f1/2 f1/2 f1/28 w(u)IS(a,(3,u,0)18dud/3da -1/2 -1/2 -1/25 1/2
1/2
1/28
w(u)
-1/2 -1/2 -1/28'0
g
IS(a,/3,u,0)l dtdud(3da
= fw(O) f1/2 f1/2 1D(t)/2 IS(a, )3,u,0)l8 dud/3da dt t =0 -1 / 2 - 1 /2 - D( t )/2
f w(O)(D(t)H4 +H5+E)dt,
(12.1.5)
t-o
where D(t) is the largest value of u for which w(u) >_ t. We take w(u) to be the upper bound for IS(a, (3, u, 0)12 in Lemma 12.1.3, for Jul >_ 1, and H2 for Jul < 1. Thus, for 8>_ 1/2H3/2
H4/t2 1/2S
for H >_ t > H3/2, for t5 H3/2 .
The integral in (12.1.5) is <<
f
H3i2 H4
dt +
fHz H 4
dt +H'+E
H7+F
H3n 2t2 2S and we deduce the result from (12.1.3) on multiplying by S. t
o
The minor arcs
259
We have not quoted Lemma 12.1.2 explicitly in establishing Theorem 12.1.4; the ideas of Lemma 12.1.2 are built into the proof.
12.2 THE MINOR ARCS For the full four-variable integral (12.1.2) we do the integrations over a and 13 first, using the Hardy-Littlewood method proper. We write
t=lul+Ivl, and suppose that T z 1. By Dirichlet's approximation theorem (Lemma 1.5.1)
there is a rational number a/q with
a=
q
+O
Q
q
(12.2.1)
Lemma 12.2.1 (Minor arcs estimate) For 1 < t < H3/2, and a in the range (12.2.1), we have
H log H
IS(a, 13, u, v)I << H 1/2t1/6 log H + q1/2t1/6
If lul<-U,IvI H/(tU)1 /3,
(12.2.2)
then
IS(a,13, u, v)I << H1/2U1/6 log H.
Proof We use the differencing step (Lemma 5.6.2) to compare S(a, /3, u, v) with the sums F, e(f(h + d) - f(h - d)). (12.2.3) h
We write
f(x + d) - f(x - d) =
4adx q
+g(x),
where, for some 0, cp between -1 and 1, 4ad
g'(x)=f'(x+d)-f'(x-d)- q =2df"(x+9d)-
4ad ,
(12.2.4)
q
g"(x) =f"(x+d) -f"(x-d)=2df(3)(x+ cpd) 3d (vH - u(x + ppd)). 4H3/2x5/2 The order of magnitude of g"(x) may be variable as x goes from H to 2H.
Hence we must divide the interval H to 2H for h into subintervals J on which
IvH - uxkx pHT
The Hardy-Littlewood method
260
for some p = 2-'. We note that either p x 1, or I vHl x luxl and u and v have the same sign. In the second case lul x lul. In both cases the interval J has length between bounded multiples of pH. To avoid an infinite dissection, we
put together the intervals (if any) on which p «t1"3/H, and estimate the sum over these intervals as O(t1"3), which is an allowable term.
We apply the differencing step separately on each interval J, so the conditions of summation in (12.2.3) are that the two numbers h ± d should both belong to J. The change in g'(x) on J is (p2dt
O(pHmaxlg"(x)I)=01 H2 ). The range of d is 1 to D, where D will be chosen with
p2Dt «H2,
(12.2.5)
and D so small that g'(x) changes by at most 1 on the interval J. We use the truncated Poisson summation formula (Lemma 5.4.3) with B -A < 1, so the error term is bounded, and the sum over r = 4ad (mod q) has at most one term. In fact in (12.2.4) 4ad d dt << g'(x) = 2df" (x + 9d) + Hz 9Q
R
The truncated Poisson summation formula gives
e(f(x+d)-f(x-d))= f el g(x)- q )dx+0(1). \
h±dEJ
(12.2.6)
I
where r is the unique integer (if any) with
r
- + y=g'()
r= 4ad(modq),
q
for some y with 1y1 < 1, and some in J. We call d a good value if r is non-zero mod q and rl
Ig'(x)-q
2
q
for x in J. Other values of d are called bad; for these ad
p2dT
q
H2
d +-. qQ
(12.2.7)
For good values of d we use the First Derivative Test (Lemma 5.1.2). The integrals in (12.2.6) sum to D q-1 -) q b=1116/qIl 1
OI1+
= 0((q +D)log q).
(12.2.8)
The minor arcs
261
For bad values of d we use the Second Derivative Test (Lemma 5.1.3) or the trivial estimate. The integral in (12.2.6) is Is
(H3
O(min( pH, 1/ min jg ((x)j )) = O min pH,
I pdT )
Whether d is bad depends on its size as well as on congruences mod q. If d lies in a range K 5 d < 2K, where K is a power of 2, then there are at most 2K/q multiples of q in the range, and
0(
K
gp2Kt
H2 + Q )
non-zero residue classes which can contain bad values of d. There are
0
gp2Kt K fl H2 +Q +q J
K
bad values of d in the range K 5 d < 2K, contributing
O (K+q)
+q(1+Q)
Ht
3K Pt
)
As K runs through powers of 2 not exceeding D, then this expression sums to the corresponding expression with K replaced by D.
When we add in the term pH corresponding to d = 0, and the terms (12.2.8) from good values of d, then the differencing step (Lemma 5.6.2) gives
e(f(h))IZ<<
D 3
X pH+(q+D)logq+(q+D)
i
Ht) +qI1+Q) T
J
(12.2.9)
1
We choose
D x pH/t113,
so that
q +D «Q. We can take D z 1, since the case p >> t1/3/H has already been dealt with trivially. To satisfy (12.2.5) for all p, we require
t
«H3"2.
The right-hand side of (12.2.9) is << t 1/3( pH + Q log Q + PQt113 +H 2 lqt213).
262
The Hardy-Littlewood method
The lemma follows on taking the square root and summing over O(log H) values of p. To simplify the notation we replace the limits of integration
lul < 1/28, Ivl < 1/20 in (12.1.2) by Iul 5 U, lvI <_ V, and we consider the integral
fl/2
/2
ru IV
J 1/2 f-' 1/2 J U -V
(S(a, /3, u, v)l10 dv du d(3 d a.
(12.2.10)
We take (12.2.2) as the definition of minor arcs in the integration over a when u and v are fixed. Lemma 12.2.2 (The minor arcs contribution) The contribution to the integral I in (12.2.10) from the region with a on a minor arc and t = l ul + l vI >_ 1 is H1+E
«U4/3VH5(1 +
U
)loge H.
Proof We have J
r1/2 J
/2
fU IV IS(a,
u)I $ du du
f-' 1/21/2 J-u -V 1
UVN$(2U' 2V) <<
(i
+
d/3 da
H1+E1 JH4,
U
by (12.1.2) and Theorem 11.2.5. We ``take out two factors ISI2 at their maximum from I, using Lemma 12.2.1, and then we have an eighth-power integral over a range which is a subset of the range for J: the standard minor arcs trick in the Hardy-Littlewood method.
12.3 THE MAJOR ARCS We consider regions which are minor arcs in the integration over u, but major arcs in the integration over a, with an approximation
a=a/q+A,
A<< t1"3/qH,
(12.3.1)
q 5 H/(tU)I13.
(12.3.2)
IKISZ.
(12.3.3)
where t = ul + Id >_ 1,
We put
f3=(b+K)/q,
We suppose that U >> H, since the case U << H corresponds to the case of a small, treated in Theorem 12.1.4.
263
The major arcs
We write bx
axe
f(x) =
+
q
+g(x),
q
so that
KX
g(x)=Ax2+
q
/x
+ul H)
K
g'(x) = 2Ax + q +
x
3/2
+v(
3ux1/2
1/2
H)
V
2H312 + 2H 1/2x1/2 3u v g" (x) = 2A + 4H 312 x 1/2 - 4h1/2 x 3/2
(12.3.4)
3(uH-ux)
(3)(x) = 8H 3/2X5/2 1 g
3(3ux - 5uH) 16H 3/2X7/2
Lemma 12.3.1 (Poisson summation on the major arcs) On a major arc with t z 1, either (12.3.5) IS(a, f3, u, v)I «H1/2U1/6 log H, or 1
IS(a,
max
u, v) <<
rx
2H
I JH e(g(x) - 4) dx ,
(12.3.6)
with
u <
(12.3.7)
and
«
t
+
1 HgU1/3
1 V << H2 + H1/2gU1/2
(12.3.8)
HZ
Proof This is another `divide and conquer' argument. We note that g"(x) has one stationary point for x > 0, so that we can divide the range H 5 h < 2H
into at most three intervals, on each of which g'(x) is monotone. We subdivide further into intervals J on which
oK:5 Ig"(x)Is2uK, where o- = 2_s for some integer s, and t
tUl/3
K=2IAI+ HZ << qH by (12.3.2). As in Lemma 12.2.1, we must show that the length of J tends to
The Hardy-Littlewood method
264
zero as
o,
tends to zero. Since x H and t >- 1, we must have either
Ig(3)(x)I >> 1/H4 or Ig(4)(x)I >> 1/H5 at each point of J. Suppose that J has length L. In both cases we have Ig(3)(x)I >>L/H5, I g" (x)I >>L2/H5 somewhere on J. The length of J is O( (o-K1Y5) ),
which tends to zero as promised. We estimate trivially on all intervals whose length is O(H1/2U1/6). This leaves O(log H) long intervals to consider. We apply Poisson summation (Lemma 5.4.6) to get ah2 + bh e
+g(h)
q
G(a, b - r; q) qA-rSgB+-',
f e(g(x) - rx) dx !
q
log(gB-qA+2)), with
qB+qA << gHsuplg"(x)I «vgHK, and by the Second Derivative Test (Lemma 5.1.3)
f e(g(x) - rx) dx <<
1
(o-K)
The error term is certainly O(HU log H), which is smaller than the right-hand side of (12.3.5). There are now two cases. Case 1
If
1, then the sum over r contributes QgHK
(vqK)
«H (K)
Intervals J for which this case holds give (12.3.5).
If aqHK is less than some sufficiently small constant, then there is at most one value of r in the sum, contributing Case 2
<<
VF
(
Either this expression is O(H1i2U1/6) or, for some bounded B, o-K < B/HgU1"3. This is the approximation step of the divide-and-conquer method.
(12.3.9)
We now combine information from the different intervals J. Since g" (x) has one maximum for x > 0, there are at most two intervals of x on which g" (x) S B/HgU'I3
The major arcs
265
Each interval corresponds to at most one value of r. For these values of r we
may extend the Fourier integral to the full range H< h < 2H, since this changes the integral by at most O(H'/ZU'/6 log H). Also, comparing (12.3.9) with the explicit expression for g" (x), we see that 1
t
(12.3.10)
<<
H2 + Hq U'/3 Since J has length at least H'/2U'/6, by the mean value theorem there is a point x in J with g(3)(x) << o K/H'/2U'/6. For this value of x we have u=
Hv
X
+ O(H3 Ig(3'(x)I) =
Hv
X
` +0 ( o.KH512 U1/6
JI
and
v 4h'/2X3/2 +O
g"(x)=2A+
UKH'/2 U1/6
so that
2A=
V
- 4H'/2X3/2 +O(
o-KH'/2 U1/6
In particular, by (12.3.9) Hv
u= X +0
(
H3/2 qU1/2
V
2A= - 4H'/2X3/2 + 0
H1/2gU1/2
which gives the bounds (12.3.7) and (12.3.8).
Lemma 12.3.2 (The major arcs integral) The regions which are minor arcs in the u-integration, but major arcs in the a-integration, contribute H' + )b0g2H << H'V'/4 + U
to the integral I in (12.2.10).
Proof The parts of the major arc on which the uniform bound (12.3.5) holds give the second term in the lemma, just as on the minor arcs. The remaining part corresponds to the centre of the major arc in a one-variable application of the Hardy-Littlewood method. Here it is twisted by the coordinates u and v. For a, b, and q fixed we must estimate
fflf qs
l g(x) If,e 2
r1
q1
to
dxl du du qK d1t,
The Hardy-Littlewood method
266
over the region where (12.3.6) holds, which is a subset of the region defined by
V
A <
1
K<<1,
+ HqU1/2 '
(12.3.11)
H3/2
u<< V+
(12.3.12)
IvkSV.
qU1121
The integer r is determined by a, b, and q, so it is constant in the integration. We make a change of variable x = Hy2, so that rx
g(x) - q = AH2y4 + uy3 +
(K - r) q
Hy2 + vy
=G(AH2,u,(K-r)H/q, v, y), where, for a vector v = (A, u, B, v), G(v,y) =Ay4 + uy3 +By2 + vy. Then
f2H
rx
e g(x)-
)
q
H'
dx=HJAH2,u,(K-r)H/q,v,
where J(v) is the integral
J(v)= f 2 e(Ay4+uy3+By2+vy)ydy. 1
For positive L, let M(L, y) be the set of vectors v corresponding to K, A, u,v satisfying (12.3.11) and (12.3.12) for which
IG'(v,y)I =14Ay3+3uy2+2By+vI
(12.3.13)
J G" (v, y)I =112Ay2 + 6uy + 2BI S L2,
(12.3.14)
IG'3)(v, y)I = 124A + 6u15 L3,
(12.3.15)
IG(4'(v, y) I = 241A1 <- L4,
(12.3.16)
and let M(L) be the union of the sets M(L, y) for 1 - Lk for some k in 15 k:5 4. We apply the kth derivative test (Lemma 5.1.4) to get J(v) << (1/Lk)1jk << 1/L.
If v lies in M(L), but not in M(L/2), then
1J(V)1
1
L
unless G(v, y) has more than one root in 1 - E
The major arcs
267
Let Vol M(L) denote the four-dimensional volume of the set M(L). Then
M(L) corresponds to a set of points (a/q + A, (b + K)/q, u, v) of volume (Vo1M(L))/H3 on which
IS(a,13,u,v)I»H/LI; here a, b, and q are fixed. By the Riesz interchange principle, these parts of the major arc contribute
H10 Vol M(L) L L10gs
(12.3.17)
H3
where L runs through powers of 2 in the range 1
T
>rH_
<< L << U1/6
corresponding to H1/2U1/6 << IS(a, /3, u, v)I << H. If L is less than one, then (12.3.13)-(12.3.16) give IvI << L.
Jul << L3,
Since we have t = luI + I v I > 1, we conclude that L is bounded below when the set M(L) is non-empty. Since G(v, y) is a quartic polynomial, G(v, y + t) agrees with G(R(t)v, y) apart from the constant term, where R(t) is the linear map with A
R(t) u
=
1
0
0
0
4t
1
0
0
A u
B
6t2
3t
1
0
B
V
4t3
3t2
2t
1
v
The set M(L, 0) is the intersection of a box, sides 2L4, 2L3, 2L2, 2L about the
origin, and the box P defined by (12.3.11) and (12.3.12). The set M(L, y) therefore consists of a parallelepiped, volume 16L10, intersected with the box
P. If v lies in M(L, y) then R(-y)v lies in M(L,0), and IAI
(12.3.18)
lu-4Ayi5L3,
(12.3.19)
IB - 3uy + 6Ay2I 5 L2,
(12.3.20)
lv-2By+3uy2-4Ay3I5L.
(12.3.21) Eliminating u and B between (12.3.19), (12.3.20), and (12.3.21), we have
Iv-4Ay31
We deduce that Vol M(L, y) << L6 min(L4, L3 + V).
(12.3.22)
The Hardy-Littlewood method
268
Next we need a finite set of M(L, y) to cover M(L). We choose a sequence of points yi distant 2/L apart. The rows of the vector 4y3
3y2
2y
1
A
12y2
6y
2
0
u
24y
6
0
0
B
24
0
0
0
vi
A U
=D(y )
B V
correspond to the inequalities (12.3.13)-(12.3.16). The rows of A
A
=D(y+t)
D(y)R(t)
B
B
v
V
give the same expression with y replaced by y + t. We also have
D(y +t) =E(t)D(y) =
1
t
t2/2
t3/6 t2/2 D(y).
0
1
t
0 0
0 0
1
t
0
1
If It151/L and v is in M(L, y), then the entries of D(y + t)v have absolute size at most
L'(1+1+Z+6+.")SeL', where e stands for the base of natural logarithms. Hence M(L, y) is a subset of M(eL, y + t), and we can cover M(L) by the sets M(eL, y.). There are at most L points yi, so by (12.3.22) Vol M(L) << L' min(L4, L3 + V).
(12.3.23)
If L is small, then all points of M(L) satisfy the bounds (12.3.11) and (12.3.12). Also, by looking at (12.3.15), we see that a point v with 124A1 >_ L4/2
cannot lie in M(L, y,) and M(L, yi+3). For small L we have
VolM(L) xLtl Substituting (12.3.23) and (12.3.17) and performing the summation gives
H7
I
H7V 1/4
V
<< q5 L min( L,1 + L3
<<
q5
L
when we sum L through powers of 2. Summing this bound over a, b, and q gives the result of the lemma. We can now give a result for medium-sized values of S.
The major arcs
269
Theorem 12.3.3 When 6 and 0 satisfy 5A?/16 H 3/2-e >> 1,
6H1+e << 1,
(12.3.24)
then
N10(6, 0) << 603/4H'.
(12.3.25)
When 6Q9/16H3/2-e << 1,
iH8/9 >> 1,
(12.3.26)
then N10(6, A) << Q3/16H11/2+e
Here e > 0 is arbitrary, but the implied constants depend on e.
Proof We fit Lemmas 12.1.2, 12.2.2, and 12.3.2 together to estimate the integral in (12.1.2). We have
0 H'
1 N10(2U«
U(Ql/4 +
U4/3
Hs+e(1+
0
N
For
H
N10(1/2U,0) «Q3/4H'/U. We pick U = 1/2 6 if possible, and, if not, then we pick U X A9/16H3/2+e
if possible, and use N10(6, A)<_ N10(1/2U, A)
for6<1/2U.
O
We can also apply these methods to the twelfth-power integral. Theorem 12.3.4 When 6 and 0 satisfy
6H1+e «1,
6(OH2-e)3/5 >> 1,
then
N12(61 0) «60H9. When 6(OH2-e)3/5
«1,
AH1/3-e>> 1,
then
N12(6, 0) << p2/5H39/5+e
Here e > 0 is arbitrary, but the implied constants depend on e.
270
The Hardy-Littlewood method
Proof The analogue of Lemma 12.1.2 is the bound O(H9) for the major arc in the u-integration. The analogue of Lemma 12.2.2 is the bound << U5/3VH7+e(i +H/U) for the minor arcs, and the analogue of Lemma 12.3.2 is the estimate << H9 + U513VH7+e(1 +H/U) for the major arcs, since the sum over L in Lemma 12.3.2 converges with L12 in the denominator. The choices of U are analogous to those in the previous theorem. 0
12.4 EXTRAPOLATION We use Lemma 11.3.3 to obtain results weaker than Theorem 12.3.3 but valid over longer ranges. Theorem 12.4.1
For e > 0 there is a constant C(E) > 0 such that N10(6, SH) << 57/4H31/4
(12.4.1)
for C(E)
ii /"-e
1
C(e)H'+e'
(12 4 2) .
.
an d
N10(S, 5H) << 328/41H260141+e
(12.4.3)
1 C(E) H28/15 S 3 S 7j33/25-e
(12 4 4)
for .
.
and
N10(S, SH) << 367/119H104/17+e +H5
(12.4.5)
for 4
1
S S 5 H28/15 H2
Corollary
(12 4 6) .
.
In all these ranges (12.4.2), (12.4.4), and (12.4.6) we have
N10(8, SH) << SH7.
Proof The result (12.4.1) is contained in Theorem 12.3.3. For smaller values of S we use the extrapolation formula of Lemma 11.3.3 (with N10 replaced by N10 using Lemma 12.1.1): N10(S, SH) <<
T
N10(ST2,2BHT) +HN8(S, BST) + 54/3T5/3H7+e (12.4.7)
Extrapolation
271
By Theorem 11.2.5
HN$(S,BST) << H5, which is negligible compared to (12.4.3), and gives the second term in (12.4.5). In order to apply Theorem 12.3.3, we choose T so that 5T2Hl+E << 1.
(12.4.8)
The second condition in (12.3.24) requires 525T41H33-E>> 1.
(12.4.9)
First we choose T so that (12.4.9) is just satisfied; this choice easily satisfies (12.4.8). The first term in (12.4.7) is << S(SHT)3/QH7
by (12.3.25) of Theorem 12.3.3, which gives the expression (12.4.3). The third term in (12.4.7) is smaller in the range (12.4.4). In the range (12.4.6) we use (12.3.26) of Theorem 12.3.3: T
No(ST 2 26HT) << S 3/16 H 91/16+e/T 13/16, ,
and equalize the first and third terms with the choice T x 1/(655H63)1/119
to get the first term in (12.4.5). The conditions (12.3.26) of Theorem 12.3.3 require the reverse of (12.4.9), which holds in the range (12.4.6), and SH17/9T >> 1,
which is easily found to be satisfied. The corollary holds with a margin to spare except at 8 x 1/H2. 0
13 The First Spacing Problem for the double sum 13.1
FAMILIES OF SOLUTIONS
The coincidence conditions for two integer vectors in the treatment of the double exponential sum of Chapter 8 are k111 +k212=k313+k414, ll kl + l2 k2 =13 k3 + 14
(13.1.1)
k4 + O(-qLI),
11+12=13+14,
(13.1.2)
(13.1.3)
11/ kl + l2/ k2 =13/ k3 + 14/ k4 + O(11LFK ),
(13.1.4)
where rt is small, with all variables within ranges
K
L
(13.1.5)
If we can write
k,=g;+h,, 11=g1-hl,
12=g2-h2,
13=h3-g3,
14=h4-g4,
then (13.1.1) and (13.1.3) become the corresponding equations in Chapter 11
with r = 4, so the ruled surface method can be applied. The inequalities (13.1.2) and (13.1.4) are different from Chapter 11, and the range (13.1.5) is
more restricted. We shall not use condition (13.1.4). We follow Watt's account (1990).
Lemma 13.1.1 (Families of solutions) The integer solutions of (13.1.1) and (13.1.3) form families: lines parametrized by
k.=mi+t
(13.1.6)
with 11, 12, and 13 fixed, 14 =11 + 12 -13, m1, m2, m3, and m4 = 0 fixed, and I1m1 + l2m2 =13m3.
(13.1.7)
The number of families which contain integer solutions in the range (13.1.5) is O(K2L2 + L3).
Proof We verify at once that (13.1.6) and (13.1.7) together give a solution for
Families of solutions
273
all t. There are at most L3 families with m1 = m2 = m3 = 0, the diagonal
solutions. For other families we write 1= (11,12,13),
m = (ml, m2, m3)
as three-dimensional vectors, and put A = Ill, µ = I ml. We write d and e for the highest common factors d = (11,12,13), e = (m1, m2, m3). We call families primitive if d and e are both one. Case 1
Families with µ/e < A/d. For fixed m, the vectors I/d lie in the
plane lattice in 083 of integer vectors orthogonal to m/e. The determinant of
this lattice, the area of its unit cell,
is
p./e, because the lattice in 083
generated by m/e and the integer vectors orthogonal tb it has index IM/ell. By counting squares (Lemma 2.1.1), the number of vectors 1/d satisfying (13.1.5) is
L eL2 « (L/d)2 + « µ/e d A*
For each power of 2, M = 2', there are O((M/e)3) triples m with highest common factor e fixed, and M5 m < 2M. These contribute L2M2 M 3 eL2 << e d2M d2e2 families, and summing M through powers of 2 gives O(K2L2/d2e2) families with fixed common factors d and e.
«
Families with µ/e < A/d. These give the same bound O(KZL2/d2e2). In both cases we sum over d and e to get O(K2L2) non-diagonal families.
Case 2
0 Lemma 13.1.2 (Counting primitive families) The number of primitive families with 11, 12, and 13 fixed, and the ratio m2/ml in a fixed interval of length 0, and
withm1 in arange M
where o,(n) denotes the sum of the divisors of the integer n. Similarly, the number of primitive families with ml, m2, and m3 fixed, m3 0 0, and 11/12 in a fixed interval of length 0, with L< 1, < 2L is << d(Im3D+OL2o (Im3D/m3.
Proof Since the highest common factor d = (11,12, l3) is one for primitive families, then we can write (non-uniquely) 13 = rs,
(11, r) = 1,
(12, s) = 1.
Suppose that 11m1 + 12m2 =13m3,
l1n1 + l2n2 = l3n3.
(13.1.8)
The First Spacing Problem for the double sum
274
Then modulo s we have m1n212 - m2n112 = -m1n111 + m1n111 = 0 (mod s),
so that m1n2 - m2n1 = 0 (mod s).
Similarly we get a congruence mod r, so that m1n2 - m2n1 = 0 (mod 13).
If the ratios m2/m1 and n2/n1 are different, then m2
n2
m1
n1
(13.1.9)
m1n1 m1n2 - m2n1
Now let f be the highest common factor f= (m1, m2). Since e= (m1, m2, m3) =1, (13.1.8) gives f 113. So each primitive family is associated
with a factor f of 13. Suppose that n1, n2, and n3 in (13.1.8) are the parameters of a family associated with the same factor f = (n1, n2) of 13. Then we can sharpen (13.1.7) to m2
n2
m1
n1
13/f
f13
>
I(m1/f)(n1/f )I = Im1n11
f13
4M2
Since m2/m1 must lie in a short interval, then there are << 1 + iM2/f13
(13.1.10)
possible rational numbers m2/m1 (this argument corresponds to Lemma 1.2.3 with an extra congruence condition). When f and the ratio m2/m1 are known, then we know m1 and m2, and thus also m3 from (13.1.8). The first result follows on summing (13.1.10) over all factors f of 13, and using
E1-LL gO-(1) fu
f
f
g fg=1
l
1
The second assertion follows by interchanging the roles of 11, 12, and 13 with those of m1, m2, and m3. 0
Lemma 13.1.3 (Solutions in one family) The number of integer solutions of (13.1.1), (13.1.2), (13.1.3), and (13.1.5) within a particular family is at most K4
min 1+0
(13.1.11)
, K ,
Im1m2m31
and also at most K4 min 1 + 0
(m1(m1 - m2)(m1 - m3)I
)K).
(13.1.12)
Families of solutions
275
Proof We use (13.1.3) to eliminate 14 from (13.1.1) and (13.1.2), getting (k1 - k4)11 + (k2 - k4)12 = (k3 - k4)l3,
kl - k4 )l1 + ( k2 - k4 )l2 =
(13.1.13)
k3 - k4 )13 +
(13.1.14)
Eliminating 13 between (13.1.13) and (13.1.14), we find that (
-
k1
k3)( kl
- k4 )11 + (
m (m 1 - M3)11 +m2 (m 2 - M3)12' 1
k2
k3)( k2 - k4 )l2 = O('qL/),
-
kl + k4)( kl + k3) _ O(LK2)
`
(
k2 + k4 )(
k2 + k3)
(13.1.15).
Thus nK2
m2(m3-m2)Q+OI 11
12
(13.1.16)
I
m2(m1 - m3)
I m1(m1 - m3)I
where Q
k+ k4)(1
(lk +
k3) = 1+OI Iml Km2)
4)(F 1)
(
1
(13.1.17) 1i
can be regarded as a function of t, depending on the family, with d
d
r
dt k1=1'
dt
(>
k
+
k
V`-
1
2 k; +
J
k; + kj
1
2k
(k;kj)
2
so that d
2dt
1
logo
1
(klk4) + (
(k1k3)
k2 - k1)( k3 + (k1k2k3k4)
1
1
(k2k4)
(k2k3)
k4 ) >> IM1-M21 z
K
Hence within a fixed family, the values of t for which (13.1.3) holds form an interval of length ,K4 O I m1(m1 - m2)(m1- m3)I
which gives the bound (13.1.12). Similarly considering the values of 14/12 leads to the bound (13.1.11).
0
The First Spacing Problem for the double sum
276
Lemma 13.1.4 (Solutions of all four equations) Suppose that k1,. .. , k4, ll, ... ,14 satisfy (13.1.1)-(13.1.5). Define 0 and .p by
kl +12 k2 -13 k3 -14 k4 = OLFK,
ll
11/ kl + 12/ k2 - l3/ k3 -14/ k4 = cpLFK (so that 0 and cp are 0(17)). Then
max rjIki-kjl<
fl Iki - kjl xIpIK4, joi and there are at most OK
mink,1+0
cp(m1 - m2)
D
solutions in the same family as the given solution. If
101 > 2K I cp1,
then,
for each i,
11Iki-kj1
joi
101K3.
Proof We write a = kl , /3 = k2 , y = k3 , 6 = k4 , so that the system of equations becomes
MT
2
-13/7
cpL
-14/S where M is the van der Monde matrix a2 /32
y2 S2
with determinant and adjoint matrix given by
0=detM=(a-/i)(a-y)(a-S)(/3-y)(/3-Sky -S), adjM=
1r1
'r2
Tr3
T4
'r10'11
X2012
'T30'13
'T40'14
7r1 X21
'T2 022
'T3 023
74 0'24
1T1 X31
r2 0`32
'r 3 X23
'T4 X24
Families of solutions
277
are the symmetric functions of /3, y, and S:
where
0"11 =0 +y+6,
0`12=Ty+yS+S/3,
//
0`13-Th'6, and 0`2J, 0`3, and 0`4j are the corresponding symmetric functions of y, 6, and a, of S, a, and /3, and of a, /3, and y, respectively. The products are
7r1= 0/f'(a),
7r2 = 0/f'(/3),
and so on, where f(x) is the polynomial
f (x) = (x - a)(x - /3)(x - y)(x - 8) whose roots are a, /3, y, and S. After this preparation, we have
0
11/a 12/a
0
= (adj M)T 9LfK 0
-13/y -14/6
cpLFK
so that 11
aLV f'(a)
12=
f(/3) ((a+y+6)8+ay6co),
13
yLFK
f'( SL
14
f'( S)
((a+/3+6)8+a/i6cp), ((a+/3+ y)9+a/3ycp),
We deduce that
kl + k2 )(kl + k3)( kl + k4) f'(a)
(k1 - k2)(k1 - k3)(k1 - k4) = (
<< K2a/3y6 I cpI << riK4.
If 191 < K I kp1/9, then
(/3+y+6)I01<s (2K)
KI 'I 9
< V3
and similarly for the other expressions. We deduce that 11
l2 = - (a-
=
y)(a- 6)
+0(I coKI))
(y2-/32)(/32-62)(a+y)(a+3) r 1+0 (a2-y2)(a2-62)(/3+y)(/3+6) (M3 2
(m1 -m3)m1
Q(I 1+O
cpKi)),
0
cpK
The First Spacing Problem for the double sum
278
in the notation of Lemma 13.1.3. Thus, by the argument of Lemma 13.1.3, there are at most K2
0
1
(P
KIm1-m 21
solutions in this family. Similarly, if 191 >- 2K I q'l, then
IaysDls
( (2K) )3191
2K
Iwl,
and
If'(a)1x K3/2191.
13.2 GOOD AND BAD FAMILIES We estimate the total number of solutions to (13.1.1), (13.1.2), and (13.1.3) in the range (13.1.5) by summing the bound of Lemma 13.1.3 over all families.
However, as in the proof of Lemma 11.1.3, we must take care to do the longest summation first. We renumber k1, ..., k4, preserving the pairing of k1 with k2 and k3 with k4. If max ki and min k, are in the same pair, then we make them {k3, k4}, and we number within each pair so that
min min I ki - ki l =Ik2 - k41.
i=1,2 j=3,4
If max ki and min ki are in opposite pairs, then we make them k1 and k4 in either order, and number so that
Ik2-k4151k1-k21. Writing m. = ki - k4, ni = I mil, we see that m1, m2, and m3 have the same sign, n1 z n2,
In1- n31 z n2,
(13.2.1)
and that (13.1.13) becomes 11n1 +12n2 =13n3.
Since L 51. < 2L, we have 11n1 <13n3 < (ll +12)n1 = (13 +14)n1,
n1/2 < n3 5 4n1.
(13.2.2)
Families with n1 = n3 are now called trivial; by (13.2.1) this implies n2 = 0. There are KU trivial families with n1 0 0, and at most L3 trivial families with n1= 0 (so that all the ki are equal). All non-trivial families have n1 and n3 non-zero by (13.2.2).
Good and bad families
279
Now we let A(K, L,,q) be the total number of solutions of (13.1.1), (13.1.2),
and (13.1.3) in the range (13.1.5), and A'(K, L,,q) be the number of nontrivial solutions with k1, ..., k4 ordered and e = 1, and A*(K, L,,q) be the number of non-trivial ordered solutions with d = e =1. Then clearly (
L
A'(K,L,17) < >. A* K. d=1
Z,
1
d
For non-trivial solutions e = (m1, m2, m3) is defined. From the proof of Lemma 13.1.1, each value of e arises from O(K2L2/e2) families. Of these, the families with less than e solutions contribute K2L2
K
<< s e e=1
<< K2L2logK
2
e
solutions. If there are more than e solutions in the family, then by Lemma 13.1.3, the solutions form an interval of values of k4, and so a proportion at least 1/2e of them have e I k4, which implies e I k, for each i. There are eight ways of renumbering each ordered solution, so
(K A(K,L,71)
<< KL3 +K2L2 to g K+
1
eA*
d5L e5L
KL e
d
ll
(13.2.3) 1
Thus we need to consider A*(K, L, ,t7), the number of ordered primitive solutions. As in Lemma 13.1.1 there will be two cases, depending on the relative sizes of K and L. Families for which the error term in (13.1.16) is always numerically less than a are called good. Since 1 < Q < 2 and Irn31 < 4 Im11, then 1m1 - m31 <- 16 1m21= 16n2
(13.2.4)
in good families. In particular, good non-trivial families have m2 0 0. Other families are bad; for these m1(m1 - m3) << 77K2.
Lemma 13.2.1 (Bad families) The number of bad non-trivial families is O(-qK2L2 log KL). They contribute O(gK3L2 log KL) solutions.
Proof We count the number of bad families with M
(13.2.5)
which implies n2 = Im21 < 2N
by (13.2.1). By Lemma 13.1.2 with 0 << N/M, there` are O(I
d(l3)+
N
MM2v(13)/13 I
(13.2.6)
280
The First Spacing Problem for the double sum
such families for each triple 11, 12, 13. We sum this bound over l1, 12, and 13 to 2L g
<< L' log L + L2MN E EE f2g2 f
13=L
8 fg=1i
«(L3 +L2MN)log L << L2MN log L +L4, (13.2.7) using Lemma 11.1.3 on the first sum, and a technique from its proof on the second sum. Similarly, by Lemma 13.1.2 with A << 1, there are
O(d(n3) +L2o-(n3)/nj) such families for each triple m1, m2, m3. We sum this bound over m1, m2 and m3 satisfying (13.2.5) and (13.2.6) to << MN 2 log M + L2MN log K <
(13.2.8)
One or other of the bounds (13.2.7) and (13.2.8) is O(L2MN log W. Summing M and N through powers of 2 with N:5 4M, MN < r)K2 gives
0
O(r&K2L2 log KL) families, and each family gives at most K solutions.
Lemma 13.2.2 (Good families with in fixed)
Each good family with m3 and
m2 in fixed ranges
MSIm31<2M,
N<M
N<-Im21 <2N,
(13.2.9)
contributes a bounded number of solutions of (13.1.1), (13.1.2), (13.1.3), and (13.1.5) unless
(13.2.10)
M2N << -qK4,
in which case these families contribute in total
N
N O( riK4 M log M+
3L2
ll
M)
(13.2.11)
solutions.
Proof By (13.2.2), (13.2.9) implies that
M/4
(13.2.12)
and (13.2.4) and (13.2.1) imply that
N51m3-m11532N.
(13.2.13)
We fix m1, m2, and m3 satisfying these conditions. By Lemma 13.1.2 with
_
z
71K 2 M + MN << K + MN from (13.1.16) and (13.1.17), the number of possible 11, 12, and 13 is
0 « I MI K m2I << d(n3) +
LZ n2n3) 3
M
2
(K
+ MN
281
Good and bad families
As in Lemma 13.2.1, this bound sums over ml, m2, and m3 satisfying (13.2.9), (13.2.12), and (13.2.13) to
«MNZIog M+LZMNZ ( 1K+MN nK 2 By (13.1.11), the number of solutions in each family is K4
11
0 `min M2N2,K uniformly in l1, 12, and 13. The total number of solutions from such families is I
rtK4
OI M2N2
MN2 log M+
N ) + gK2L2 M K ,
L2MN2 1
K
which gives the` result of the lemma.
O
Lemma 13.2.3 (Good families with 1 fixed) Irrespective of (13.2.10), the good families with m3 and m2 in the ranges (13.2.9) contribute
N
N
11
0 lrtK3L2 M+KL3 MlogL)
(13.2.14)
solutions.
Proof We fix 11, 12, and l3. From (13.1.17)
I1m1(m1 - m3) + 12m2(m2 - m3) «OK2L,
(13.2.15)
where, by (13.2.4),
A=77+
m1(m1- m2)(m3 - m1) K3
M2N
«77+ K3
(13.2.16)
Eliminating m3 between (13.1.13) and (13.2.14) gives 11(11-13)m1 + 21112m1m2 + 13(13 -12)m2 <<
K2L2.
(13.2.17)
We multiply both sides by 1112 + V, where D is the discriminant of the quadratic, D=(1112)2-1112(13-11)(13-12)=111213(11+12-13) =11121314.
We can now factorize (13.2.17) as
(11(13 -11)m1 + (1112 + D)m2)((1112 + rD-)m1 + 12(13 -12)m2) «OK2L4. (13.2.18)
Now m1 and m2 have the same sign, and even if 13 - l2 is negative, then
1112+FD z2Ll2z2l2(12-13).
The First Spacing Problem for the double sum
282
Hence the second factor on the left of (13.2.18) is at least L2 Im1I in order of magnitude, and so m2
11(11-13)
MI1
11 12 +Vi
AK2
qK2
«
m21
M2
+
N K
(13 . 2 . 19)
we note that (13.2.18) implies
LN LN -qK 2L LN + + << M , M M2 since the family is good. By Lemma 13.1.2 there are 11
K
-13 <<
(13 . 2 . 20)
N riK2 0'(13) (d(l3+_+IM2 ) 12 3
of these families, and by (13.1.11) of Lemma 13.1.3 they give O
-qK20'(13)1 + r1K4 M2N 0'(13)
M2N K
12
<
13
-qK30' (13) 123
1
solutions. We sum over 11 satisfying (13.2.20), then over 12 and 13 to get the result of the lemma. Theorem 13.2.4 The system of equations and inequalities (13.1.1), (13.1.2), (13.1.3), and (13.1.5) has OW + K 2 L 2 log K + ,gK3L2 log K) integer solutions.
Proof First we estimate A*(K, L, -q). We count good families with m fixed unless K log K >> L2, (13.2.21) when the bound (13.2.14) of Lemma 13.2.3 is
OI gK3L2 M +KLM logK(KlogK)1/2) =O('qK3L2
M +K2L2 M (13.2.22)
1
When (13.2.21) is false, then the bound (13.2.11) of Lemma 13.2.2 is
(
O K2L2
N + K3L2 N M) M
also. Summing M and N over powers of 2 with N << M << K now gives O(K2L2 log K+ i1K3L2 log K)
(13.2.23)
The problem with a perturbing term
283
solutions from good families with M2N << -qK4. These terms dominate the contribution from good families with M2N >> qK4 and from bad families. Thus (13.2.23) is an upper bound for A*(K, L, r1). The theorem now follows from the reduction formula (13.2.3). O
13.3 THE PROBLEM WITH A PERTURBING TERM The coincidence conditions in the treatment of the exponential sum in Chapter 9 are more elaborate, but if the parameters of the construction satisfy
H5 <
(13.3.1)
then we have
L
K
L3
<<
R
XQ
S
IQ
I
Q)
<<
I
Hence for L < 1 < 2L, K5 k < 2K we have 13
k2 =19(0)+
01(0) + O
(12)
(R /(I)
3
= l + 24k312 +0 R (Q
)
and 2
k e(k2)
n
12
413
T372
+O
9 (k2)
R
(N)
Thus the coincidence conditions include (13.1.1), (13.1.3), and (13.1.4) to the same order of accuracy as before, but (13.1.2) is replaced by ll k, 1 +
12
24k2
12
k2 1 +
+ 12
24k2
12
=13 k3 1 +
12
24k3
2
+ l4 k4 1 +
24k4
2
+ O(-qL1k). (13.3.2)
We regard (13.3.2) as a perturbed version of (13.1.3), and we try to count the solutions of the new system of equations by the same method.
The First Spacing Problem for the double sum
284
Lemma 13.3.1 (A small perturbation) The system of equations and inequalities (13.1.1), (13.1.3), (13.3.2), and (13.1.5) has
O((KL4 +K2L2 + gK'L2)log K) integer solutions.
Proof We write (13.3.2) as 11
L2
k1 +12 k2 =13 k3 +14 k4
)Lv),
and apply Theorem 13.2.4.
Finally we consider the coincidence conditions for the most complicated version of Jutila's method in Chapter 10. The argument is simpler, but we must keep track of the coefficients b(l). Lemma 13.3.2 (Two each side, with coefficients) Let b(l) be any sequence of complex coefficients, and let L be a positive integer. The number of solutions of
11-13=12- 14,
l1 -
12 -
13 =
14 + O(0F
with L < li < 2L, counted with weight b(11)b(l2)b(l3)b(14) is 2L-1
2(
2
2L - 1
(
Ib(l)12) +0 I (1 + AL2 log L) L
E
Ib(l)14)
L
Proof The trivial solutions with l1 = 12 or 11 = 13 give 2L-1
2(
2
Ib(1)12) L
2L-1
- E Ib(l)I'. L
For the other solutions we use I b(l1)b(l2)b(l3)b(14)I 5 4(16(11)14 + ... +Ib(14)14).
By symmetry we can replace the weight by Ib(11)14. As in Lemma 11.2.4, the solutions with 11 fixed have
(11-13)(11-14) = O(OL2).
By Lemma 11.1.3, there are O(iL2 log L) choices for 13 and 14 different from 11, and 12 is given by 12 = 11 - 13 + 14.
Part IV The Second Spacing Problem: `rational' vectors
14 The First and Second Conditions 14.1 THE COINCIDENCE CONDITIONS First we set up a uniform notation for the Second Spacing Problem in Chapters 7, 8, and 9.
Lemma 14.1.1 (The coincidence conditions) The conditions for the coincidence of two y vectors corresponding to rationale a/q and a'/q' in Lemma 7.4.2 and Lemma 8.4.1 imply four conditions of the form a
a'
S A1,
(14.1.1)
5 A2,
(14.1.2)
5 03,
(14.1.3)
IK-K'I5A4.
(14.1.4)
ab
a'b'
q
q
In Lemma 7.4.2 we have R4
01xNZQ2V,
R2
Q
02XNZ
R2
03XNQ,
A4XN.
R2
03_HQ' R2
Q A4x H
In Lemma 8.4.1 we have
01 x
R4
HNQ2V'
A2
HN'
In both cases we have the chain of inequalities R2
R2
Al S A1Vx n202 << A2 << A3 X and
O2PQ>> 1.
T A4 << A4,
The First and Second Conditions
288
Proof The condition 1
IY1-y1'1 s
2r(2H)2V
in Lemma 7.4.2 gives in case 1 4a'
4a
R4 <<
q
q'
N2Q2V'
and a'
54 q
4a
4a'
q
q'
q'
<4
4a
4a'
q
q'
and similarly in cases 0 and 2. We argue similarly for y3 -y3. The condition 2
2
1
I
(27µ'q'3)
(27µq3)
2r(2H)3"2
gives Q3
Vµ,q,3
-1 It
R2
R6
NR2 N3Q3
µq3
)
«
N2'
and (14.1.2) follows on multiplication by the conjugate factor 1 + 3) . The condition I
3K
3K'
1
5
1/(27µ'q'3)
(27µq3)
2r(2H)1/2
gives
K - K'
1
(27µ,q,3)
1
1
(27µ,q,3)
2 µq3)
2r(2H) /2
so that µ,q,3
IK - K'ls
4L27ttk'q,,3
+
A
«NR2
Q32
+
(
R2
NR NQ
H Q «N.
We argue similarly for the condition of Lemma 8.4.1.
In the case M>> FT, the rational number a/q is less than one, and so the numerator a runs through a shorter range. If M2 - M< M/C2C3, (14.1.5)
Magic matrices
289
then the numerator a lies in a range P< l al < 2P with
QT MZ,..NR2. MQ
P
(14
.
1 6) .
We lose no generality in supposing that (14.1.5) holds, since we can take a
finite subdivision of the sum S into ranges M< M< M2, M2 + 1 < M5 M3,... . On the second range we replace M in the inequalities by M2 + 1, and so on.
Lemma 14.1.2 (Inverting the fractions) In the case M >> T , the First Coincidence Condition (14.1.1) implies 1
(14.1.7)
where q and q' are defined by
qq + as =1,
q'q' + a'a' = 1.
Proof We have q
q'
1
1
a
a'
aq
a'q'
which gives (14.1.7) since lal > P, I ql >- Q.
O
14.2 MAGIC MATRICES One of Bombieri and Iwaniec's deepest insights was that the first condition (14.1.1) was linear in the action of the modular group on primitive integer vectors, described in Section 1.3 of the first chapter. To simplify the notation, we write r/q = l a/ql. Lemma 14.2.1 (The magic matrix) in their lowest teens with r
Suppose that r/q and r'/q' are rationals r'
liq
g111
where r and r' are defined by if = 1(mod q), r'r' = 1(mod q'). Then b
(q') - (c d)(q) for some integer matrix of determinant ad - be = 1 with
Icl < Digq'.
(14.2.1)
290
The First and Second Conditions
The entries of the matrix are related by
cr'
bq
q
r'
a = q, + q, = -r - + r-, q'
cr
d=- q
+
bq'
= Y, +
q
(14.2.2)
r
(14.2.3)
Y'.
Proof Since r is only defined mod q, and r' mod q', then we can suppose that
fir'/q' - r/qI < A1. Let q and q' be defined by qq + if = 1, q'q' + r'r' = 1. Then r'
r'
q,
= q,
- q'
r
q
r
r'r + qq'
r'q - rq'
r
r'
-q
r
q
q'r - qr' q' q - rr'
q
which defines the matrix. The bound for c is immediate, and the relations (14.2.2) and (14.2.3) are restatements of (qr
c
qr
f
abq')-(aqd
'-cr' 0
Next we classify magic matrices. In the first chapter we saw the classification by trace. When calculating Fourier coefficients of modular forms (see Kuznietsov 1980), the right classification is into the identity, the upper
triangular matrices, and all other matrices. Our argument uses a third classification.
Matrices with I cl >- BQ/P or I bI >- BP/Q, where B is a sufficiently large constant. Type 3
Type 2
Upper and lower triangular matrices other than the identity.
Type 1
The identity matrix, and any matrix not already classified.
We write
SM=M2-M,
(14.2.4)
or, for a family of sums with varying endpoints,
SM = max M2(i) - M. i
Lemma 14.2.2 (Type 1 matrices) (in terms of the constant B).
The number of type 1 matrices is bounded
Magic matrices
291
Proof Type 1 matrices other than the identity have
O/lbcl5B2. -1, then
Then Ibl, Icl s B2. If be
0 0 ladl 5 B2 + 1,
and at, Idl < B2 + 1. If be = -1, then ad = 0. If a = 0, then (14.2.2) gives r' = bq, so b =1, c = -1, r' = q, and (14.2.3) gives
r d=-q +-. r Hence d is positive and d 5 4. Similarly, if be =
1 and `d = 0, then a = 1, 2,
3,or4.
0
Lemma 14.2.3 (Type 2 matrices) In the construction of Chapter 7, upper triangular matrices have
15 I b I 5 min(1, 28C2C3)P/Q. Lower triangular matrices have
15Icl S min(1,88C2C3)Q/P. There are corresponding bounds in the constructions of Chapters 8 and 9.
Proof For an upper triangular matrix q' = q, r' = r + bq and I r' - rI 5 P, so that
IbI 5 Ir' - rI/q 5 P/Q. For a lower triangular matrix r' = r, q' = cr + q, and I q' - qI 5 Q, so
IcI<
Q/P similarly. In the construction of Chapter 7, the fractions r/q and r'/q in the upper triangular case, or r/q and r/q' in the lower triangular case, are values of f"(x)/2. By the conditions (7.2.1) of chapter 7, the values of f"(x)/2 lie in an interval of length 5 SC3T/2M2, whose lower endpoint is
z T/2C2M2. Hence we must have
P T -Q -4C2 M21 so that
r
r'
Ibl= q-q 5
SC3T
2M2 5
26C2C3P Q
The First and Second Conditions
292
Similarly, for lower triangular matrices
2SC2C3P/Q ? I r/q - r/q'I = I cr2/qq'I > 4P2 ICI/Q2. Lemma 14.2.4 (Type 3 matrices)
For type 3 matrices
Idl x
lal
P Q
Q
Icl x
IbI.
The number of type 3 matrices with K5 I c I < 2K is 2
O min
Q
SZQZPZ
2
2
KP 2)
+ min K, 2+ 1+ Q
(
(
SQ )
log K
Proof We suppose that B z 8 in the definition of type 3 matrices. Then in (14.2.2) and (14.2.3), for I cl BQ/P we have B P r' q< 2< 4 s 4Q IcI 5 2q, Icl, q
r' 3r' 3P P IcI <- 2q, IcI <- lal s Zq, IcI s Q 4Q
IcI,
and similarly for Idl. Then I bcI s ladl + 1 s
3P
2
p
(Q IcI) + ( 8Q
2
IcI)
s
loP2 c2, Q2
and z
2
Ibcl > I adl - 1 >_ (
Q
IcI)
2
c2.
- (8Q ICI) 64Q2
We argue similarly when IbI z BP/Q.
To count type 3 matrices in the case P z Q, we note that if c and d are fixed, and c and d are integers with cc + dd = 1, then the relation ad - be = 1 implies that
b= - c + dt (14.2.5) for some integer t<< P/Q. When K5IcI<2K, then Idl KP/Q, so there a = d + ct,
are O(K2P2/Q2) type 3 matrices in this range. In the case P< Q we express c and d in terms of a, b, d, and b, where as + bb = 1, and argue similarly. For small 8, we write A = J- (M)12.
By the conditions (7.2.1) of Chapter 7, or their analogues in the other chapters,
r/q, r'/q' = A(1 + 0(6)).
(14.2.6)
Magic matrices
293
Put 0 = q/q'. Adding (14.2.2) and (14.2.3) gives
a + d - 2 = 0 + 1/0 - 2 + O(6A Icl).
(14.2.7)
Since z < 0 5 2, we have 0:5 0 + 1/0 - 2 5 1. If a + d = 2, then the magic is a transvection with some eigenvector (w, v), a primitive integer matrix
vector, and, for some integer u, c
d)(uv2 w 1
u 1-UVw
Substituting these values in (14.2.2), and using (14.2.6), we find that
=A+O SA+
-). 1
Icl
V
For K < Icl < 2K and fixed v and w, the number of choices for u is at most 2K/v2. The number of possible rationals w/v with v in a range
V
<<
1+(K+5A)V2,
(14.2.8)
by Lemma 1.2.3, giving a total of O(K min(1, A2) + (1 + SAK)log K)
(14.2.9)
transvection matrices with K< I cl < 2K when we sum V through powers of 2. The remaining type 3 matrices have a + d 0 2, so (14.2.7) gives (14.2.10)
6AK>> 1.
We conjugate the magic matrix into a matrix with smaller entries. Let J be the interval [(1 - S)A, (1 + S)A], and let 1/n be the rational number in J with least denominator. By Euclid's algorithm (Lemma 1.5.2) there are integers k and m with
kn-lm=e= ±1,
0<m
(14.2.11)
If m > n/2, then we replace k by l - k, m by n - m. If m > 0, then since
neither k/m nor (l - k)/(n - m) lies in J, but I/n, which is in J, lies between these two rationals, then we have
1-k
k
2SA<m -- n-m
1
m(n-m)'
so that SAmn < 1;
this inequality holds trivially if m = 0. We rewrite (14.2.6) as r
r'
1
9 , q, _ - + O(6A). n
(14.2.12)
The First and Second Conditions
294
Then (14.2.2) and (14.2.3) give
an - c1= n9+ O(SAnK) = O(SAnK), do + c1= n/O + O(SAnK) = O(SAnK). Hence by (14.2.11)
m
ce
ce
n
n
n
am - ck= -(an-cl)- - =m6- - +O(SAKm), and similarly ce
ck+dm =m/9+ - +O(SAKm). n
The magic matrix is conjugate to
d')
(c'
n)-1(c
=(k
l n1
(m
d)
akn + bmn - ckl - dim - akm - dm2 + ck2 + dkm
aln + bn2 - c12 - din
-aim - bmn + ckl + dkn
1
(an - cl)(ck + dm) - mn
c
-(am-ck)(ck+dm)+m2
(an-cl)(cl+dn)-n2 -(am-ck)(cl+dm)+mn
Here mn
a'
c
SAnK
+O
Icl <<
lm+
K n
+SAKm
K n
SAnlm+ - +SAKm << SAK,
by (14.2.12). Similar calculations give b' «S2A2Kn2, c' << K/n2, d' << SAK.
The trace is unchanged by conjugation, so a' + d' # f 2, and b' and c' are both non-zero. If 6An2 >> 1, then, for fixed c' and d', a' = d' + c't, b' = - c' + d't by (14.2.5) applied to the conjugate matrix, where t is an integer with (SAK S 2A2Kn2 t << minl Ic,l , Id'I
Summing over c' and d' gives SAK
O
1sc'«K/n2 52A2K 2 1 s c' 4 d'/SAn2
d'
=O(S2A2K2)
The Second Condition
295
possibilities for the conjugate matrix. If SAn2 << 1, then we count the number of choices of c' and d' when a' and b' are fixed, and we get the same bound. Adding the two cases gives the second term in the minimum. O
14.3 THE SECOND CONDITION Whilst the First Condition is algebraic, the Second Condition is analytic. First
we simplify it, using calculations like those following the inversion step (lemma 5.5.3).
Lemma 14.3.1 (The Second Condition)
The Second Coincidence Condition
(14.1.2) implies
q'
h(a/q)
q
h(a'/q')
<< -2,
(14.3.1)
where h(z) is a function constructed from the original function F(x) in the exponential sum and the size parameters T and M. In the case of a family of sums, corresponding to F(x, y) for different values y = y;, then h(z) becomes h(z, y), h(a/q) becomes h(a/q, y), and h(a'/q') becomes h(a'/q', y'). We have
h'(z) h(z)
M2
Q
(14.3.2)
T P' provided that F(3) and F(4) are non zero in Chapters 7 and 9, or F" and F(3) are non zero in Chapter 8, and h'(z) 1 M2 _ Q (14.3.3) T h(z) z P' provided that F(3) and F" F(4) are non zero in Chapters 7 and 9, or F" and F'F(3) - 3F" 2 are non-zero in Chapter 8. In Chapter 10 we have (14.3.2)
-
3F(3)2
and (14.3.3) with M2/T replaced by M/T, provided that F" is non zero for (14.3.2), and F', F", and F' + 2xF" are non zero for (14.3.3).
Proof For the simple exponential sum of Chapter 7, we first consider the inverse function g(z) of f"(x)/2 (in the notation of (7.1.2)). Let f" (g(z)) = 2z, so that
g'(z) = 2/f(3)(g(z)). Then h(z) is defined by
h3(z) = f (')(g (z))/6, and (14.1.2) becomes h3(Z')q'3
h3(z)g3 - 1
S A2,
The First and Second Conditions
296
with z = a/q, z' = a'/q', from which (14.3.1) follows. For a family of sums with parameter y we write f(x, y), g(z, y), and h(z, y), and we replace f" and f (3) by the corresponding partial derivatives with respect to x. We estimate the logarithmic derivatives from
3h2(z)h'(z) = f (4)(g(z))g'(z)/6, with
h'(z) h(z)
2 f (4)(g(z)) 3f(3)(g(z)) f(3) (g(z))
2M2
F(4)(g(z)/M)
3T
(F(3)(g(z)/M))2
For the double exponential sum of Chapter 8, the construction is
h3(z) =f"(g(z))/2,
f'(g(z)) =z, with
h'(z) h(z)
f(3)(g(z))
M2
3(f (g(z)))2
3T
F(3)(g(z)/M)
(F" (g(z)/M))2 In Chapter 9 we have the corresponding formulae with f(x) replaced by f(x,0), and differentiation replaced by partial differentiation with respect to X.
For Chapter 10 the construction is
f'(g(z)) =z,
h2(z) = 1/g(z),
with
h'(z)
1
h(z)
2g(z) f" (g(z))
M
1
2T (g(z)/M)F" (g(z)/M)
Type 2 matrices can give rise to many coincidences; they are the most troublesome to control.
Lemma 14.3.2 (Restrictions on type 2 matrices) When (14.3.2) holds, then upper triangular matrices giving coincidences between minor arcs of the same sum in a family of sums have b <<
2 P/Q.
When (14.3.3) holds, then lower triangular matrices giving coincidences between minor arcs of the same sum in a family of sums have
c «A2Q/P. Proof For upper triangular matrices
r'
q,=
r + bq q
so that q' = q, and (14.3.1) gives 1
O2>>
q
for some 6 between r/q and r'/q.
logh(rll =I q
bh'(
h()
)
The Second Condition
297
For lower triangular matrices r
r'
q'
cr+q'
so that r' = r, and (14.3.1) gives h(q1))-log(-h(q))I
A2>> log(
h'(1/z) d `z2h(1/z) evaluated at some value z = q between q/r and q'/r. c
log(zh(z))I
11
cI
,
z
J
o
Lemma 14.3.3 (The domain of a type 3 matrix) If either (14.3.2) or (14.3.3) holds, then for all pairs of minor arc vectors which satisfy the First and Second Conditions with a given type 3 magic matrix (and with given parameters y and y' in the case of a family of sums), the rational numbers a/q and a'/q' lie in fixed intervals of length A2/1C11
where c is the lower left entry of the magic matrix.
Proof We follow Kolesnik's account in Graham and Kolesnik (1991), which is non-constructive. Huxley and Watt (1988) gave an iterative construction of a nested family of intervals containing a/q. As in Lemma 14.2.1 we write
r/q, r'/q' for a/q, a'/q' to avoid confusion with the usual notation for matrices. In the previous lemma and Lemma 14.2.1 we have
0 » I q'h(r'/q') qh(r/q)
I -I (cz+d)h((az+b)/(cz+ d))
2
-
I'
h(z)
where z = r/q. The functions h(z) for which the right-hand side is constant are modular forms, and are undefined on the real axis, whilst our h(z) is continuous, differentiable, and non-zero. We have log(cz + d) + log h (
az + b cz + d )
logh(z) «O2.
-
(14.3.4)
The derivative of the expression on the left of (14.3.4) is c
cz+d
-T
h' az + b h(z) cz+d) (cz+d)2 h ( - h(z) 1
We need only consider values of z with
The First and Second Conditions
298
matrices is chosen so large that the term c(cz + d) dominates the values of h'/h when (14.3.2) holds. We deduce that (14.3.4) holds for an interval of z of length O(i 2/Icl). In the case when (14.3.3) holds, then we write (14.3.4) as az + b h(z) log(az + b) + log h (cz + d) - log ( z
<<
2.
The derivative is c
h'
1
cz+d + (cz+d)2h
(
az + b
cz + d
cz+d
az+b
h'(z)
)-
h(z)
1
Z.
Again, we need only consider values of z with
P/2Qslaz+bI S2P/Q, since az + b = r'/q when z = r/q. By Lemma 14.2.4 lal is bounded below for type 3 matrices in terms of the constant B, and, as above, we see that (14.3.4) holds for an interval of z of length
O A2
1P
1
lal Q =
O
jcI
0
where we have used Lemma 14.2.4 again.
Lemma 14.3.4 (Restrictions on type 3 transvections) When (14.3.2) holds, then type 3 matrices of the transvection form
r1+uvw
-uw2
uv2
1 - uvw
which give coincidences have l uvl <<
2Q.
Let Bo be the implied constant in Lemma 14.3.3, and let wo/vo be a rational number of least denominator for which, for some 0 between -Bo and Bo and some x in M5x5M2, we have wo
f"(x)
vo
2
+902 vo
in the construction of Chapter 7. Then the number of type 3 transvection matrices with K:5 Id < 2K which fix wo/vo (so that K:!-, i 2Qvo) is
O(K/vo ), and the number of those matrices which fix some other rational number (so that K :! 2Q2) is O((SKP/Q + O2 )log K/vo ), and
vo z max(1, O(Q/P)).
The Second Condition
299
Proof Since the transvection matrix gives a trivial coincidence at w/v, then we have 1 r w 1 =I q - (SBoAa
q
uv2
v
which gives the first inequality. We deduce that the fixed point w/v of the transvection must lie within BOA2/v2 of some value of f"(x)/2. Also, if K< uv2 < 2K, then w/v lies in an interval I of length O(SA + O2/K), where A is as in Lemma 14.2.4. For each L, the number of rational points of denominator at most L in the interval I is
< 1 + O((SA + O2/K)L2), by Lemma 1.2.3. When w/v is chosen, then the number of non-zero choices for the integer u is at most 2K/v2. We have
vo» max(1,1/A),
v
0
and the result follows by partial summation.
Lemma 14.3.5 (Coincidences in the First and Second Conditions) If either P >> Q and (14.3.2) holds, or P << Q and (14.3.3) holds, then (for minor arc
vectors from the same sum in the case of a family of sums), the number of coincidences in the First and Second Conditions with P :!g lal < 2P, Q :!g
q<2Qis
O(PQ + Q2(P2 + Q2) + A1(A1 +A2 )p2Q2).
(14.3.5)
If S is small, but SPQ >> 1, then we have the better estimate
O(SPQ + min(SO2, S2)(P2 + Q2)) + O(SL2 PQ loge PQ)
+ O min S A 1
PQ2 , vo
S01
QZ
2
vo
,
PQ3
A2 2 log Q) + O(5201(O1 + vo
O2)P2Q2),
)
(14.3.6) where vo was defined in the previous lemma. The implied constant depends on the bounds for the derivatives of the function F(x).
Proof The rational numbers a/q lie in an interval of length O(SP/Q). By Lemma 1.2.3 there are O(SPQ) of them. There are 0(1 + O2P/Q + t2Q/P) magic matrices of types 1 and 2 by Lemmas 14.2.2 and 14.3.2, which gives the first two terms in the bounds. If 55 O21 then we can use Lemma 14.2.3, and replace A2 by S. For type 3 matrices we consider separately each range K5 I cl < 2K, where K is a power of 2 with vo << K<< O1Q2.
The First and Second Conditions
300
In Lemma 14.3.3 the rational r/q lies in an interval of length O(O2/K). By Lemma 1.2.3 there are < 1 + O(O2Q2/K) << (Al + A2)QZ/K
(14.3.7)
possibilities for r/q. The first result (14.3.5) follows. For a < 1 we distinguish
three cases. In the proof of Lemma 14.2.4, the number of type 3 matrices which are not transvections is O(52A2K2). They give the last term in (14.3.6). For the transvection matrices we subtract one from (14.3.7), since a transvec-
tion matrix gives a trivial coincidence at w/v. For K << A2Q/P we replace (14.3.7) by the trivial estimate O(6PQ) from Lemma 1.2.3. The number of
0
type 3 transvections is estimated in the previous lemma. 14.4
A FAMILY OF SUMS
A family of exponential sums formed with functions Fi(x) is a very general situation. We suppose that
Fi(x) =F(x, yi), where, for i =1, ... , I, the points yi lie in 05 y < 1 with
yi+1 -yi ? 1/J;
(14.4.1)
this implies that 15 J + 1. In Lemma 14.3.1 the function h(z) in the Second Condition becomes h(z, y). Lemma 14.4.1 (Shifting the parameter) Suppose that (d/dz)log h(z, y) is continuous and bounded away from zero. Then a particular matrix M gives a coincidence in the Second Condition for a particular minor arc vector for
O(O2J+ 1) values of the parameter yi. The condition on h(z, y) corresponds to d F112
1
dx F111
ZC
0
for some constant Co in the constructions of Chapters 7 and 9, and to
- >-
d F12
1
dx F11
Co
in the constructions of Chapters 8 and 10.
Proof We write (14.3.1) as
q' q
h(a/q,y) «A2, h(a'/q', y)
A family of sums
301
to display explicitly the dependence on the parameter. When the rational a/q, the value of y, and the magic matrix are fixed, then a'/q' is fixed, and log h(a'/q', y') lies within an interval of length O(O2). The bound follows from the spacing condition (14.4.1). In the construction of Chapter 7 we have
a
ay
logh(z,Y)=
1
a
3 ay
logF,,I
( g(z, y)
M ,Y
g(z,y)
Full ay
M a and we eliminate g2 by differentiating the definition 3F111 ( 1
a12 F'11
T
M ,Y
+ F1112
=2z
g(z' y)
to get
F111 921M + F112 = 0.
Thus
a ay
- log h(z,Y) _
F1112
3F111
F112 F1111 2
3F111
a F112 3 ax F111 1
There is a similar calculation in the other cases.
0
To control the triangular matrices we require further differentiability conditions to ensure the uniform continuity of the mixed partial derivative h12. There is also a condition that the points y; lie in a shorter interval, analogous to (14.1.5). As with (14.1.5), no generality is lost, but we must subdivide the sum over i into a bounded number of shorter sums before applying the large sieve.
Lemma 14.4.2 (The domain of a type 2 matrix) Suppose that h(z, y) is non-zero and three times continuously differentiable. We state the conditions for upper and lower triangular matrices together: for P > Q put H(z, y) = h(z, y),
0=02, Z=P/Q, butforP
z' I a; a2H(z,Y)I :! Co I H(z,Y)I
(14.4.2)
for 05m<3,05n52,m+nS3,and COZm la; a2H(z,Y)I z I H(z,Y)I
(14.4.3)
form=0, n=1 andn=0, m=1, and COZ2 I H1(z, y)H12(z, y) -H11(z, y)H2(z, y)I Z IH(z, Y)I2 (14.4.4)
for the closed intervals K1 of z and K2 of y containing the rationals r/q (for
The First and Second Conditions
302
P z Q) or q/r (for P < Q) and the points yi, respectively. Here a1 and the suffix
1 in H1 denote partial differentiation with respect to the first variable, and similarly for a2 and the suffix 2. Suppose that the length of K2 is
D<1/40Co, and that
0
1
1-A<20C' Then for all pairs of minor arc vectors which satisfy the First and Second Conditions with a given upper triangular magic matrix and with given parameters
y and y', the rational numbers a/q and a'/q lie in fixed intervals of length
1 P2
X O2
IbI Q2
,
and similarly, with a given lower triangular magic matrix, the rational numbers a/q and a/q' lie in fixed intervals of length A2/ICI.
In the constructions of Chapters 7 and 9 the condition (14.4.4) corresponds to the non-vanishing of the determinants D1
=l Fl
1111
F11112
F1111 I
F1112
forP>-Q, and 3F1211 + 4F11F1111
3F11F111
F lt a
F11111
F1111
F111
F11112
F1112
F112
D2 =
for P < Q. In the construction of Chapter 8 we require the non-vanishing of the corresponding determinants with one less partial differentiation with respect to the first variable throughout. In the construction of Chapter 10 we require the non-vanishing of F112 for P z Q and D3 =
2xF11 + 4F1F11
F1
-2x
F1F111
F11
1
F1 F112
F12
0
forP- Q. For these matrices (14.3.1) becomes
I H(r/q + b, y') - H(r/q, y)1:5 A I H(r/q, y)I This gives
H(r/q + b, y') = B1H(r/q, y),
where 1-0
(14.4.5)
303
A family of sums
We approximate H(z', y) on a neighbourhood of the point (r/q, y)
in
terms of the partial derivatives. By the upper bound (14.4.2). log
H(z', y') H(z, y') + log H(z, y') H(z, y) Iz -z'I Z + CO l y -Y'15 CO
H(z', y')
5 I log
H(z, y)
Thus
H(r/q + b, y') = B2H(r/q + b, y) with log B215 COD.
We now have B1
log
log
B2
H(r/q + b, y)
H(r/q, y)
by the lower bound for H1 in (14.4.3). Thus
Z 'C2 D + Co log (1 and for Iz-z'I5b, Iy - y'I S D, log
H(z',y')
1
C1= 2COD + Co log
H(z, y)
1
1- 0 J .
(14.4.6)
In (14.4.5) we look at the interval of values of z for which (14.4.7) H(z + b, y') - H(z, y) = eH(z, y) for some e in -A 5 E :!g A. Let this interval have length E. By the mean
value theorem there is some z in the interval with
E
d H(z+b,y') dz
H(z, y)
5 2A,
so that
E
I H(z, y)I2 >- I H1(z + b, y')H(z, y) - H(z + b, y')H1(z, y)I
Rearranging the inequality, we have
IH1(z+b,y')-H1(z,y)IS
20 E,
0 IH(z,y)I+COZIH(z,y)I,
and
H1(z + b, y') - H1(z, y) = 'qH(z, y)/Z, with
InI <- (2 + CO) OZ/E.
(14.4.8)
The First and Second Conditions
304
For some 9 in 0 < 0 < 1, the left-hand side of (14.4.7) is
bH,(z+9b,y')-(y-y')H2(z,y'+ 0(y-y')), and, for some cp in 0 < cp < 1, the left-hand side of (14.4.8) is
(pb, y') - (y -y')H12(z, y' + (p(y -y')). Thus
b IH1(z + Bb, y')H12(z, y' + (P(y - y'))
-Hll(z + (pb, y')H2(z, y + 0(y -y')) I <- IH(z, y)I(A IH12(z, y' + (P(y -y'))I
+(2 + CO)
E
< A IH(z, Y)12.
IH2(z, y' + 0(y - y'))I) CO(3E CO)
exp COD.
(14.4.9)
Now for any z', y" with I z - z' 15 b, l y -y" I < D, by (14.4.2) we have b
IH(z,Y)I Zm
y)II/Zm,
< Clec'IH(z, for m < 2, n < 1, and m + n<- 2. The left-hand side of (14.4.9) is at least b I H1(z, y)H12(z, y) -H11(z,Y)H2(z,Y)I - 4bC1e2C, I H(z, y)I2/Z2 >
b
2b 0
IH(z, y)I 2
V
since C1 < 1/10CO. We deduce that
E-4Co(3+Co)
OZ2 b
which is the required result. The same argument works for lower triangular matrices when we write c for b. The lower bound for H1 is the condition (14.3.2) or (14.3.3) of Lemma 14.3.1, and the lower bound for H2 is the condition of Lemma 14.4.1. The condition (14.4.4) is new. In the case P< Q we find that H1H12 - H11H2
4M2
D1 1o/3
9T413
(F111)
T2/3 9M2
D2
and in the case P > Q H1H12-H11H2
10_3
(F111)
A family of sums
305
In the construction of Chapter 10 for P5 Q M F112 H H12 H11H2 4T2 x3(F11)" 1
and in the case P > Q 1
4M
H1H12 -H11H2
D3 x3(F11)3
Lemma 14.4.3 (Coincidences from a family of sums) Suppose that the family of sums is formed with functions F(x, y,) which satisfy (14.4.1) and the conditions of Lemmas 14.3.1, 14.3.5, and 14.4.1. Then the number of coincidences in the First and Second Conditions among minor arc vectors from the whole family
of sums withP> 1 in (14.2.4), then we have the alternative estimate
O(S(02J+ 1)IPQ + z2I2((P2 + Q2)log PQ + SPQ log2 PQ) + 5201(01 +A2 )j2p1Q2 )-
The implied constant is constructed from the bounds for the derivatives of the function F(x, y).
Proof The type 3 matrices are treated by Lemma 14.3.3 for each pair of values y and y' of the parameter, as in Lemma 14.3.5. For type 1 matrices we
fix a/q and y and use Lemma 14.4.1 to count the number of values of y'. We use Lemma 14.4.2 for type 2 matrices. For each y and y' the upper triangular matrices give
E
0 Q2 minA2
151b1« SP/Q
1 p2 SP Ibl Q2'
Q) + 1
= O(min(02 log PQ, S2)P2) +O(SP/Q) coincidences, and similarly for lower triangular matrices. The term O(SP/Q) may be omitted since SPQ >> 1 and A2 PQ >> 1 by Lemma 14.1.1.
In Lemma 10.6.5 we need the maximum number of coincidences involving a particular minor arc. Lemma 14.4.4 (Coincidences with a fixed minor arc) Under the hypotheses of Lemma 14.4.3, the number of coincidences involving a fixed minor arc with
P
The First and Second Conditions
306
Proof We treat the type 1 and 2 matrices in two ways. The first term in the minimum comes from Lemma 14.4.1. For the second estimate, we note that two minor arcs which coincide with the fixed minor arc for the same value of the parameter must coincide with one another for slightly larger values of 01 and 02. The matrix giving the coincidence is the product of a type 1 matrix and a type 2 matrix that obeys Lemma 14.3.2, since the parameter takes the same value for both minor arcs. For type 3 matrices we fix the parameter on the variable minor arc. The magic matrix has c << 01Q2, and r/q fixed, with
r
d
q'
q
c
cq
in (14.2.3). The integer d lies in an interval of length 3/2, so there are at most two possible values for d, and the magic matrix is fixed up to a triangular factor:
(a c
b1=r1 tao bol dJ
0
1)(c
d,
with t << 02P/Q by Lemma 14.3.2 again. The number of type 3 matrices which can give a coincidence at r/q is
0 01Q2 1 + 02
P )) =
O( 1Q2 + 01A2PQ).
A similar argument using (14.2.2) gives
O(01P2 + A102PQ), and we take the minimum.
0
15 Consecutive minor arcs 15.1 PARAMETRIZING RATIONAL POINTS The elementary methods for the First Spacing Problem used the divide-and-
conquer method, dividing the solutions of the inequalities into families parametrized by a real number t. The expressions in the different inequalities were related by differentiation with respect to t, and if one inequality held for
a long interval of t, then the inequality corresponding to its derivative became sharper. The method for the Second Spacing Problem is again divide-and-conquer, dividing the solutions of the inequalities according to the magic matrix. The minor arc vectors are parametrized by rational numbers. We find a similar phenomenon of inequalities related by differentiation, and a long interval of solutions of one inequality giving a sharpening in the inequality corresponding to the derivative: a phenomenon as unexpected as it is beautiful. We give the details for the construction of Chapters 7 and 9. The construction of Chapter 8 differs only in the numbering of the derivatives and in the numerical factors in the Taylor series. After Lemmas 14.1.1 and 14.1.2, the coincidence conditions for a pair of
minor arcs J(a/q), J(al/ql) with Q:5 q, q1 < 2Q can be written as follows:
(')=( C the magic matrix M = (A
ICIStl1ggi;
)(),
(15.1.1)
n ) will be fixed in this discussion; N-1gi
5 02,
p43 ab a1b1 q
< 03,
(15.1.2)
(15.1.3)
q1
IK-K11S'&4 -
(15.1.4)
R2 R2 01 X n2 02 << 1 3 X n2 D4.
(15.1.5)
Here
in all cases.
Consecutive minor arcs
308
In (15.1.1) we have
C «D1Q2 <
a2
at D1Q2 +
42
q
,
jat
A,D<< -C+1. q
The set of possible magic matrices
is
independent of the order of
magnitude Q. We consider an interval on the real line with endpoints e/r, f/s, consecutive fractions in some Farey sequence, with
fr - es = 1.
(15.1.6)
By Lemma 1.2.2, the rationals a/q in the interval can be written as
a=eu+ft,
q=ru+st
with (t, u) = 1. Writing Mf
e
(s r)
ell ,
__(fl
rl
Si
we have
eiu +flt al = M eu + ft _ qt
(ru+st) - riu+sIt
We suppose that the Coincidence Conditions (15.1.2), (15.1.3), and (15.1.4)
hold at a/q, but not necessarily at e/r or at f/s. Since (t, u) = 1, there are integers t, u with
tt + uu =1.
(15.1.7)
(eu + ft) (fit -su) - (et - fu)(ru +st) = 1,
(15.1.8)
We note the identities and
rt-su
u
ru +st + t
r
t(ru +st)
(15.1.9)
We think of a/q and the coefficients of the Taylor polynomial as functions of t and u, so we write b(t, u), K(t, U), A(t, u), µ(t, u), v(t, u), and m(t, u) for the quantities corresponding to a/q in the construction of Chapter 7: f" (m (t, u)) 2
=
a q
+ A(t, u),
µ(t, u) = f (3)(m(t, u))/6,
qf'(m(t, u)) = b(t, u) + K(t, u), v(t, u) = f (4)(m(t, u))/24.
Parametrizing rational points
309
We use a suffix 1 for the corresponding quantities constructed from a1/qi. We use the letters b, m, K, A, µ, v (and these only) without argument for the values at t = 0, u = 1, which occur frequently in the formulae that follow. We introduce a new variable by
n =m(t,u) -m,
n1=m1(t,u) -m1.
How n and n1 depend on t and u will be elucidated in (15.1.12) below. We
take Taylor series about m and m1 with n and n1 as the variables of expansion. These expansions cover U consecutive minor arcs, with
U< R2/rs,
(15.1.10)
n << NU.
The Taylor series given are about the left-hand endpoint; similar expansions are possible about the right-hand endpoint, with sign changes which are left to the curious reader. The order of magnitude assumptions f(3)(x) X 1/NR2, fl4)(x) << I f (3)(x)I/M << 1/MNR2,
guide us as to which terms to keep, and which terms to treat as perturbations. The mean value theorem in the form
f" (m + n) =f"(m) + nf (3)(m) + O(n2 max If (4)(x)I) gives
2(eu+ft)
(ru+st)
+2A(t,u)=
2e
+2A+6nµ+O
r
r
n2
1
MNR2 '
so that, by (15.1.6)
!n
3nµ 1 + Ol M)
t
r(ru +st)
+ A(t, u) - A.
(15.1.11)
Approximating further, we have n
3µr(ru + st)
+O( l+G - (t) +O 1+ M) ( M)
(15.1.12)
where G(x) is the continuous function
G(x) = 1/3µr(rx+s). (15.1.13) We make constant use of (15.1.12) to pass from the discrete to the continuous. The mean value theorem for the first derivative gives
f'(m+n)=f'(m)+nf"(m)+
2
f(3)(m)+O(n3maxIf(4'(x)D,
Consecutive minor arcs
310
so that b(t, u) + K
ru+st
u)
2en
b+K
r
+
b+K
+
r
2en
+
n3
r
nt
+nA(t,u)+nA+ r(ru
2en
b+K
r
r
r +2nA+3n1µ+01 MNRZ)
r
n3
(
+0I
.
-
nt (15.1.14)
+r(ru+st)+ru+st
say, where we have used (15.1.11) to eliminate the non-arithmetic term in A.
The non-arithmetic residual term y in (15.1.14) depends on t and u. We analyse y later. In order of magnitude we have y n n3 ru + st - NR1 + MNR
n1 (1+)((t)+NR1(
1+M)) (15.1.15)
by (15.1.12). We shall see that y is negligible when the right-hand side of (15.1.15) is 0(1/N). We still need some further notation. We have +st)(b + 2en) + K(TU +st) + nt + b(t, u) + K(t, u) _ ( r r r bst 2nt K(ru +st) nt + r (fr-1)+ =(b+2en)u+ + +y
r
= bu + 2n(eu +ft) + H,
r
r
(15.1.16)
where
H=H(t,u)=
(bs - n)t + K(rU+st) +y. r r
(15.1.17)
When I K(t, u)I < -11, we have
b(t, u) = bu + 2n(eu +ft) + h, K(t, u) = 771,
with
h =h(t,u)=[[H(t,u)]], ,q= rt(t, u) _ ((H(t, u))). If K(t, u) is within O(Q/N) of ± Z, then we have to consider two minor arc vectors with b(t, u) differing by one, and K(t, u) taking both values ± z + O(Q/N). The numbers h and i are treated to correspond. We count a coincidence if either possible vector belonging to the arc J(a/q) coincides with a vector belonging to the arc J(al/qi) We can now state the subtle identity at the heart of this chapter.
Coincidence over a short interval
311
Lemma 15.1.1 (Rational coincidence conditions) The Third and Fourth Conditions at a/q can be put into the form n
n1
ru +st X
K - K1
u(h - h1)
t
t
(rlu +s1t) + rl
rlu+slt
r ru+st
'n1 - y1
+
t
r 5 L3,
I-i-7111SO4
(15.1.18)
(15.1.19)
Proof By (15.1.8) we have
(rt - su)b(t, u) = bu(rt - su) + 2n(eu + ft)(rt - su) + (rt - su)h = brut + sit - s) + 2n(ru + st)(et u) + 2n + (rt - su)h. Hence
rt-su
II
l nL+stb(t,u)J
=
-su ru+st +ru+st
((2n_bs
-r(2n-bs
r
-u
11
I`I1
-
ru+st +I1t(ru+st) t)hli1 2n - bs r(H - rl) uh (( ru+st + t(ru+st) t 2n-bs bs-n K r(y - q) ru +st
+-+ ru +st t t(ru +st)
+
n
K
uh
r(y- q)
ru +st
t
t
t(ru +st)
uh
t
We subtract two expressions of this type to get the Third Condition (15.1.3). The Fourth Condition (15.1.4) in this notation is 177 - 7111 «04,
so we can suppose that
Iii - -ql I
is less than 1, and that H and H1 are
approximately congruent modulo one. Thus we can write
[[H1-H]] = [[H1]] - [[H]] =h1-h, to get (15.1.18).
15.2 COINCIDENCE OVER A SHORT INTERVAL We continue the approximation step of the divide-and-conquer method. Our task is now to show that coincidence is a property of the minor arc, not of the
312
Consecutive minor arcs
particular rational point a/q chosen. First we introduce a function g(x) depending on the two sets of reference fractions e/r, f/s and el/rl, fl/sl. We may call it the Fortean function after Charles Fort (1973), the investigator of coincidences. The function g(x) plays the same role in this chapter as the function D3/2(t) in Chapter 11. Lemma 15.2.1 (The Fortean function) Let
g(x)
-
G(x) r
G1(x)
-
rl
1
1
- 3µr2(rx +s)
3µ1ri (r1x +s1)
(15.2.1)
Then g(x) and its derivatives can be written as
g(k)(x) =
(-1)kk! 3µr3
ri+1
k+1
(s1/r,
(rlx
s/r +x ,
+S1)k+l
µr3
µlri
and if the Second Condition holds at a/q = (eu + ft)/(u + st), then, for k > 0 and x = u/t, (Q\k+1(Q2+NRZ01). NR2 nQ2
g(k)(t)<<
(15.2.2)
The last factor may be simplified to 02 in the important case k = 2.
Proof The first assertion follows from (-1)krk-2k!
(-1)kr1-2k!
g (k)(x) +s)k+1
31x(rx
3p1(rlx +s1)k+l
The Second Condition at a/q is µ1(t, u)(r1u + slt)3
µ(t,u)(ru +st)
-1 «A2.
(15.2.3)
From the bound for the fourth derivative f (4)(x) we have
µ(t, u)/µ = 1 + O(n/M),
(15.2.4)
and similarly for µ1(t, u). By (15.1.1) applied to e/r and (eu + ft)/(ru + st) we have
r1u +s1t
r1
Ct
ru +st
r
r(ru +st)
-
« Qtr D1,
(15.2.5)
and (15.1.12) gives tR2
n
rQ
N'
(15.2.6)
We can write (15.2.5) as
ru+st ru+st
r
r
1+Of
n
2
RZ01,
.
(15.2.7)
Coincidence over a short interval
313
Combining (15.2.3), (15.2.4), and (15.2.7) gives (15.2.2), when we notice that A2>> N/M (15.2.8) by the condition N3 << MR2, N << MT-1/4, which was (7.2.5). In the construction of Chapter 8, we find that (15.2.8) still holds because (8.1.5) gives HN2 << MR 2, and
A2 R2/HN. The inequality (15.2.7) is not used for k = 2, when the powers of r and
(ru + st) agree.
O
Lemma 15.2.2 (The arc of coincidence) Let J(e/r) be a minor arc, with e/r the rational of least denominator subject to r >- R. If the Coincidence Conditions
hold at a/r, then they hold at (eu + ft)/(ru + st) whenever n(t, u) << N, but possible weakened by a constant factor.
Proof The First Condition, with the particular magic matrix M, is assumed throughout this chapter. Since r may have order of magnitude smaller than Q, we write the Second, Third, and Fourth Conditions as Al r3
µr 3 -1<<
bs
b1s1
r
r1
2, Q
(15.2.9)
03,
(15.2.10)
K - K1 << n A4,
(15.2.11)
<<
r r
where A2, A3, and A4 are the sizes appropriate for denominators in the range Q5 q < 2Q. First we consider the Second Condition. We reverse part of the calculation of Lemma 15.2.1, deducing (15.2.3) from (15.2.9), (15.2.7), (15.2.4), and (15.2.8) when n << N.
In the Fourth Condition we have
(t r - b1sl bs
n1 )) _
-t( n
- Y1) + ru 11
1
+K1
ru+st
- rlu+s1t
st
r
(K- K1)
(15.2.12)
r1
r
By (15.2.10) and (15.2.6) II
( bs t r
- b,sl 1 II S t II bs - blsl II rl
r
rl
<<
Q
n RZ n A3 « N A 3 << N 04. r QZ
Consecutive minor arcs
314
By (15.1.12), (15.2.1), and (15.2.2) with k = 0 n nll ul rrt n2
rr
tY-Y)=t8(t)+OIY(1+M) 1
NR2 rt2
n2
Q
(tR2)2QN rQ
t
r2 02 +
Qn
R2 N
n
n
NZD4+RZD4<< NL14.
<<
Since K1 is bounded, (15.2.7) gives
ru+st K1(
r
- r1u+s1t r
<<
Qn r
1
n QN
NVA2«N
R2
n n2«No4.
By (15.1.15)
y-yI
nQ /
n2
n
NR1+M)<< N04.
(15.2.13)
At this stage we have
st(K-K3)))+o(N114),
(15.2.14)
and the first term is O(04) by (15.2.11). Since n «N, we deduce that 71 -771«D4,
with the proviso that when q is close to ± 2, then we consider both choices of rl; one of them is close to i7i, the other is close to -,q1. Even in this case, the Fourth Condition holds at a/q for one of the two choices of 17. The Third Condition in the form (15.1.18) is still more complicated, but we have done much of the work already. Since 71- 71, and K - K1 are small, we can remove the fractional part brackets in (15.2.14), and deduce that
K-Kl t
r n r(rt-711) R2 GG - - A4 << 2 D4 «03, t(ru + st) Q Qt N
We have
H-H1=
ti-r bs
bls1 111
rl
+O(L4),
and (15.2.10) gives
bs/r - b1s1/r1= d + O(Q03/r) for some integer d. Thus t
h-h1= [{dt+o(-A3)II1
=dt+O R2 N 113) =dt+O(N L4).
Linearizing the Fourth Condition
315
If D4 is sufficiently small, then h - h1 = dt, and
u(h-hl)/t=ud, an integer. The other terms in (15.1.18) are treated as above. By (15.2.13)
r(y- y1) r tR2 « tQ rQ 04 «D3. t(ru +st) Since -71 - y, is bounded, (15.2.7) gives r1 (,q
yl) t(r1u +s1t)
r t(ru +st)
R2
n
r
<< tQ
NY
O2 <<
02 rGG
3.
QZ
Finally, by (15.1.12), (15.2.1), (15.2.2), and (15.2.6) n
nl
ru+st
r1u+slt
1
(1
u
n2
tg,(t)+OI NR2t << rQ2
n
n`
1
N
02(1+N)+Q+QA2 1
N
<< 2+Q+-L)2<< 3. Thus the Third Condition (15.3.2) continues to hold at a/q, possible weakened by a constant factor. 0 After Lemma 15.2.2 we can, in a sense, drop Modification 1. If f/s and e/r are the two rational points of smallest denominator on a given minor arc, then, up to constant factors, the Coincidence Conditions of Lemma 14.1.1 hold for the y vectors corresponding to f/r and fl/r1 if and only if they hold for the y vectors corresponding to e/s and e1/sl. 15.3
LINEARIZING THE FOURTH CONDITION
After Lemma 15.2.2, we can suppose that the Coincidence Conditions (15.1.1)-(15.1.4) hold for a positive proportion of the rationale a/q with Q5 q < 2Q which are values of f" (x)/2 on a particular minor arc, but with the '5 ' inequality signs replaced by '<<' signs, with some bounded implied constant. Our object is to remove the term in u from (15.1.18), and to replace the absolute value modulo one by an ordinary absolute value. We make a delicate approximation in the Fourth Condition (15.1.4), and use the results of Chapter 4. Lemma 15.3.1 (Analytic coincidence conditions) Let J(a/q) be a minor arc,
Consecutive minor arcs
316
on which a/q = (eu + ft)/(ru + st) is the rational number of smallest denominator subject to q >- R. If Q S q < 2Q, and the Coincidence Conditions hold at
a/q, and if
G -t Smin
N
,(MR2)1/3
(15.3.1)
B3Q
where B3 is a certain absolute constant, then x = u/t satisfies
II(a-Ox +/3-g(x)II«R2/rG(x),
(15.3.2)
la - c -g'(x)I << G(x)r/N2
(15.3.3)
for some integer c, with
a= K- K1,
(15.3.4)
(b + K)S
(b1 + K1)s1
r
r1
2A
2A1
(15.3.5)
+ 3µr
3µ1r1
Proof The rational values of f"(x)/2, a
eu + ft
e
q
ru+st
r
1
r(ru/t+s)'
lie in an interval of length x 1/R2, which corresponds to an interval in x = u/t of length (r2x2 +s2)/R2 for
t>> rQ/R2,
n>> N.
We impose the condition n 4 NU of (15.1.10), where U ::g R2/rs. By (15.1.12)
t x nrQ/NR2 << rQU/R2. As in (15.2.12), the Fourth Condition (15.1.4) is
r1u +s1t bs n n1) ru +st ---+ K1+y-y1 «A4. IIt\---)-t Kr rl r r b1s1 r1
r1
(15.3.6)
Three of the five terms on the left of (15.3.6) are linear in t and u. We combine the other two terms. By (15.1.14) we have
y(t, u)
b(t, u) + K(t, u)
b+K
2en
ru + st
ru + st
r
r
=2nA+3n2µ-
nt
nt r(ru + st ) n3
r(ru+st) +OMNRZ
= n(2A + 3µ(n - G)) + 0(n3/MNR2),
(15.3.7)
Linearizing the Fourth Condition
317
where we have written G for G(u/t). The error term in (15.3.7) is 0(1/N) when G(u/t) is less than the second term in the minimum in Lemma 15.3.1. We use (15.1.12) repeatedly to get
ru(+st =(2A+3µ(n-G))G+OIl N /I. Thus t
(
1
y(t, u) = 3µr (2A + 3µ(n - G)) + OI N )
=r(
M 3A)
When we substitute into (15.3.6), then the only non-linear term is
rG`t)+ 1G1(t) -tI
t
We write (15.3.6) as
Ilau + /3t - tg(u/t)II «04,
(15.3.8)
where a and /3 are defined by (15.3.4) and (15.3.5). Let xo be a value of u/t on the minor arc. We take a Taylor approximation to g(xo +y) with
y << (r2x2 +s2)/R2. In the remainder term we have, for some 9 in 0 < 0 < 1, ty2g"(xo + ey) <<
t(r2x2 +52)2 NR2 R4 a
r3
rt
3
(Q) A2
N
<< 4 _ R Q3 O2 « 041 by Lemma 15.2.1.
For values of u/t corresponding to a positive proportion of the minor arc under consideration, we have a linear approximation of the form
H-Hl = a'u + /3't+O(A4), with
a' = a -g'(xa),
(15.3.9)
and
R -g(xo) +xog'(x(,).
(15.3.10)
Then
IIa'u+/3'tII<<&
.
(15.3.11)
Consecutive minor arcs
318
Let ao/qo be the rational number of smallest denominator which is a value of f" (x)/2 on the minor arc. By Lemma 1.6.3, for Q > B1 max(go, R2/qo)
(15.3.12)
there are R' rational values a/q of f" (x)/2 with Q1165 q < 2Q on the minor arc, lying between ao/qo and an adjacent fraction in some Farey sequence, and
R' Q2/B2R2. Here B1 and B2 are certain constants. By (15.1.12) and the assumption n >> N, the denominator t has fixed order of magnitude Q', say, with
t< Q' x rQG(u/t)/NR2, and u < 2Q/r, so that u Q _<M'> rQ'
t
We can rewrite (15.3.11) as
a'u
6'
v
t + R, - t 5 Q' for some integer v, where 6' «D4. We have set up the conditions of Theorem 4.3.3 with parameters a', (3',
6', M', N', and R'. Either R' ::g max(86'Q',1285'M'Q'2) + 1, (15.3.13) which is impossible if G(u/t) is less than the first term in the minimum in (15.3.1) of the lemma with B3 sufficiently large, or else all the solutions lie on a rational line ey = cx + d, with c, d, and e integers with no common factor, and l
ea
86'Q' rG « N2, - c sR'-1
(15 . 3 . 14)
l
and
1e13-dl <
126'M'Q' R, - 1
R2 << N r
(15.3.15)
.
By the construction in Lemma 1.6.3, the sequence of rationals u/t includes two consecutive Farey fractions u1/t1, u2/t2, so ev1 = cul + dt1,
eve = cue + dt2,
and
c = c(t1u2 - t2u1) = e(t1v2 - t2v1).
Hence e I c, and similarly e I d, and e must be 1. By (15.3.14) and (15.3.15)
Ia'u+(3't-cu-dtl <
206'M'Q'2
5
R' - 1
5 32'
Linearizing the Fourth Condition
319
since the second term in (15.3.13) gives the maximum provided that B1 is sufficiently large in (15.3.12). Thus we have
h-h1=[[H-H1]]=v=cu+dt. When we subtract cu from (15.3.8), then the nearest integer is a multiple of t, and so we can divide by t to get (15.3.2). In (15.1.18) we have
u
t(h-h1)=
c(1 - ft) t
C
+du= r - c+du.
(15.3.16)
The bounds (15.2.7) and (15.2.6) give X11
r1
t
r1u +s1t
r ru +st
1
rlu +slt
t
R2
n
r
<<
r ru +st
r1
I
A2 <<
Qt N
02 «A3.
Q2
When the Fourth Condition (15.1.19) holds, then
r6q1 - ri)
t(ru +st)
R2 N <
By (15.1.15) and (15.3.1)
yr t(nt + St)
n « rtt NR2
1+
n2
1
M < < Q<<
3.
The Second Condition in the form (15.1.18) has now been simplified to nl n K-K1-CI (15.3.17)
<< 03. ru + st
r1u + s1t
t
11
By (15.1.12), (15.2.1), (15.2.6), and (15.3.1) we have
n
nl
ru+st
r1u+sltl
(1 n211 Q(1+MJI tgl(t)+OI
1
G
u
B5 N2 R2
B4-A 25-
B5 2
2
<
B3 Q 1 B3 for some absolute constants B4 and B5, provided that we have chosen B1 Q
sufficiently large. Since
c = [[a -g'(xo)]], by (15.2.2) we have
c<<1+
NR2 r3
rt 12 R2
1-I Q
N2
n2r
<< 1+N3.
(15.3.18)
Consecutive minor arcs
320 Then
K - K1 -C
1
InI R2
t
I«t+NZQ'
and
I(K-K1-C)/tl
the left of (15.3.17) is the absolute value modulo one of a number that is numerically less than one-half, so it is the ordinary absolute value. We deduce that
K-K1-C t and multiplying by t gives (15.3.3).
t
t << D3, g'(u)I
15.4 COINCIDENCE OVER A LONG INTERVAL So far we have considered only one minor arc at a time. The parametrization
was set up so that we could consider a whole interval of minor arcs, each corresponding to an interval of values of x = u/t. Our next lemma is one of interpretation rather than calculation. Lemma 15.4.1 (The resonance curve)
The conditions (15.3.2) and (15.3.3) of
Lemma 15.3.1 mean that there is an integer point (c, d) close to the curve z = K(y). The function K(y) is constructed from g(x). We define k(y) implicitly by
g'(k(y)) = a -y,
(15.4.1)
z = K(y) = (a-y)k(y) -g(k(y)) +(3,
(15.4.2)
and K(y) by The functions k(y) and K(y) are algebraic, with finitely many branches, and each branch has finitely many maxima and minima. When a coincidence occurs
at (eu + ft)/(ru + st), then for y defined by k(y) = u/t, and some integers c, d,
y = c + O(rG(x)/N2), z = d + O(R2/rN).
(15.4.3) (15.4.4)
Proof The equation (15.4.3) is (15.3.3), and (15.4.4) follows on substituting x times (15.3.3) into (15.3.2).
We call z = K(y) the resonance curve because integer points on or close to it correspond to coincidences of vectors in the large sieve, and so to pairs of minor arcs which produce similar sums after Poisson summation; they may differ in length or by a phase factor. The resonance curve depends on the
Coincidence over a long interval
321
magic matrix and on the two reference fractions e/r and f/s used to define the interval. The accuracy in (15.4.3) decreases as we go further from a/r. If the domain of the magic matrix is long, then we must subdivide it. We have a set of resonance curves forming short arcs in the plane, and we want to count
how often a resonance curve comes close to an integer point. In the First Spacing Problem in Chapter 12, most families do not contain an integer solution. We expect that most resonance curves do not approach an integer point.
The functions g(x) and K(y) are complementary functions in the sense of Chapter 4, with a shift of the origin to y = a. We have
K'(y) = -k(y), K" (y) = -k'(y) =1/g" (k(y)), so that formally
y = f dy
= - f g"(x)dx,
K(y)= - f k(y)dy= - f xdy= f xg"(x)dx. The resonance curve has a cusp when g" (x) changes sign.
Lemma 15.4.2 (The long arc of coincidence) There is a constant B7 such that, if there are L minor arcs corresponding to rational numbers between e/r and f/s with Q 5 q < 2Q, with
G(x) 5 min
N ,
B7 04
(MRZ)1/3
(15.4.5)
on which the Coincidence Conditions hold, then there is a particular a/q = (eu + ft)/(ru + st) for which the Coincidence Conditions hold in the stronger form ( 1 l C <<minl AjQ2, 1 O2R2J,
µ1(t, u)(rlu + s1t)3 +st)3
µ(t,u)(ru
(15.4.6)
1
- 1 << L2 '
,
(15.4.7)
and, for some integers c and d u
a - c - g'I t
<<
rG(u/t) LN2
(a - c)t +13-g(t)-d«rG(u/t)'
(15.4.8) (15.4.9)
Proof We choose B7 so that (15.4.5) implies (15.3.1). From the proof of
Consecutive minor arcs
322
Lemma 15.3.1, all the rational points at which the Coincidence Conditions
hold correspond to points y = u/t, z = v/t on a line z = cy + d. We can suppose that L >- 32 (by absorbing powers of 2 into the order-of-magnitude constants in (15.4.6)-(15.4.8), if necessary, in order to cover the cases with L :!g 31). We number these minor arcs Jd in order, and pick points x, = u,/ti in I with Q5 ru; + sti < 2Q, corresponding to minor arcs Jd(i), with i =1, ... , 8 for which
d(i + 1) - d(i) >- L/16, so that
G(xi+1)-G(xi)>> LN. By Cauchy's mean value theorem, there are points y11 ... , y4 with
x2i_ 1
<
yi < x2i for which
a-c-g'(yi)
R2
1
< LN rG(yi) ,
G'(y,) and thus
a-c-g'(yi)<<
R2
LN(ryi + s) '
a sharpening of (15.4.3) by a factor 1/L.
We repeat the process to eliminate a - c. There are points z1, z2 with Y21- 1
g"(zi) G'(zi)
R2
1
L2N2 rzi+s'
and thus R4
1
g" (zi) << L2
N(rzi + s)'s
From the formula for g"(x) in Lemma 15.2.1 we see that
s1/r1 +zi
s/r+zi
3
µr3
(rzi +S)3
µ1r1
NR2
R4
1
L2 N(rzi+s)3
1
R2
L2 N2
which implies that
µl(r1x+S1)3 µ(rx +S)3
R2 -1«LL2 N2 1
(15.4.10)
for x = z1 and x = z2. The left-hand side of (15.4.10) is monotone, since
r1x+s1
r1
rx+s
r
C
r(rx+s)'
Extending the Taylor series
323
where C is an entry in the magin matrix M. Thus (15.4.10) holds for z1 S x < z2, an interval corresponding to at least L/32 complete minor arcs. Giving x a rational value u/t in this interval, we have p.1(r1u +s1t)3
µ(ru +st)3
-1<<
1 R2 L2 N2 <<
1 2
02
(15.4.11)
Hence the left-hand side of (15.4.7) has order n
1
<
2
+
1
<<
j2
02
n3
+
1
l
j
«jy2
A 2,
by the second inequality of (15.3.1). We obtain the corresponding strengthening (15.4.6) of the First Condition on observing that r1z2 +Sl
r1z1 +S1
C
C
rz2+s
rz1+s
r(rz2+s)
r(rz1+s)
= 3µC(G(z2) - G(zl)) >> CL/R2. Thus (15.4.10) gives 1
C
R4
1
L3 N2 << L3 A2R2 <<
o
as asserted.
15.5 EXTENDING THE TAYLOR SERIES We want the parametrization to take in as many consecutive minor arcs as possible. The upper bounds on G(x) are u
N2
N
in (15.3.1), which corresponds to the fact that we only use the Fourth Condition when applying Theorem 4.3.3, and G
tu < (MR2)1/3
(15.5.1)
in (15.3.1), which allowed us to neglect the terms in the fourth derivative. This last condition can be relaxed, at the cost of further complication. Suppose that f(x) is five times differentiable, with the natural bound f's)(x) << I f (3)(x)I/M2 << 1/M2NR2.
Consecutive minor arcs
324
We take the Taylor polynomials in the first section to one degree higher, with explicit terms in v = f (4)(m)/24. We replace (15.5.1) with the bound n2 << MR,
(15.5.2)
n4fcs>(m) << 1/N
(15.5.3)
which implies that
in the Taylor series. The mean value theorem for the third derivative gives f (3)(M + n) =f (3)(m) + of (4)(m) + O(n2 Max If (1)(X)I),
so that 61k(t, u) = 6,u + 24nv+ O(n2/ M2NR 2).
To eliminate µ(t, u) from the Seco nd Condition, we w rite 1
1+4nv
1
p.(ru + st)3 1
11(n, +st)
-1 /A.
4nv
1-
3
-I
n2
+O m2
n2 1
+0
I
M
1 3
p.(ru +st)
-
4vt
NR2
3µ3r(ru +st)
4
+O M 3 I
Q
n2 11+ 1
M )) , (15.5.4)
where we have used (15.1.12). The Second Condition can be expressed as 1
NR2
1
µ(t, u)(ru +st)3
µ1(t, u)(rlu +slt)3
<<
A2'
(15.5.5)
Q3
the error term in (15.5.5) dominates the error term in (15.5.4) because we assume (15.5.2). We substitute (15.5.4) and the corresponding relation for µ1(t, u) to make the left-hand side of (15.5.5) into u
3
(NR2
u
Q3
1
n2
(M+MZ
where h(x) is a correction to the Fortean function g(x), defined by h( x )
4v
4v1
27p.1.3r3(rx+s)2
27µi3(rlx+sl)2
(15 . 5 . 6)
The Second Condition implies u
g
(t)-h( _u )t«
NR2
t3NR2 3
_6T_
02
which corresponds to (15.2.2) of Lemma 15.2.1.
3A 2 ,
(ru/t _+d
(15.5.7)
Extending the Taylor series
325
Next we examine the mean value theorem for the second derivative:
f" (m + n) =f" (m) + of (3)(m) +
2
f (4)(m) + O(n3 max If (')(x) 1),
which gives
2(eu+ft)
ru+st
2e
+2A(t,u)=
r
n3
(
+2A+6nµ+12n2v+Ol`M2NR2
1
and
n-
t
2n v
3µr(ru +st)
t
=G(U)
n3
+O(1 + M2 )
Vt2
-
n3
(15.5.8)
9µ3r2(nt + st)
This gives n
nl
ru+st
rlu+slt
u u --g'(-) + -h'(-) +O(1 3), t t 4t t 3
1
(15.5.9)
where we have used (15.5.3) to estimate the error term. Finally we expand the first derivative by the mean value theorem:
f'(m + n) =f'(m) + nf" (m) +
2
f (3)(M) + 6 f(m) + O(n4 max 1(5)(x)D,
which gives
b(t, u) + K(t, u)
b+K
ru+st
r
+
(1 2en' +2nA+3n2µ+4n3v+Ol N ) r
by (15.5.4). From the definition (15.1.14) of y we have
y
ru +st
=2nA+3n2µ+4n3v-
nt
r(ru +st)
+O
1
(N)
= G(2A + 3µ(n - G) + 4n2v) + 0(1/N). We use (15.1.12) repeatedly to get
ru+st
=G(2A+3µ(n-G)+4G2v)+OI N1.
Thus t
2A 1
u y=-In-G(-) +-J + r t 3µ
4vt3 27µ3r3(nc +St)2
+0(i14). (15.5.10)
Consecutive minor arcs
326
We can also substitute from (15.5.8) to obtain tat 2vt3
3µr
27µ3r3(ru +st)
Continuing as in Section 15.3, we have
yr t(ru +st)
u I +O(A3). -h'(4t t 1
y1r1
t(r1u +s1t)
(15.5.11)
We are now ready to consider the Fourth Condition. From (15.5.10) u 2ltt u nt nit 2A1t
Y- y1= r -
tg(t) + 3µr
r1
3µ1r1
+th(t) +O(04).
To linearize as in Section 15.3 we require
ty2(g"(x) -h"(x)) «04, where x
(15.5.12)
a value corresponding to the minor arc considered, and
is
y << (r2x2 +s2)/R2. The estimate (15.5.7) has the same strength as Lemma 15.2.1, so (15.5.12) follows from the Second Condition in the form (15.5.7). We have a linear approximation of the same form as in Section 15.3 with
a' = a -g'(xo) +h'(xo), 0' _ /3 -g(xo) +h(xo) +xog'(xo) -xoh'(xo). If (15.3.11) and the first inequality of (15.3.1) hold, then we recover the bounds (15.3.14) and (15.3.15) for 11 a'11 and 11 011. We can still divide by t in
the Fourth Condition. We turn to the Third Condition in the form (15.1.18). When the Fourth Condition holds, then the terms involving -1 and -q1 are O(A3), and the term in u is given by (15.3.16). The left-hand side of (15.1.18) is
II
n
n1
ru +st
r1u +s1t 1
u
+
a-c t
+
u
1
-
yr t(ru +st)
tg'(t+th'(t+
y1r1
T(-r1u -+s it)
a-c t
(15.5.13)
+O(A3),
by (15.5.9) and (15.5.11). As in Section 15.3, we have
t
gl
( t ) I < 8'
and
I(a - c)/tI < s, if B1 and B3 are sufficiently large and the first inequality of (15.3.1) holds. By (15.5.6) and (15.5.3) we have u « -h'(-) t t 1
vt2
µ3r2(ru +st)3
I
u «-G2(-) <<MQ B1 t 1
1
Extending the Taylor series
327
and so if B1 is sufficiently large, then the sum of the three terms in (15.5.13) is less than one-half, and we may replace the absolute value modulo one in (15.5.13) by the ordinary absolute value. We can now reassert Lemma 15.3.1 in a modified form. Again, a similar calculation works for the construction of
Chapter 8, but with different numerical factors associated with the nth derivatives.
Lemma 15.5.1 (Higher analytic coincidence conditions) Let J(a/q) be a minor arc, on which a/q = (eu + ft)/(ru + st) is the rational number of smallest denominator subject to q z R. If Q5 q < 2Q, and the Coincidence Conditions hold at a /q, and if u
G
t
2
< min
-
B3Q
,
(MR)1/2 ,
(15.5.14)
where B3 is a certain absolute constant, then x = u/t satisfies
II(a - c)x + /3 -g(x) +h(x)II «R2/rG(x), I a - c -g'(x) + h'(x)I << rG(x)/N2
(15.5.15) (15.5.16)
0
for some integer c, with a and /3 as in Lemma 15.3.1.
Lemma 15.4.1 continues to hold with the modification that g(x) is replaced by the more accurate Fortean function g(x) - h(x) at all occurrences. There is one change in Lemma 15.4.2. Because we have a more accurate Fortean function, (15.4.10) in the proof of Lemma 15.4.2 is replaced by
µ1(t, u)(rlu +s1t)3
µ(t, u)(ru +st)
3
1
<
R2
1(
NZ+M
1+M n2
using (15.5.7). This is just what we want to prove (15.4.7) of the lemma, but not so convenient for (15.4.6). From the bound for the fourth derivative we have
µ(t1, u1) p.(t2,u2)
-1<
and similarly for µ1(t, u). Because we have replaced the bound (15.3.1) by (15.5.14), we cannot absorb O(LN/M) into O(02/L2). The new lemma is as follows.
Lemma 15.5.2 (The very long arc of coincidence) There is a constant B7 such that, if there are L minor arcs corresponding to rational numbers between e/r and f/s with Q < q < 2Q, with
G(x) < min
N B 704 7A4
(MR)1"2
(15.5.17)
Consecutive minor arcs
328
on which the Coincidence Conditions hold, then there is a particular a/q = (eu + ft)/(ru + st) for which the Coincidence Conditions hold in the stronger form
C<< minOlQ2,
1
L3
µ1(t, u)(rlu + Slt)3
µ(t, u)(ru +st)3
A2R2+
NR2
(15.5.18)
M
1 « 02, -L2 1
(15.5.19)
and for some integers c and d
a - c - g'( `l
(a-c) t +13-8(
t1
u
) + h'(
/
u
t)«
+h(t) -d<<
rG(u/t)
l
(15.5.20)
z
t)
(15.5.21)
0
16 The Third and Fourth Conditions 16.1
COUNTING COINCIDENT PAIRS OF MINOR ARCS
We put together the ideas of the two previous chapters, We want to count coincident pairs among the vectors y constructed from polynomial approximations on the minor arcs. The parameters of the method, M and T, and also H in Chapters 8 and 9 are immutable. The choice of N, the length of the short sums, was made at the outset, and determines R, and so which arcs are major, which are minor. Some fine tuning can be done before applying the large sieve. In each range Qs q 5 2Q for the denominator of the rational
number a/q, we can subdivide into a finite number of cases in which P< a < (1 + 6)P for some S» 1. We have also taken an extra parameter V>_ 1 in the First Condition in order to restrict the set of magic matrices.
The first type of magic matrix consists of the identity (and perhaps a bounded number of other matrices), which gives a coincidence on every Farey
arc. For each magic matrix of types 2 and 3, our aim is to show that minor arcs on which the Second, Third, and Fourth Conditions hold, and the First Condition with this particular magic matrix, are widely separated. We con-
struct a set of reference fractions as in Section 1.6 of Chapter 1, with consecutive reference fractions being about U/RZ apart (where the integer U is a free parameter), so that the interval between two consecutive reference
fractions corresponds to (parts of) at least U consecutive Farey arcs. We could make U depend on the length of the arc of coincidence, in the sense of Chapter 15, sharpening the counting argument, but not in the dominant term. We call the intervals between consecutive reference fractions reference intervals. The reference intervals will be chosen no longer than the domain of the magic matrix (in the sense of Chapter 14).
We apply Chapter 15 to the rationals lying in a reference interval. The reference fraction with the larger denominator will be a/r. By the construc-
tion of Chapter 1, if e/r is a left-hand endpoint, then there is another
reference fraction f/s with f/s - e/r = 1/rs, and similarly if e/r is a right-hand endpoint. We have
r», R
U
1
RZ
rs
(16.1.1)
In the construction of Chapter 15 we have G(x) << NU. Hence we can satisfy
The Third and Fourth Conditions
330
the conditions of Lemma 15.4.2 when (
1
N
r
NUSB1-min -, -,(MR2) 1/3
(16.1.2)
A2 A4
for some constant B1. We assume that R
1
2/3
U SB2mi n ( NA 2 )
1/3
MR2
1
'A4' (
(16 . 1 . 3)
N3 )
for some constant B2 z B1. By (16.1.1) we see that (16.1.3) implies (16.1.2) uniformly in the reference fraction e/r when B2 is sufficiently large. We use the notation
S=(M2-M)/M of Chapter 14, and we suppose that 6M/N >> U, (16.1.4) so that the reference intervals form a non-trivial subdivision of the range for
f"(x)/2. Lemma 16.1.1 (Type 2 matrices) The number of pairs of minor arcs that coincide in all four conditions, given by triangular matrices, is O min(U
A2,
62) R (P2 + Q2)
Proof We use the Riesz interchange principle. Instead of counting for each type 2 matrix how many coincidences it gives in the Second, Third, and
Fourth Conditions, we consider, for each reference interval, how many matrices give many coincidences. There are O(6M/NU) reference intervals, each corresponding to x U minor arcs. For a particular reference interval and a particular magic matrix, suppose that the number of minor arcs in the reference interval for which the matrix gives a coincidence lies in the range L,. .. , 2L - 1. By Lemma 15.4.2, the Second Condition holds for one of these minor arcs with A2 replaced by O(i2/L2). By Lemma 14.3.2, we have B<<
min(
02, 6
L2
C<<
Q
mint
j
26)
for upper and lower triangular matrices respectively. In the upper triangular case, these matrices give O
P 1 L-min(L0216L
coincidences. Summing L through powers of 2 with L << U, we find that there are at most P Q min(02, 6U)
0
Counting coincident pairs of minor arcs
331
coincidences for minor arcs in this reference interval given by upper triangular matrices. The result follows on summing over reference intervals. El Lemma 16.1.2 (Type 3 matrices) The number of coincidences between pairs of minor arcs given by type 3 matrices is 1
P2R4
1
I W 0/3 )A2
11 2 r(+)62
If S is small, then we can replace this bound by O
P2R4
r1
PR2 log2 R Q
1
Q2 + I U+ v2/3) A2
in the notation of Lemma 14.3.4.
`
Proof We argue as in the last lemma. Suppose that a type 3 matrix gives between L and 2L - 1 coincidences for the minor arcs belonging to some reference interval. Lemma 15.4.2 tells us that the Second Condition holds with 02 replaced by O(02/L2). The domain of the magic matrix in Lemma 14.3.3 extends over 0(/2 R2/L2 ICI) consecutive minor arcs, spread over 0(1 + Q2R2/L2U ICI)
reference intervals. The number of magic matrices with C in a range K5 ICI < 2K is O(K2P2/Q2), and for 6 small it is also
O
62K2P2 Q2
SKP log R Q
+
+tl2logR+
52K2P2
K
=0
v2
+
Q2
KP log R Q
in the notation of Lemma 14.3.4. Here K is a power of 2 satisfying
(1 1 r R21 K<< min(L11Q2,L12I'3)«02R2min( V,L). For fixed K and L, there are R2 DLU Or(L+2-k)
l 52K2P2
KPlogR 11
(16.1.5)
Q + 11 pairs of coincident minor arcs in this range. For fixed L with L << V 1/3 we sum K over powers of 2 with K<< A2R2/V to get
0I
(L+) V
r
Q2
PR2log2R
P2R4
S2L12
Q2V2 +
L12
QV
, 1
and then sum L over powers` of 2 with L << V'1' to get 1
0 (V5/3 +
1
P2R4 Q2
+L12
PR2V log R QV
The Third and Fourth Conditions
332
pairs of coincident minor arcs. For fixed L with L >> V 1"3 we sum K in (16.1.5) over powers of two with K << O2 R2/L3 to get (
OL+
5 2 2 P2R4 Q2L6 +
L2
A2
U
and then sum L over powers of 2 with L >> 1
O(
1
) 82A2 SO2
P2R4 QZ
PR2log2R QL3
V113 2
+L)
to get PR2 loge R Q
pairs of coincident minor arcs. These two estimates give the second result of
the lemma. We can also replace S by 1, and drop the second term in the second bracket of (14.1.5) and the expressions that follow. This gives the other result of the lemma. Lemma 16.1.3 (Coincidences in all four conditions) Suppose that the conditions of Lemma 14.3.5 hold, and that the system of reference fractions is chosen
so that each reference interval corresponds to at least U and to at most C1U consecutive Farey arcs (where the constant C1 may be chosen), and that R
1
U< B3 min
2/3
MRZ
1
, Q4 ,
N3
A2N)
1'3
SM
,V2'3, N
,
(16.1.6)
where B3 is a sufficiently large constant, depending on C1. Then (for minor arc
vectors from the same sum in the case of a family of sums) the number of coincidences between minor arcs which can be written as J(a/q) with
P5IaI<2P,
Qsq<2Q,
is
U02(P2+Q2)+ UA102P2Q2I).
(16.1.7)
I
We also have the estimate 2
O n2 (6PQ+minlU- A2) 82)(P2+Q2) + U O2PQ log2 R +
U 0102P2Q2)
(16.1.8)
in the notation of Lemma 14.3.4, and this is better when S is small. Our next two bounds do not require the condition U5 V 2/3 in (16.1.6). For A so small that 401Q2 5 1, the bound becomes
/1 l1
(PQ + min( U O2, 62)P2 I I,
(16.1.9)
Counting coincident pairs of minor arcs
333
and for Al so small that B401P2 < 1, where B4 is a constant constructed from the bounds for the derivatives of the function F(x), then the bound becomes rS R2 (16.1.10) O Q2I SPQ+mint UD2,S2Q211.
The implied constants are constructed from the derivatives of the function F(x).
Proof This follows from Lemmas 16.1.1 and 16.1.2 when we note that A1Q2 X A2RZ/V. If 4A1Q2 < 1, then all magic matrices are upper triangular, and if t11P2 is sufficiently small, then all magic matrices are lower triangular. In both these cases the terms from Lemma 16.1.2 do not occur. Lemma 16.1.4 (Coincidences from a family of sums) Suppose that SPQ >> 1, and that the family of sums is formed with functions F(x, y,) satisfying (14.4.1), and the conditions of Lemmas 14.3.1, 14.3.5, and 14.4.1. Suppose that the system of reference fractions is chosen so that each reference interval corresponds to at least U and to at most C1U consecutive Farey arcs (where the constant C1 may be chosen), and that
R 2/3 1 mR U<_B5 min (-) , Q4,N2 1
1/2
M 2/3
SN1
SM
log M, N
(16.1.11)
where B5 is a sufficiently large constant, depending on C1. Then the number of coincidences between minor arcs which can be written as J(a/q) with
P
Q2Q,
is 2
UA2J)+UAZIZ(PZ+QZ)logPQ+ UAl02I2P2Q2)J. (16.1.12) We also have the estimate involving 8: z
O QZ (ISPQI 1 + U OZJ) + U t2I2((P2 + Q2)log PQ +SPQ log2 PQ) 1
SZ
1
+ U Al A2J2p2Q2)
.
(16.1.13)
Our next two bounds do not require the condition U< V 2/3 in (16.1.11). For Al so small that 401Q2 < 1, the bound becomes / (RZ 1 OI Q2 SIPQI 1 + U A2J) + min( U O2 log PQ, 62) I2P211, (16.1.14) I
The Third and Fourth Conditions
334
and for Al so small that B4 A, P2 S 1, where B4 is a constant constructed from the bounds for the derivatives of the function F(x, y), then the bound becomes
0 (Q21 SIPQ(1+ UA2J) +min(UA2logPQ,S2JI2Q2I). (16.1.15) The implied constants are constructed from the derivatives of the function F(x, y). Proof In the argument leading to the previous lemma, we use Lemma 15.5.2 in place of Lemma 15.4.2 and Lemmas 14.4.1 and 14.4.2 in place of Lemma 14.3.2 as in the proof of Lemma 14.4.3. Lemma 14.4.2 introduces a new case: triangular matrices with a large entry off the diagonal, with a domain smaller than the whole range for f"(x)/2. By analogy with (16.1.5) of Lemma 16.1.2, triangular matrices with K< Cl I5 2K - 1 have K << min
,
P DZ R2 L3 SQ
,
and they continue KLU
))
coincident pairs belonging to sets of between L and 2L - 1 coincidences in
intervals between consecutive reference fractions. We sum over L << (A2R
2/K)1/3 to get
0((K2A2R2)1/3
+ Q2R2/U),
and then over K << SQ/P to get 2Q
O
Q 2R2
(s
)1/3
+ U log Q R2
coincidences. The condition U << (A2M/6N)2/3 log M
in (16.1.11) ensures that the first term in this bound can be omitted. We argue similarly for upper triangular matrices: the same condition is necessary. We have simplified the upper bounds by replacing the minimum denominator vo by its lower bound vo >> max(1, Q/P). o
Lemma 16.1.5 (Limitation result) There is a constant B6 such that for S=
MZMM >_ Bb min
( i, Ql
(16.1.16)
Counting coincident pairs of minor arcs
335
the number of coincidences between minor arc vectors corresponding to rationals
a/q with Q5 q < 4Q has order of magnitude >> SPQ + S min(6, (12)A3 Q4(P2 + Q2) + S2 A1 A2 L3
A4P2Q2.
(16.1.17)
Proof We consider the construction of Chapter 7 and the case P >_ Q. For Q < P we count rationals q/a and consider lower triangular matrices instead
of upper triangular matrices. By Lemma 1.6.2, the number of possible rational numbers a/q in the range of f"(x)/2 is 6TQ2
2C3M2
- 4Q,
where C3 is the constant defined in Chapter 7. If B6 is large enough in (16.1.16), then this number is >> 6PQ. The first term in (16.1.17) is the contribution of the identity matrix. The third term in (16.1.17) comes from the usual box argument. Since the numbers 0; are bounded, then we can divide the appropriate region of four-dimensional space into K boxes, in such a way that two minor arc vectors in the same box must register a coincidence, and K << 1/O1 O2 A3 04. Suppose that Wk vectors fall into the kth box. The number of coincidences is at least Wk > K
Wk)2 > Al 2 03 A4(SPQ)2.
To consider upper triangular matrices, we divide three-dimensional space corresponding to a/q and the third and fourth entries of the minor arc vector into K boxes, such that if two minor arc vectors with the same denominator q fall into the same box, then there is a coincidence in the Second, Third, and Fourth Conditions. The number of boxes satisfies
K « 6/02 A3 A4 for a >_ A2, and
K << 1/03 A4
for S < A2. Suppose that Wk(b, q) vectors corresponding to rational numbers
a/q with a = b(mod q) in the range of f"(x)/2 fall into the k th box. The number of coincidences is at least
F, E E Wk (b,q)>- (KL E 1)-1(I F, E Wk(b,q)) k
q
q
b mod q
b mod q
S2P2Q2 >>
Q2
k
q
2
b mod q
1
L3 A4 min 11 S O2).
We argue similarly with lower triangular matrices.
The form (16.1.17) of the result follows because the maximum of four expressions is at least one-quarter of their sum.
11
The Third and Fourth Conditions
336
16.2 SUMS WITH CONGRUENCE CONDITIONS Investigations in number theory often produce sums of the form
E a(m)e(f(m)), m
where a(m) is a periodic function that depends only on the residue of a modulo some fixed integer k. We can write the sum as
E a(l) F, e(f(kn+l)),
(16.2.1)
n
l mod k
or, by the finite Fourier transform, as
k E a(l) E el [ mod k
r
1
k l I E el f(m) +
h mod k`
hm
1
(16.2.2)
IM
When m has size m x M, then (16.2.1) gives a family of k exponential sums with summand n of size n x M/k, and (16.2.2) gives a family of k sums with
summand m x M. The sums are shorter in (16.2.1), but there may be cancellation in the sum over l in (16.2.2) when we evaluate the finite Fourier transform. In each case we have a family of sums which differ by a very small perturbation, so small that the centres of the Farey arcs J(a/q) are the same for each sum of the family. The corresponding minor arc vectors differ only in
the terms from the first derivative. The sum (16.2.1) corresponds (with a change of notation) to the sums M2
Sh = F, e(f(m +h/k)),
(16.2.3)
m=M and the sum (16.2.2) to M2
Sh = E e(f(m) +hm/k).
(16.2.4)
m=M
We call (16.2.3) and (16.2.4) congruence families of sums. The first derivative
in So, f'(m) = (b + x)/q, corresponds to the first derivative in Sh, being b+K h b+K 2ah 1 1 q + kq +ONRZ q
in (16.2.3), and b + K
h
k
q
in (16.2.4). Hence K is replaced by k 2ah
K+
Sums with congruence conditions
337
modulo one in the sum (16.2.3), and by
in the sum (16.2.4). The condition corresponding to SPQ >> 1 is (16.2.5)
SPQ >> k.
Lemma 16.2.1 (Type 1 matrices acting on congruence families) For a congruence family of sums of the form (16.2.3) or (16.2.4), satisfying (16.2.5), the number of coincidences between pairs of minor arc vectors indexed by rational
numbers a/gwith PS al <2P,Q5q<2Qis O(S(04k+d(k))kPQ). Proof We consider the sums (16.2.4). The division of the sum into Farey arcs, and the choice of rational approximations a/q to f"(x)/2, is the same
for each sum of the family. In the Fourth Condition, K in the sum So corresponds to ((K + hq/k)) modulo one in the sum Sh. Let 1= (k, q). There
are k/l different numbers of the form ((K + hq/k)), spaced 1/k apart modulo one, each occurring l times. The number of coincidences between different sums Sh with the same a/q (given by the identity matrix) is
0 11 +t14 l
) =O(1+A4k).
For a type 1 matrix other than the identity, we argue similarly. If the vector
y(a/q, h) coincides with both y(a'/q', h') and y(a'/q', h"), then
K+hk -K'- kq]5a4, and similarly for h", so that
(h' k
(I
)q' 5
204,
and, for given a/q and h, a'/q' is fixed by the magic matrix, and there are 0(1' + A4k) choices for h', where 1' = (k, q'). For each factor 1 of k, let M(1) be the number of minor arc vectors with 1 I q, and let N(I) be the number of minor arc vectors with (k, q) = 1. Then, for r 11,
M(r) _ E N(l), Ilk, rIl and
F, 1N(1) = F, N(l) F, cp(r) = F, cp(r)M(r). Ilk
Ilk
rll
rlk
The Third and Fourth Conditions
338
By Lemma 1.2.3 we have
r ram (7Q)2
M(r)=0I
+1J
WRY
=0(
SPQ
+1).
We multiply by l + 04k, and sum over the factors 1 of k, to get
E 0((l + A4k)N(l)) = 0(E cp(r)M(r) + A4kM(1)) rlk
ilk
= O(SPQd(k) + k + S&4kPQ + 04k). For the family of sums (16.2.3), we argue similarly with 1 = (2a, k). The number of minor arc vectors with r 12a is Q2+1)=0(SPQ+1
0(m r again, by Lemma 1.2.3.
)
0
/
Lemma 16.2.2 (Coincidences for a congruence family) Suppose that (16.2.5) and the conditions of Lemma 16.1.3 hold. Then the number of coincidences between minor arcs among sums of a congruence family of the form (16.2.3) or (16.2.4) which can be written as J(a/q) with P< Jal < 2P, Q5 q < 2Q, is
O(Q ((A4k + d(k))kPQ + U Azkz(Pz + Q2) + U Al O2k2PZQz)) . (16.2.6) We also have the estimate z
O
(
Q2I
S(L14k+d(k))kPQ+min(Uz2,52)kz(P2+Qz) z
1
+ U O2k2PQ log2 R +
1
U A, A2k2PZQ2J
,
(16.2.7)
in the notation of Lemma 14.3.4, and this is better when S is small. For O1 so small that 401Q2 < 1, the bound becomes
O(Qz (S(04k+d(k))kPQ+min(U
O21
82)k2P2) I,
(16.2.8)
and for 01 so small that B4 01 Pz < 1, where B4 is a constant constructed from the derivatives of the function F(x) as in Lemma 16.1.3, then the bound becomes
(A4k+d(k))kPQ+min(UDz,Sz)
(16.2.9)
The implied constants are constructed from the derivatives of the function F(x).
Eliminating the centres of the arcs
339
Proof For type 1 matrices we use the previous lemma. For types 2 and 3 matrices, we argue as in Lemma 16.1.3 for each pair of values h and h'. 16.3
ELIMINATING THE CENTRES OF THE ARCS
The construction of the approximating polynomial depends on the choice of a
centre for the Farey arc. Surprisingly, the Coincidence Conditions can be restated in a slightly weaker form in which the centre m of the Farey arc is not implicit. In the construction of Chapter 7 with f(x) = TF(x/M) as usual, we define a function g(y) by
2y =f"(g(y))
(16.3.1)
The centre of the Farey arc I(a/q) is the integer m with
m = [[g(a/q)]].
(16.3.2)
First we note that
µ=6.(3)(m)=bf(3)(g(a))(1+0(N1
)).
We can change the notation to (16.3.3)
µ-6f(3) (g(9
with a negligible change in the y vector. Hence we can define p. by (16.3.3) in the Coincidence Conditions.
Kolesnik noticed that the implicit m can be eliminated from the Fourth Condition, and, with some loss of accuracy, from the Third Condition also. Lemma 16.3.1 (Semi-analytic coincidence conditions) ditions of Lemma 14.1.1 imply
af'(g(q )) + 2gg(q) -a'f l (g(q )) 1
«01+A3+ 04++ q
The Coincidence Con-
_27g(4)
dal
(16.3.4)
NR'
q
and
a
of
(g(q))-2ag(a) <<
4+
NR2
q'f'(g(q'))+2a'g(a')
(16.3.5)
The Third and Fourth Conditions
340
Proof By the mean value theorem we have
(g(q))+(m_g())P(g())+O(If(3)(x)I)
f'(m)=fl
-f(gI a
I
g(
I + (m
q
) 2a +O(1/NRz).
q
q
Hence q K((qf'(m)))= ((qf'(g()) -2ag(gll +O(NRz
((qf'(g()) - 2ag(q ))) +O(NR q z) modulo one. Thus (for one of the two choices of K if ((K)) is close to ± z) we have
K=
((qf'(g()) -2ag(q))) +O( NR qz )
and so the Fourth Condition implies (16.3.5). Similarly we have =a(f(M)
q)
q af(g(a
m-g(a
)
+
q) a
q =af'((a))
2aa
aK
-+0(laI/NR2)
q
q
(2 2)
K
- `+O(lal/NRz) g-+ (m g(-)) q q q q aK lalz -2m--+O df - f (g(a9 )) q(g(a9 )) q + NR 1
We deduce that Zib
((q ))-((afl(g(q))+2q(g(q))
q
))+O(q+NRz)
modulo one. When we subtract two terms of this type in the Third Condition, then we have aK q
5 q'
a
a'
q
q'
+
K'
a
q(K-K') 501+14
a
q
by the First and Fourth Conditions. Hence the Third Condition implies (16.3.4).
Eliminating the centres of the arcs
341
Lemma 16.3.2 (The arc of analytic coincidence)
Let
G(y) =f'(g(y)) - 2yg(y). Suppose that the Third and Fourth Conditions hold at a/r. Define the rational
f / s b y es - 1(modr),1 5 s S r - 1 and fr = es - 1. Then, fort and u integers with t>-O,u>-O,t+u>-1, we have
/e'u+f't
(eu+ft
(r'u + st)G I
(ru + st)G I
1ru+stil
A5,
ru+s,t
where f'/s' is constructed similarly from e'/r', and
(ru+st+tR2/r)
05 <<
N
(
t2R4
l+
,
r2(ru +st)2
JI
If e
eu + ft
t
r
ru+st
r(ru+st)
1
<< -, RZ
then we have
ru + st
AS <<
N
and the Fourth Condition in the form (16.3.5) holds at (eu + ft)/(ru + st). A similar result holds with f/s constructed by es = - 1(modr), fr = es + 1.
Proof The functions f'(x)/2, -G(y)/2 are complementary in the sense that their derivatives are inverse functions. We have
G'(y) = f"(g(y))g'(y) - 2g(y) - 2yg'(y)- - 2g(y), g'(y) =
(16.3.6)
2
(16.3.7)
f(3)(g(y))
- M « RN 2
(4)
g"(y)
(f )(g(y)))3
4
6
(16.3.8)
,
where we have used (7.2.5). The Third and Fourth Conditions at e/r in the form (16.3.4) and (16.3.5) give -f)g(r)-s1G(r,)
sG(e)+2(r R2
R2
N2
rN
+2(ers
s
1
r
1
Nr
r
NR 2
N
r
(
-f)g(r')II
s+
R2
r
and
rG(r) - rG(
r
r'
r
r «N+NR2 «N.
The Third and Fourth Conditions
342
Hence, for t and u non-negative,
ref
2t
(ru+st)GI Y)+
e'
rg(e)
e'
2t
r-(r'u+s't)GI r) r'g(r'
ru+st+tR2/r
(16.3.9)
N By the mean value theorem and (16.3.6), we have
eu+ft e G(ru +st) =G ( r)
t
e
el
t2
r(ru +st) G (r)
+ 2r2(ru +st)2
rl
t3
+0 I
max IG(3)(y)I) r3(rU + St)3
2t
e
a
t2
-G(r)+r(ru+st)g(r) +0
t3
r3(ru + st)3
r2(ru+st)2glr
max lg" (y)IJ
Hence, by (16.3.7) and (16.3.8) we have
(ru
2t -+ r r
(eu + ft +st)G) (ru 1ru+st _
+st)G(e)
t2
`r R6
t3
+
3µr2(ru +st)
e
-gI-)
r (ru +st)2
and we can write (16.3.9) as
(ru+st)G
(e'u +f It
(eu + ft
-(r'u+s't)GI` r'u+s't)
+st)
+t2
3µr (ru +st) 3,u'ri2 (r'u +s't) 1
1
2
<<
ru+st+tR2/r N
+
t3
R6
r3(ru +st)2 N
(16.3.10)
The third term on the left of (16.3.10) involves the Fortean function of Lemma 15.2.1, and we find that t
2(
1
3µr2(ru +st)
-
1
3µ'ri2(r'u +s't) )
We deduce the result of the lemma.
R4 t2(ru+st+tR2/r) N
r2(ru +st)2
0
Eliminating the centres of the arcs
343
We express the condition (16.3.5) in terms of rational points close to a curve y = h(x). Lemma 16.3.3 (The Fortean function) The condition (16.3.5) inplies that there is an integer c with (16.3.11)
where 6 is a positive real number depending only on D4, and
h(x) = G(x) - (Cx +D)G(
Cx +B Y)
Cx+D
is a three times differentiable function depending only on the magic matrix, with
23µ µq (l,q ,3 3
h"
(a) q
=
1
(16.3.12)
.
Proof We deduce (16.3.11) directly from (16.3.5) with c = [[qh(a/q)]]. For (16.3.12) we observe that
h" (x) = -T(3)
4
4
(g(x)) + (Cx + D)3f (3)(g((Ax + B)/(Cx + D))) '
and we substitute the relations 6µ =f(3)( g(alq)) (similarly for µ') and Ca /q + D = q' /q. Lemma 16.3.4 (Linearizing the analytic condition)
There is a constant Bl
such that, for r < N/B1, the rational points (a/q, c/q) in the notation of Lemma 16.3.3 given by Lemma 16.3.2 with a/q = (eu + ft)/(ru + st) satisfying a
e
r 5
1
(16.3.13)
l
q
RZ
lie on a line y = lx + n, where 1 and n are integers with
h(
h,( re
r)
r
) -nl «
1
er)
N The conclusion remains true if (16.3.13) is replaced by 2(min(N'NR
q
rl5B'
with a suitable constant B2.
2
1/3
The Third and Fourth Conditions
344
Proof Since we assume the Second Condition, (16.3.12) gives
h"(a/q) z z 2NR2 << R4/N, and with x = a/q = e/r + z, we have e
h(x)=hI Y +zl =h{ r I +zh'I e I +0 -N4 `
r
lJ
In (4.3.1) we take
/
`
f(x)=ax+/3=xh'I er )+hI In the notation of (16.3.11)
/
`
1
YI +Yh'I Ye `
`
----R C
as
1 +z2R4
q
q
N
Let R' be the number of rational points with q < 2Q given by the construction of Lemma 16.3.2, and let
M' = min(,/R2,1/rs). We have set up the conditions of Lemma 4.3.1 with M and R replaced by
M' and R', and 8 <<
Q
(1 +Mr2R4) << Q
«O4.
If the points (a/q, c/q) do not all lie on a straight line, then Lemma 4.3.1 gives
R' << 1 + M'Q3/N. By Lemma 1.6.3, for Q > 256r we have R'>> M'Q2>> 1. Choosing Q = 256r + 1 gives r >> N, contradicting the assumption r < NIB, if the constant B1 has been chosen sufficiently large. Hence the points do lie
on a straight line, which we can write as Ix + my + n = 0 with integer coefficients satisfying m <- 0, (1, m, n) = 1. As in the proof of Lemma 15.3.1, we see that m = -1. The inequalities for 1 and n follow from Lemma 4.3.2.
If we drop the condition 0:5-.z< M', and replace it by I zI S M", with M" > 1/R2, then we must take 8 <<M"2QR4/N. We still obtain a contradiction unless M'Q2 << R' << 8M" Q2 << Q3Mi 2R4/N.
When we take Q = 256r + 1, then this gives
N 1/3
11/3
(r R2 M"»R2(r) mini,Irsl 1
and we deduce the last assertion of/the lemma.,
Eliminating the centres of the arcs
345
Whilst the constant term n of the line y = lx + n is localized in Lemma 16.3.4, the gradient 1 is not determined accurately. In general h"(x) changes sign, and we are close to a point of inflection. If h"(x) (which must be small
by the Second Condition) is bounded away from zero, then we may use Lemma 4.2.1 in place of Lemma 4.3.1.
Lemma 16.3.5 (A resonance curve for a triangular matrix) Suppose that the condition of Lemmas 16.3.2 and 16.3.4 hold, and also that the magic matrix is triangular, and that the minor arcs come from sums with the same function F(x), satisfying the conditions of Lemma 14.3.1. Let k(y) be the function complementary to h(x), defined by
k(y) = yj(y) - h(j(y)), where x = j(y) is the function inverse toy = h'(x). Then, in the upper triangular case,
M(R2 + rs)2 Ik(1) + n I «A2 IBIN2R4Q2 , and, in the lower triangular case,
Ik(1)+ n1«02
(R2 + rs)2 . ICIMQ2
If the same rational line contains points corresponding to L >- 5 different minor arcs, then we replace (R2 + rs)2 by R4/L2.
Proof For an upper triangular matrix (
I
B), the Second Condition in the
form (16.3.12) gives (aq)
h "(a)
=3
µ
-
1
µ
2 3
(f l3)(g(a/q +B)) 4B
=-3
f (3)(g(a/q))
f (4)(g(g)) (f(3)(g(, )))3
for some q. Upper triangular matrices occur only for M < FT, when we have assumed that IF(4)(x)I is non-zero, so that /
h"(allxA=IBIM2R 4
(16.3.14) q
The Third and Fourth Conditions
346
Lower triangular matrices
('
° ) only occur for M >- FT, when we have
assumed that 3(F(3))2 - F" F(4) is non-zero, and a similar calculation gives
xA=ICIM.
(16.3.15)
We have set up the conditions of Lemma 4.2.1, with h(x) in place of f(x),O(A4) in place of 3, and the distance A between the rational points a;/q; satisfying
A >> min(1/R2,1/rs). We have Ik(1) +nl << A4/Q + L
4/oA2Q2.
The first term is negligible by Lemma 14.3.2, and the second term gives the bounds stated. If the Coincidence Conditions hold for L minor arcs, then we can take A >> L/R2, and in (4.1.14) of Lemma 4.1.5 we can take both factors to be of size >> L/R2, so that
L2/R4 «04/i Q, and we have
A4/Q «A4R4/OLZQ2, and we still omit the term 04/Q. As in Chapter 15, we shall apply these lemmas with either e/r or f/s as the rational number of smallest denominator that is a value of f" (x)/2 on the minor arc. In the second case, e/r is the rational number of second
smallest denominator, and (e - f)/(r - s) is outside the minor arc, but adjacent to f/s in some Farey sequence. In both cases we find that rs 5 3R2.
The construction of Chapter 8 gives corresponding results, but with different derivatives and factorials in (16.3.1) and (16.3.3), and, for example, O5 << (nc + st)/H in Lemma 16.3.2.
Part V Results and applications
17 Exponential sum theorems 17.1
THE SIMPLE EXPONENTIAL SUM
We resume the treatment of the simple exponential sum in one variable. r
M2
S=
r m
eI TFI M))'
(17.1.1)
ME
from Chapter 7, obtaining the results of Huxley (1993a), extended to short intervals and congruence families. We use the notation of Chapter 7, and assume all the restrictions made there on the parameters N (length of short sums) and R (expected size of the denominator). We have estimated the numbers A and B in Lemma 7.4.3. Lemma 17.1.1 (Coincident `integer' vectors) Write A in Lemma 7.4.3 as A where r = 3, 4, 5, or 6 is the exponent of Lemma 7.4.2. Then we have A3 <
R2
H5+
NZ
R2
forQ<<
NZH7
N1-E'
As H7+E
NZ
for Q>> N1-E.
The implied constants depend on E where appropriate.
Proof The coincidence conditions in Lemma 7.4.3 give the inequalities (11.1.1)-(11.1.5) of Chapter 11 with
H X NQ/R2,
SH3/2 X (Q3/NR2)1/2,
so that
S x R2/N2,
SH2 X Q2/R2 >> 1,
SH x Q/N.
We do not change the order of magnitude of S by increasing S to 4/H2 when S is very small. The bounds quoted come from Lemma 11.2.1, Theorem 11.2.5, Theorem 12.4.1, and Theorem 12.1.4.
Exponential sum theorems
350
Lemma 17.1.2 (Choosing parameter sizes) Suppose that
M2 5 (1 + 8)M
(17.1.2)
for some 8< 1, and that the conditions of Lemma 14.3.1 hold. Then we have in the Second Spacing Problem BV << 82L110204/3P2R2V
(17.1.3)
U= (N/B1Q)213, V= U3/2
(17.1.4)
when
(17.1.5)
for some sufficiently large constant B1, for a choice of N and R in the ranges
Nx
Rx
52/19MT-1757,
6-1119MT-20157,
(17.1.6)
when
8» T-s/s1(log
T)3s/27,
(17.1.7)
and
8-24T49 «M114 << 824T65
(17.1.8)
BV << 8010204'3P2R2V,
(17.1.9)
We also have
when U is given by (17.1.4), but V is chosen suitably with A1Q2 x 1, for a choice of N and R in the ranges NX
R X M514T-11 24,
M112T-1/12,
(17.1.10)
with
8» T1/6/M,
Ts/12 << M << T 1/2.
(17.1.11)
We also have (17.1.9) when U is given by (17.1.4), and V is chosen suitably with
A P2 x 1, for a choice of N and R in the ranges R X M314 T- 5/24,
(17.1.12)
T1/2 << M << T7/12.
(17.1.13)
NxM3/2T-7/12,
with
6>>M/T5'6,
Finally we have (17.1.9) with U given by (17.1.4), and V = 1, for a choice of N and R in the ranges (17.1.14) N x R2 x M3/2/T1/2 with
8 >>M/T 2/3,
T1/3 << M << T5112,
(17.1.15)
and also for a choice of N and R in the ranges
NxM1'2,
R
xM514T-1/2
(17.1.16)
with
3 » T 1/3/M,
T7/12 << M << T 2/3.
(17.1.17)
The simple exponential sum
351
When Modification 2 operates, then the bounds for BV must be multiplied by
Q2/R2. The implied constants are constructed from the derivatives of the function F(x), and the order-of-magnitude constants in the ranges for M and 6.
Proof We substitute the values of
.,
into Lemma 16.1.3. The choices
(17.1.4) and (17.1.5) of U and V are valid in (16.1.6) of Lemma 16.1.13 if N5 <<MR4
(17.1.18)
(an easy consequence of (17.1.6)), and if
S» (NS/M3Rz)1/3.
(17.1.19)
The bound (17.1.3) corresponds to the last term in (16.1.8) of Lemma 16.1.3, which dominates the first term in the same bound if 5MN"20/3R4Q5/3,
1 << 6/ 1l202/3PQ x
(17.1.20)
and dominates the second term if 1 << 6 i
m in (Pz , Q z)
SR4Q
x N3 mi n
( M2
N2R4 , 1
(17 . 1 . 21)
and dominates the thir d term if l og e
R «S20 PQ x 1
S 2MQRz
N4
(17 . 1 . 22)
Since N and R satisfy NR2 x M3/T, the choice (17.1.6) is extremal in (17.1.20). The condition (17.1.21) gives the simultaneous conditions (17.1.8), and the condition (17.1.22) gives the condition (17.1.7). If S = 1, then we use (16.1.7) of Lemma 16.1.3 in place of (16.1.8), in which the term that requires
(17.1.22) is absent. We find that (17.1.7) implies (17.1.9), and that the conditions (7.2.3)-(7.2.8) of Chapter 7 can be satisfied by a suitable choice of N and R in the range (17.1.6). In (16.1.9) of Lemma 16.1.3, the second term dominates if 1 <<
2 L4/3P/Q
x MN-11"3Q213
The extremal choice of N and R is (17.1.10). We find that V>_ 1 for M>> Ts/1z The conditions (17.1.11) ensure that (17.1.18), (17.1.19), and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.10). For M << T 'l 12, we take V= 1, and choose N and R so that
1x01Q2 xR4/N2, which puts N and R into the ranges (17.1.14). The conditions (17.1.15) ensure that (17.1.18), (17.1.19), and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.14).
Exponential sum theorems
352
Similarly in (16.1.10) of Lemma 16.1.3, the second term dominates if 1 << L1204/3Q/P x M-1N-5/3Q2/3R4
The extremal choice of N and R is (17.1.12). We find that V >_ 1 for M << T'/12. The conditions (17.1.13) ensure that (17.1.18), (17.1.19) and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.12). For M >> T'/ 12, we take V = 1, and choose N and R so that 1 x O1P2 x M2/N4, which puts N and R into the ranges (17.1.16). The conditions (17.1.17) ensure that (17.1.18), (17.1.19) and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.16).
We can now substitute the estimates for A and B into Lemma 7.4.3, then the result into Lemma 7.4.2, and add up the cases. A complication is that the bound for A5 changes when Q x N1- E, and that we must change the bound for B from Lemma 16.1.3, valid for Q << N, to Lemma 14.3.5, valid for all Q. Our next lemma appears as simple methodology, but it embodies one of the driving ideas of the method. It also prevents the exponent S of Lemma 7.4.2 and the ratio S of Lemma 16.1.3 from appearing in the same formula. Lemma 17.1.3 (Normal-sized denominators) Suppose that Modification 2 is used in the choice of approximating polynomials on the minor arcs. Let a be the largest exponent of Q in any term in the bound for By. There is a choice of S in Lemma 7.4.2 for which the contribution of ranges corresponding to different powers of Q = 2k can be majorized by convergent geometric progressions, whose sum is in order of magnitude the bound obtained for the smallest size range Q x R, provided that a < r + 7 in the range R << Q << N, and a < r + 6 in the range Q >> N. The result still holds if the bound for ABV increases by a factor Q'' as we pass
from Q = Q0 to 2Q0 (S N), provided that =Q1-2*1/(r+7-a)
R << Q1
(17.1.23)
If R >> Q,, then a valid upper bound is given by the order of magnitude for Q x R, multiplied by R'".
Proof In Lemma 7.2.2, the largest power of Q which occurs in W(Q) is Q-1. Hence in Lemma 7.4.2 there is a factor q-3r/2-1+s outside the expression from the large sieve. When Modification 2 operates, then we can insert a
factor R2/Q2, since the sum over a/q contains >> Q2/R2 approximating polynomials corresponding to each minor arc. In the expression ABH5NR2V 3
Q
(17.1.24)
The simple exponential sum
353
of Lemma 7.4.3, we can ignore the factor 1 + Q/N for small Q. Since H NQ/R2, and the largest exponent of H in the bound for A is 2r - 3 + E, then the largest exponent of Q in (17.1.24) is 2r - 1 +a+ e for Q << N, 2r + a + e for Q >> N. We can now work out the largest exponent of Q in Lemma 7.4.2 as a + E
2
-$-
r + 7
2
for Q << N, and larger by z for Q >> N. Hence a positive choice of 6 is possible under the conditions given. If the upper bound for ABV has a discontinuity at Q0Yby a factor Qo, then we define an exponent 0 by (Qo/R)B ^ Q'7,
and insert an extra factor (Q/R)B into (17.1.24). We can still choose a if (17.1.23) holds: otherwise we use the larger estimate with the factor Q'I over the whole range Q >> R.
We can now give bounds for the simple exponential sum S of Chapter 7. Further complications arise when we consider a family of sums Si. Theorem 17.1.4 Let F(x) be a function four times continuously differentiable on the interval 1 <x :!g 2, whose derivatives satisfy the following conditions:
I F(')(x)I 5 Cl
(17.1.25)
I F(')(x)I >_ 1/C1
(17.1.26)
for r = 3, 4,
for r = 3, where Cl is a positive constant. Let 6:5 1 be a positive real number, and let M2
e(TF(mM)),
S= ME
where M and M2 are positive integers, T is a large real number, and M <M2 < (1 + 5)M < 2M < T. Suppose that for some C2 >- 1 either case 1 or case 2 holds:
Case 1 M 5
and (17.1.26) holds for r = 4 also;
Case 2 M >- C21 FT and (17.1.25) and (17.1.26) holds for r = 2 also and IF" (x)F(4)(x) for some positive constant C3.
-
3(F(')(X))21
>- C3
(17.1.27)
Exponential sum theorems
354
If M is sufficiently large in terms of C1, then we have S << 592/95M1/2T89/57olog T
(17.1.28)
for
5-24T49 <<M114 «$24T65, and
S» T-s/81(log T)3s/27 Secondly, we have S << 69/10(M7/20T13/60 +M13/20TH/12o)logT
(17.1.29)
when either S = 1 or S» M-1/4T- 11/108 +M- 1/4T-67/216 +M3/4T- 13/24,
(17.1.30)
for T1/3+E <<M<< T1/2,
and also for M << T1/3+E with log T replaced by T E in (17.1.29). Thirdly, we have S << 69/10(M7/20T 29/120 + M 13/20 T 1115 )log T
(17.1.31)
when either S = 1 or
S»
M1/4T- 19/54 + M1/4T-13/216 + M- 3/4T5/24,
(17.1.32)
for
T1/2 << M<<
T2/3-E,
and also for M >> T 2/ 3 - E with log T replaced by T E in (17.1.31). The implied constants are constructed from C1, C21 C31 from the implied constants in the various ranges for M, and, where appropriate, from e.
Proof The conditions on F(x) are those of Lemma 14.3.1. Lemmas 17.1.1 and 17.1.2 give the estimates for A and B in Lemma 7.4.3. The bound of Lemma 7.4.3 must be substituted into Lemma 7.4.2. Since the bound is uniform in q, we can perform the integration over 77 in Lemma 7.4.2 to get an extra factor log N. We take r = 5, so that the bound for A5 changes form at Q0 = N 1 + E in Lemma 17.1.1. With a slight loss of accuracy, we change the bound for B from Lemma 16.1.3 to Lemma 14.3.5 for Q > Q0. For Q0 < Q << N we have D4 >> N-E,
U << N2E/3,
log PQ << NE/12,
and so the terms of Lemma 14.3.5 correspond to those of (16.1.7) and (16.1.8) of Lemma 16.1.3 with the choice of U in Lemma 17.1.2. We saw in Chapter
15 that the set of possible magic matrices is independent of Q. When there are no type 3 matrices for small Q in (16.1.9) and (16.1.10) of Lemma 16.1.3,
355
The simple exponential sum
then we may omit the terms in Lemma 14.3.5 arising from type 3 matrices also. Hence as Q increases past the value Q0, then the bounds for BV pass smoothly from those of Lemma 16.1.3 to those of Lemma 14.3.5 (with type 3 matrices omitted where appropriate), apart from a factor of the form O(NE). We can now apply Lemma 17.1.3 to get ISIS <<
( M2
-M
5 R 5/2 +
I
N)
(1A5
`+R5/2(W(R))4
logs N
/IBHSNV/R) log5 N,
where the bounds for H and BV are those from the range Q R, provided that (17.1.33)
R << N1-E.
If (17.1.33) is false, then we insert a factor NE after ASBV inside the square root. Hence either S << 8MN-3/2R1/2 log N,
(17.1.34)
S « V-(-MR/N) log N,
(17.1.35)
g N, S «R1/2(W(R))4/5(RH12BV/N)1/10 log
(17.1.36)
or
or
with log N replaced by NE if (17.1.33) does not hold. We suppose that
(1 NR
8>> mint
M)
(17.1.37)
to simplify the expression for W(R) in Lemma 7.2.2, so that (17.1.36) becomes S << 3415M415N311OR -3/5 (BV)11
10
log N.
The estimate (17.1.3) of Lemma 17.1.2 gives SM log N S << N1/30R2/15 f
(17.1.38)
whilst (17.1.9) gives (17.1.38) with 6 replaced by 89/10
The bounds (17.1.28), (17.1.29), and (17.1.31) are given by the various choices of N, R, and V in Lemma 17.1.2. The term (17.1.34) is always smaller than (17.1.38). The term (17.1.35) is smaller than (17.1.38) for large 6. The choice (17.1.6) gives (17.1.28). The conditions on 6 in Lemma 17.1.2, repeated here, imply (17.1.37), and they keep 8 so large that the contribution (17.1.35) is negligible. The two terms of (17.1.29) come from the choices (17.1.10) and (17.1.14) in
Lemma 17.1.2. The first two terms in the lower bound for 6 in (17.1.30)
Exponential sum theorems
356
ensure that the term (17.1.35) cannot dominate. The third term comes from (17.1.37). The lower bounds for S in (17.1.11) and (17.1.14) follow from (17.1.30). We have R< Q1 in Lemma 17.1.3 provided that M>> T'13+E for some e> 0. For M<< T1/3+E, the bound (17.1.29) with a factor T E follows from Lemma 7.3.4. Similarly, the two terms of (17.1.31) come from the choices (17.1.12) and
(17.1.16) in Lemma 17.1.2. The first two terms in the lower bound for 5 in (17.1.32) ensure that the term (17.1.35) cannot dominate. The third term comes from (17.1.37). The lower bounds for S in (17.1.13) and (17.1.17) follow from (17.1.32). We have R5 Q1 in Lemma 17.1.3 provided that M << T2/3- E for some e > 0. For M >> T 2/3- , the bound (17.1.31) with a factor T E follows from Lemma 7.3.4. O
When S = 1, then the bounds of Theorem 17.1.4 give the five possible cases, depending on the size of (log M)/log T. At the end of the range (17.1.8), the bound (17.1.28) agrees with (17.1.29) or (17.1.31). However, for S < 1, the powers of S are different. To fill the gap, we should consider two extra cases in Lemma 17.1.2. There is a different incompleteness for small 6, when the bounds (17.1.30)
and (17.1.32) are determined by edge effects: the short interval may be dominated by a single major arc I(a/q) with q small. If such a major arc occurs, then we should treat the whole sum as a long major arc as in Section 7.5 of Chapter 7. If no such major arc occurs, then the bounds of Theorem 17.1.4 may be extended to smaller values of S. Our next lemma is methodological: for fixed S and T, a bound valid for
one size range M x T' gives some information for other sizes of M. For reasons of notations, the value of S is reduced by 1/M, compared with Theorem 17.1.4.
Lemma 17.1.5 (Extending ranges trivially) Let S be given in 0 < 8< 1. Suppose that there are constants C4 >_ 2, C5 (which may depend on S), a and /3 such that, for a class C of functions closed under the addition of linear terms and under linear changes of variable, we have in the notation of Theorem 17.1.4 ISI
whenever F(x) is in the class C, M2 < (1 + S)M, and M lies in the range T a/C4 < MS C4T a.
Then, for M:5 T
we have ISI 5 C5Tv,
and, for M > C4T ", we have
ISI52C5MTa-a.
Exponential sums with a parameter
357
Proof For M5 T" we pick an integer q > 1 so that
gMST"<(q+1)M. We note that
T"
For a=0,...,q-1, let Fa(x) =F(x) +aMx/T. Then
amodq
e(TFa('j
))=eITFI j
q )
The inner sum over a is q if q I n, zero if `not. Hence qM2
S= 1 F,
F, e(TFF(n )1,
q amodq n=qM
qM
which gives the first result.
For M> C4T" we pick integers q and N with
(q- 1)N<M
T"
Here M = qN - b for some integer bin 0< b < q. We write q-b-1
S= E Ee(TF(gMa)) a=-b
n
JJ
Since q > 2, we have
T"/C4 <M/q < C4T". The range for n begins at n = N, and runs through at most 6N consecutive integers. Hence, for each a, the inner sum has size at most C5T R, and M T P. ISI < gC5T a 5 2C5(q - 1)T,6 5 2C5 N We deduce the second result.
0
This lemma would be extremely useful if it could be sharpened, with cancellation between the q sums over the different residue classes a modulo q. These sums form a congruence family in the sense of Chapter 16.
17.2 EXPONENTIAL SUMS WITH A PARAMETER We consider a family of sums formed with a function F(x, y) for different
Exponential sum theorems
358
values y =y,, as in Section 14.4 of Chapter 14. For simplicity we only consider S = 1 in Lemma 16.1.4. Taking S < 1 would correspond to a family of sums over the same short interval. We do not have an analogue of Lemma 16.1.4
for the commoner case of a family of sums over different short intervals. Even with 5 = 1, there are ten important cases. We have not tried to optimize the logarithm power. Lemma 17.2.1 (Choosing parameter sizes for a family of sums) Suppose that the conditions of Lemmas 14.3.1, 14.4.1, and 14.4.2 hold. Then we have in the Second Spacing Problem (17.2.1)
BV << I2L110202/3P2R2 V
when U and V are given by (17.1.4) and (17.1.5), for suitable choices of N and R in the following ranges: RX1
N x 12119MT - 17/57,
-1119MT- 20157
(17.2.2)
for T$ I33 j
1/57
» JI+( TT7
M2l
T l log M,
(17.2.3)
and
Nx
Rx
(J/I)-2/11MT_3/11,
(J/I)1/11MT-411
(17.2.4)
for T1/3 >>
J
1/57
(T8
>> f 133
M2 T + (Mz + 7 Jlog M,
(17.2.5)
1
We also have, for M << T 1/`2, (
BV<
JQ 7)
(17.2.6)
when U is given by (17.1.4), but V is chosen suitably with A1Q2 x 1, and N and R are suitably chosen, in the following cases. Let
IT l1 -t
T2/3
1
E1 = T8133 + -TUT + ( J + M2
.
When E1 x M2/IT << 1, then we choose N and R in the ranges R X I-118M514T-11/24. NX I1/4M112T-1/12,
(17 . 2 . 7)
(17.2.8)
When E1 x 1/J, then we take N x J114MT- 1/3,
R xJ-118MT-1/3.
(17.2.9)
When E1 x 1/T8"33, then we take
N ,x MT-311,
RX
MT-411.
(17.2.10)
Exponential sums with a parameter
359
When El x T2/3/M2 << 1, then we choose the ranges (17.1.14). ForM>> T1/2 we have
BV «I I +
JP
)iA2i/3P2R2v log PQ,
(17.2.11)
where U is given by (17.1.3), but V is chosen suitably with L1P2 x 1, and N and R are suitably chosen, in the following cases. Let
IM2 -1
M2
1
E2= T8133 +7.,43 +J+ 7
(17.2.12)
When E2 x T/IM2 << 1, then we choose N and R in the ranges R X I- 1/8M3/4T-5/2a N x 11/4M3/2T-7/12,
(17.2.13) When E2 x 1/J, then we use the ranges (17.2.9). When E2 x 1/T8/33, then we
use the ranges (17.2.10). When E2 xM2/T4/3 << 1, then we use the ranges (17.1.16).
When Modification 2 operates, then the bounds for BV must be multiplied by
Q2/R2. The implied constants are constructed from the derivatives of the function F(x), and from the order-of-magnitude constants in the ranges for M, I, and J.
Proof We substitute the values of A j into Lemma 16.1.4. The choices (17.1.4) and (17.1.5) of U and V are valid in (16.1.11) of Lemma 16.1.4 if
N4 z MR3 log M. (17.2.14) This is a weaker condition than (17.1.18) of Lemma 17.1.2, which was never critical. However, we save more in this lemma, as N can be taken larger, and R smaller, than in Lemma 17.1.2. When I is very large, then (17.2.4) becomes the critical condition, and we choose the ranges (17.2.10). There is an extra term involving A 2 J in (17.1.12), (17.1.14), and (17.1.15). The choices of ranges
(17.2.4) and (17.2.9) correspond to the case when this term is the second largest. The other choices of ranges for N and R correspond to cases in Lemma 17.1.2.
Theorem 17.2.2 Suppose that the function F(x, y) is six times differentiable for 1 < x < 2, 0 5 y 5 1, and for some constants C1 >_ 1, C2 > 0 1,91 a2F(x, y)I <_ C1
for 25r<6, 0<sS2, r+s<6, and 1/C15Ia{F(x,y)I
(17.2.15)
for r = 2, 3, and a1
a,F(x, y)
z
C2
I
for r = 3. Suppose also that either Case 1 or Case 2 holds:
(17 . 2 . 16)
Exponential sum theorems
360
Case 1 M << T 1/2 and (17.2.15) and (17.2.16) hold for r = 4;
Case 2 M >> T'I' and 101 >- C4,
13F1211 - F11F11111 >- C31
for some positive constants C31 CO where 3F11z1 + 4F11F1111
3F11F111
F121
F11111
F1111
F111
F11112
F1112
F112
A=
Let Si be the sum e (TF( m
M
mMZ(`) =MI(i)
y;)),
,
where M< MI(i) S M2(i) < 2M, and y,,..., y, lie in 0
yi+1 -yi ? 1/J for i < I - 1. Let e > 0 be arbitrary. Then in both cases, for T8157
T1/3 E»
111/19
(T
J
+ I >> I
Mz 11
Mz
+
T
JlogM,
(17.2.17)
we have
I IS;15 <
Also in case 1 with M << T 1/2 I
JM2 )1/2 1
+
i=1
(log T)
IT
(17.2.19)
when 1
Tz/3
E1 = T8133 + Mz
/
IT `-1
+ J+ Mz
JI
1
« TE,
(17.2.20)
and in case 2 for M >> T 1/2 1/2
r r JT L ISIS << IM5/2T5/6E3/8I 1 + IMz
i=1
1
(log
T)11/2,
(17.2.21)
11
when 1
Mz
(
E2 = T8,33 + T4/3 + I J +
IM2)-' T
I
1
« TE .
(17.2.22)
A congruence family of sums
361
When the inequalities involving e in (17.2.17), (17.2.20), and (17.2.22) do not hold, then the upper bounds hold with the power of log T replaced by T E. The implied constants are constructed from C11 ... , C41 from E, and from the implied order-of-magnitude constants in the ranges for M, I, and J.
Proof We follow the proof of Theorem 17.1.4 with two simplifications: when we apply Lemma 7.4.2, then the first term is negligible, and we do not take a fifth root afterwards. p 17.3
A CONGRUENCE FAMILY OF SUMS
Theorem 17.2.2 does not cover the case of a congruence family of sums as defined in Chapter 16. An analogous result is true, with the same exponents appearing in the bounds. Theorem 17.3.1 Let F(x) satisfy the conditions of Theorem 17.1.4, and let k be a positive integer. Let the sums Sh, for h = 0,..., k - 1, be defined either by / M2
5,,= E el
3k,
f
S,, _ F, e f(m) +
hm )
, (17.3.2) k where M and M2 are positive integers, T is a large real number, and M5 M2 < 2M < T. Suppose that either case 1 or case 2 of Theorem 17.1.4 holds. If M is sufficiently large in terms of C1, then we have k-1 E IShls << (d(k))3/'19k16/19M5/2T89/114 logs T
m=M
0
for k/d(k) << T'/16 and 33
k
T49«M114«(d(k))33 T6s
d(k))
Secondly, we have in case 1 k-1 IShI' << (d(k))3/8k5/8M13/4T11/24 logs T F, 0
+ k(M7/4T13/12 + M5/2T16/21)log5 T
(17.3.3)
for k
d(k)
« min
/ T l
M2)
3/5 ,
T11 14 ,
and
T1/3+e <<M <
M314
TI/4
(17 . 3 . 4)
Exponential sum theorems
362
and also forM << T 1/3+, with log T replaced by T E in (17.3.3). Thirdly, we have
in case 2 k-1
E I SKIS << (d(k))3/$k5/8M7/4T29/24 logs T 0
+k( M1314 T113 + M512 T16/21)logs T
(17.3.5)
for k
d(k)
z min (
M2
3/s ,
T1/14,
T )
T1/2 M314 '
(17.3.6)
and T2/3- E'
T'/2 << M << and also for M >> T 2/ 3 - E with log T replaced by T E in (17.3.6). If the function
F(x) is five times continuously differentiable, with the bound (17.1.25) of Theorem 17.1.4 true for the fifth derivative also, then we may omit the terms M512T16/2' in (17.3.3) and (17.3.5), and the term T1114 in (17.3.4) and (17.3.6). The implied constants are constructed as in Theorem 17.1.4. Corollary Let X(m) be a proper Dirichlet character modulo k, and let M2
Sx= E X(m)e(f(m)) m=M
Then if case 1 holds for M << (kT) , not merely for M << FT, and case 2 holds for M << (T/k) , not merely for M << FT, then we have (d(k))3/95M1/2(k3T)89/570
SX <<
log T
for and
(k)4/i9 d(k)
T23/57 « M
Proof We follow the proof of Lemma 17.1.2 and Theorem 17.1.4, using Lemma 16.2.2 in place of Lemma 16.1.3. For the corollary, when M << (kT) then we write k-1 1 M2 hm 1 k l as in the proof of Lemma 5.4.4. When M >> (kT) , then we write h=0
M
k-1
X(h)
SX =
e(f(nk+h)),
(M-h)/ksns(M2-h)/k and the congruence family has the parameter M replaced by [M/k]. h=0
Sums with T large
363
For larger k we can turn the Second Spacing Problem around, fixing two
minor arcs I(a/q) and I(a'/q') independently of h, and asking for how many pairs h and h' can coincidence occur between the two minor arcs. Another treatment is to consider EIShl as a two-dimensional exponential sum, and to apply a two-dimensional form of the differencing step, Lemma 5.6.2. For a family of the form (17.3.2), the parameter h disappears after the differencing step. The same happens for the sums (17.3.1), apart from edge effects, if k is small compared with the differencing parameter.
17.4 SUMS WITH T LARGE Our method for estimating exponential sums works best when T is close to M2 in order of magnitude. In the double sum, which we shall consider next, M is a length and T has the dimensions of area, so it is natural that T should
be about M2. In the simple exponential sum, T can take any size. The rounding error sums in numerical integration correspond to lattice point problems with T larger than M2. For T > M2 we may need more terms of the Taylor expansion of F(x). The problem becomes more delicate, and the saving that we get is smaller. Sums with T in size ranges between M and M2 are also delicate. We use Poisson summation (Lemma 5.4.3), transforming them to sums of a length M', which is small compared with T, and obtain a saving using the bounds for sums with T large. Our main tool is the differencing step of Lemma 5.6.2, used repeatedly. An analogue of the reduction step of Chapter 3 with r > 1 would be very useful. For any function g(x) we write
0(d)g(x) =g(x + d) -g(x - d). Lemma 5.6.2 expresses the exponential sum S of (17.1.1) formed with f(x) = TF(x/M) in terms of a weighted average of exponential sums formed with the function 0(d)f(x). Unfortunately these include the `diagonal terms' with d = 0, in which every term is 1. The estimate for S must be at least as big as M/ VD in order of magnitude, where D is the largest value of the differencing parameter d. Let E be the average size of the upper bound for the non-diagonal terms with d 0 0. Lemma 5.6.2 gives S<<
(EM).
We iterate by differencing repeatedly with D = D1, D2, ..., so
S «M/ Dl + (E1M) , E1 «M/ DZ + (EZM) where E2 is the average size of the exponential sums formed with 0(d1)A(d2)f(x), and so on. Clearly we choose parameters so that
D,XM/Er,
DD+1 xD, .
(17.4.1)
Exponential sum theorems
364
In an R-step iteration of the form (17.4.1), we see from Theorem 6.1.1 that ER >> tom, so DR << VMW. In Theorem 17.2.2 the root mean fifth power bound is always >> M1/2T47/330 >> M157/220
in the non-trivial range M << T 213. For M >> T 2"3, the bound is >> M112T 1/6 >> M2/3.
Hence DR Z M113, and the estimate for the original sum using R differencing steps must be >> Ml- 1/3X28+'
For T large, we take R large, so we are far from the root mean square size
V.
Lemma 17.4.1 (Smoothing the differences) Let D, L, Q, and R be positive integers. Let A be a set of integers lying between L and 2L - 1. Suppose that for each 1 in A we are given R positive integers d1(l),... , dR(l) with
d1(l)d2(l) ... dR(l) =1, d1(l) + +dR(l)
If (r)(x)I S COT/Mr,
where CO is a constant, for 1 5 x 5 2, r< Q + R + 1. Then there is a function f(x, y), with the second derivative in y existing piecewise, with
f I X, L
-1) = 0(dl(l))... 0(dR(l))f(x)
for 1 in A, whilst a1 2f
(x, y) = 2RLa2(y + I) f ([t+r)(x) + 0
( 2 RDLT ) 11
MR+r+1
(17.4.2)
forr
0(dl) ... L(dr)f (x) =
dR
dR
...
f
d,
f(R)(x + t1 + ... +tR)dtl ... dtR
d,
= 2Rlf (R)(x + r) = 2Rlf (R)(x) +0(2 RjDTIMR + 1)
for some r with ITI S D, and similarly for higher derivatives of f(x).
Sums with T large
365
We define f(x, y) on each interval
(1-L- 2)/Lsys(l-L+ 2)/L for 1= L, ... , 2L. If I is not in the set A, then
f(x,y) = 2R1(y+ 1)f (')(x). If 1 lies in the set A, then f (x, y) = 2R1(y + 1)f (R)(x)
+
L(y + 1) 1
(1- 4(Ly +L -1)2)z
X (0(d1) ... A(dR)f (x) - 2RIf (R)(x)) = 2R1(y + 1)f(R)(x) + 0(2RLDT/MR+1), and 02 f (x, y) = 2RLf (r)(x) + O(2RLDT/Mr+1)a2f(x,y)=O(2RLDT/MR+1)
Theorem 17.4.2 Let R be a positive integer. Let F(x) be a real function R + 6 times continuously differentiable for 1 <x S 2, with F(r)(x) 96 0
(17.4.3)
forr=R+2, R+3, d2
dx2
lo g F(r)
(17.4.4)
0
forr=R+2, 3(F(R+3))2 - F(R+2)F(R+4) 0 0, and
3(F(R+3))2 + 4F(R+2)F(R+4)
F(R+4) F(R+5)
3F (R + 2)F(R + 3)
(F(R+2))2
F(R+3)
F(R+2)
F(R+4)
F(R+3)
0.
I
I
Let the sum S be defined as in Theorem 17.1.4, with M sufficiently large. Let Q = 2R. In the range M5 N6 we also require (17.4.3) for r = R + 4 and (17.4.4) for r = R + 3. Then for any e > 0 we have S <<
M(86Q- 13R-65)/(86Q-26)T13/(86Q-26)(log
T)3(5R2+6)/(43Q-13)
for N1 < M 5 min(N1, N3), and also for M S Nl when we replace the power of logT by TE, S <<
M(428Q- 49R- 263)/(428Q- 98)T49/(428Q- 98)(log
T)33(5R2+ 6)/(428Q- 98)
Exponential sum theorems
366
for N2 5 M 5 N4, and also for N8 5 M 5 N10 if either of these ranges is non-empty, «M(124Q-11R-46)/(124Q-4)T111(124Q-4) (log T)3(5R2+6)/(31Q-1)
S
for max(N3, N4)5MSN5, S <<M(712Q-89R-427)/(712Q- 142)T 89/(712 Q - 142) (log T )57(5R2 + 6)/(1424Q- 284)
forN55M5N7, S
<<M(160Q- 29R- 118)/40(4Q-1)T29/40(4Q-1)(log
T)12(5R2+6)/40(4Q- 1)
for N7 5 M :!g min(N8, N9), S <<M(68Q-4R-29)/(68Q-8)T1/(17Q-2)(log T)3(5R2+6)/(34Q-4)
for max(N9, N10) 5 M 5 N11, and also for M >_ N11 when we replace the logarithm power by T e. The ranges are defined by NR+3-6e = T 1
,
N25QR+139Q+26 = T75Q(logT)-15(5 R2+6)(Q-1) N337QR + 68Q + 13R + 52 = T37Q+13(log T)-16(5R2+6)(Q-1) N476QR + 138Q + 49R + 192 = T76Q+49(log T)-58(5R2+6XQ- 1)
N578QR+302Q+67R+268 = T178Q+67(log
T)-82(5R2+6)(Q-1)+356Q-71
r
N6267QR + 427Q + 18R + 143 = T267Q+18(log T)-57(5R2+6XQ-1)
T356Q - 31(logT)- 32(5R2 + 6XQ - 1) + 356Q - 71 N8254QR+388Q-49R-58 =
T 254Q-49 (log T) - 8(5R2 + 6)(Q - 1)
N937QR+54Q-2R+6 = T37Q-2(log
T)-4(5R2+6)(Q-1)
r
N 900QR + 124 Q + 41 =
T90Q(logT)-18(582+6XQ-1)
N2QR+2Q+4Qe = T2Q 11
The implied constants in the upper bounds are constructed from R and the derivatives of the function F(x), and, where appropriate, from E.
Proof This is the R-step iteration with the parameters satisfying (17.4.1). We
want an upper bound for the sum over d, 5 D, r = 1, 2,..., R, of the exponential sums over subintervals of M5 m < 2M formed with the functions
A(dl)...0(dR)f(x). We write l for the product 1= d1 d2 ... dR, and let S, be the sum of largest
Sums with T large
367
modulus among the exponential sums with d1 d2 ... dR = 1. The average ER in (17.4.1) satisfies
ERs
1
DIDZ ... DR
DID2 ... DR
1=1
F,
dR(1)ISII.
We use Holder's inequality to get rid of the divisor function dR(l): 2L-1
E dR(l) I SII < (E (dR(l))5/4)4/5( IS1I5)1/5 1=L
< (L3/8(Ed2(1))5/$4/5(F'{51151/5 << L4/5(log
L)cRZ- 1)/2(F
IS115)1/5,
where we have used the bounds for divisor functions in Lemmas 11.1.2 and 11.1.3. The power of Log L can be reduced for the important range where 1 is close to D1D2 ... DR by using the fact that d1,..., dR lie in restricted ranges. The divisor function dR(l) can be replaced by a small multiple of the divisor concentration function 1 R(l), whose averages were estimated in Lemma 11.1.4.
We have now set up the hypotheses of Lemma 17.4.1 with A the set of integers I between L and 2L - 1 which occur as products d1d2 ... dR. We can suppose that D1,. .. , DR are powers of 2 with Dr+ 1 = D'2. Then L 5 D1D2
... DR = DR/D1,
that the error term in Lemma 17.4.1 is DR negligible, provided that D1 is sufficiently large. For Theorem 17.2.2 we need to know accurately the derivatives a; f(x, y) for r5 5, and an orderof-magnitude bound for a; f(x, y). This corresponds to taking Q = 5 in Lemma 17.4.1.
We apply Theorem 17.2.2 with I = J = L, and yl =1/L - 1, and f (x, y1) = T'F(x/M, y1), where
T' = LT/MR. We run L through the powers of 2. The largest value of L should dominate the sum: from the structure of the bounds in Theorem 17.2.2, we see that the largest value of L dominates except possibly when the order of magnitude increases as we pass from (17.2.18) to (17.2.21), and when the extra factor T'E appears for M >> T'2/3- E. Since we do not intend to optimize the logarithm powers, we can remove the first discontinuity by replacing the fifth power of log T in (17.2.18) by the 11/2-th power. The discontinuity at can be overcome by using Lemma 17.1.5 for each value of yl, to compare with sums T'2/3-E
368
Exponential sum theorems
of length M', where M' is smaller than
so large that M'2/T'4/3 still the dominant term in (17.2.22). We can use a trivial comparison argument because there is no saving on the sum over the parameter in this T'2/3-E, but
is
range.
First we consider the case when (17.2.17) holds. Then L < T'8133, so the first term dominates in (17.2.18), and T)11/2.
E IS115 << L16/19M5/2TP89/114(log
In the notation of (17.4.1), ER : M1/2L- 18/570T,89/570(log
T)(5R2+6)/10
and the largest value of L satisfies LQ/(2Q-R) ^ DR ^ M112L18/570T,-89/570(log
T)-(5R2+6)/10
giving
D1 xDQ Q
^M(89R+285)/(712Q-142)T89/(712Q-142)(log T)57(5R2+6)/(1424Q-284)
The conditions (17.2.17) correspond to N5 5 M5 N7, and the bound in this
range is S << M/ D1 . For the largest value of L, the condition M2 « T' corresponds to M 5 N6. For smaller values of M, Case 1 will hold for large L and Case 2 for small L. Hence we always need the hypotheses of Case 2 of Theorem 17.2.2. There are similar calculations in the other six cases. 0
a3 I
a4
a8
a'2
1
1
a5
FIG. 17.1
a7
a10 I
a9
I
Sums with T large
369
Table 17.1 Bounds for exponential sums Steps C
C C
Exponent 0
AC AC
A2C
7
517
60
12
873 =
29 + 42a
65
7
120
114
12 =
89 + 285a
49
65
570
114
114 =
11+78a
5
49
120
12
114
13 + 21a
356
5
873
12 =
0.5922...
0.5833...
0.5702.. .
=0.4298...
4+103a
12
356
128
31
873 =
29 + 173a
227
12
280
601
31 =
89 + 908a
423
227
1282
1295
601 =
11+191a
87
423
244
275
1295 =
13 + 94a
1424
87
146
4747
275 =
4+235a
120
1424
264
419
4747 =
0.4167.. . 0.4078.. .
0.3871... 0.3777.. .
0.3266.. . 0.3164.. . 0.3000.. .
49 + 1351 a
967
120
1614
3428
419 =
29 + 464a
199
967
600
716
3428 =
89 + 2243 a
19
199
2706
74
716 =
11 + 428a
161
19
492
646
74 = 0.2568.. .
13 + 253a
2848
161
318
12173
646 =
A2C
A2C
To a=
4+39a
60
AC
From a =
0.2864.. . 0.2821.. .
0.2779...
0.2492...
If F'(x) is a monomial function in the sense of Chapter 3, with monomial part x_S for some s > 0, then the conditions of Theorems 17.1.4 and 17.4.2 hold for each R. We write M = T a. The bounds take the form S << TR(log T)'y
Exponential sum theorems
370
for some y< 1. The implied constant depends on the bounds for the derivatives of F(x), but not on a. In the first column of Table 17.1, C indicates Theorem 17.1.4, and ARC indicates Theorem 17.4.2 with R differencing steps.
Rows of the table bracketed together indicate an upper bound with several terms. The last two columns give the range of a for which each term is used. There are two types of critical values of a. Within a set of bracketed rows we
meet values of a at which one term takes over from another as dominant term. These correspond to re-entrant (concave) vertices of the piecewise linear polygonal graph of 6 against a. Between two unbracketed rows we meet values of a at which the argument giving the best estimate changes. These correspond to convex vertices of the polygonal graph. Figure 17.1 (not
to scale) illustrates the portion of the graph corresponding to ARC. The abscissae a; are given by a1 = (log N)/(log T). For values of a > '1 not in the table we apply step B first. In Section 19.2 we sketch an iterative idea which cuts away some of the corners on the graph. In many applications of exponential sums we have to consider different
order-of-magnitude ranges for M and T. The most important ranges are
those in which a= (log M)/(log T) takes one of the values in the last column of Table 17.1 which is a convex vertex of the graph, and which lies
above the lines adjoining other pairs of convex vertices. Examples are 49/114, 356/873, and 1424/4747; we call these the outstanding vertices. Joining consecutive outstanding vertices gives another polygonal line. Each side of this new polygon corresponds to a bound of the form S << T P log T,
(3 = (a + ba)/c,
(17.4.5)
which is valid for all a > 0 (even for a >_ 1). The classical notation states bounds as S << (T/M)KM'1,
(17.4.6)
where the numbers (K, A) form the exponent-pair as in Lemma 7.3.4. Table 17.2 Outstanding vertices a
/3
712Q(R - 1) + 865Q + 338
712QR + 865Q
712QR + 865Q + 338
1424QR + 1730Q + 676
65
503
114
1140
49
141
114
380
712Q
712Q - 169
712QR + 865Q + 338
712QR + 865Q + 338
Sums with T large
371
Table 17.3 Exponent pairs A- E
K - E 1
169
712R + 1577
1
169
2
1424Q - 338
712
2
1424Q - 338
7577
28229
43860
43860
89
187
570
285
6299
29507
43860
43860
169
1424Q - 338
1
169
712R + 1577
1424Q - 338
712
Because of the logarithm, (17.4.5) corresponds to K = a/c + E, y = (a + b)/c + c, for any e > 0. The integer R in Tables 17.2 and 17.3 takes the values 1, 2, 3,..., with Q = 2 K
18 Lattice points and area 18.1 EXPONENTIAL SUMS The analytic method for estimating the discrepancy between the number of lattice points inside a closed curve involves dividing the curve into five or more pieces, taking local coordinates y =f(x) on each piece, and estimating
the contribution E p(f(m)) that each piece of the curve makes to the discrepancy. The classical estimate is Corollary 2 to Lemma 5.4.3. We use the
Fourier expansion of Lemma 5.3.1, which leads to the sums considered in Chapter 8. Our first task is to use the estimates for the First Spacing Problem in Chapter 13 and the Second Spacing Problem in Chapter 16 to complete
the estimation of the exponential sum, and deduce the bounds for the rounding error sum. As in the last section, we require that certain derivatives and determinants of derivatives do not vanish. We shall show that the local coordinate systems can be chosen to satisfy these conditions. In Chapter 8 the sum divided into Farey arcs. The arc I(a/q) corresponds to a side of the Voronoi-Sierpinski polygon with the same gradient a/q. The saving over Lemma 5.4.3 comes from the large sieve and good estimates in the First and Second Spacing Problems. The three different approaches to the discrepancy of a closed curve in Chapters 2 and 6, and in this chapter, all analyse the curve in terms of tangents in rational directions. We obtain the results of Huxley (1993b), and new rounding error estimates improving those of Huxley (1991). Theorem 18.2.3 corresponds to the result of Iwaniec and Mozzochi (1988).
Lemma 18.1.1 (Choosing parameter sizes) Suppose that the conditions of Lemma 14.3.1 hold. In the Second Spacing Problem we have BV << Al 02 &4/3PZR2 V
(18.1.1)
U= (H/B1Q)2/3
(18.1.2)
with
for some sufficiently large constant B1, in the following cases.
Exponential sums
Case (a)
373
We choose V= U312, for a choice of N and R in the ranges
Nx
m ( M22T18 1/35 T I H22 ) ,
1/35
H11
R
M( M11T9
)
,
(18 . 1 . 3)
provided that MT-318 << H< B2 1MT- 17/57,
(18.1.4)
H>> max(T4/M9, M11/T6),
(18.1.5)
and
Case (b) For M< B3T 1/2, we choose V so that A1Q2 x 1. There are two subcases. If
H> T11/8/B2M3,
(18.1.6)
then (18.1.1) holds for a choice of N and R in the ranges
NxM/(HMT)1/7,
RxM(HSM/T6)1/14,
(18.1.7)
provided that (18.1.8)
M513/T2/3 << H <M112/B2T1/12
If (18.1.6) is reversed, then (18.1.1) holds for a choice of N and R in the ranges
NxM2/(HT2)1/3,
R x (HM3/T)1/6
(18.1.9)
provided that both
M3/5/T1/5 << H _ M3/2/B2T1/2,
(18.1.10)
H5 T4/B2M9.
(18.1.11)
and
Case (c) For M >_ B3 1 T1/2, we choose V so that 01P2 x 1. There are two subcases. If (18.1.12)
H;-> B2M5/T21/8,
then (18.1.1) holds for a choice of N and R in the ranges
Nx (M1/8/H5T7)1/7,
RX
(H5M3)1/14,
(18.1.13)
provided that M1/3 << H
,
M3/2/B2T7/12.
(18.1.14)
Lattice points and area
374
If (18.1.12) is reversed, then (18.1.1) holds for a choice of N and R in the ranges (HM7/T3)1/6
Rx
Nx (M2/H)11'3,
(18.1.15)
provided that both M715/T3/5 << H<M1/2/B2,
(18.1.16)
HsM11/B2T6.
(18.1.17)
an d
The parameter B3 may be chosen to be any positive real number. The constant B2 and the implied constant in (18.1.1) are constructed from the derivatives of F(x) and from B3. When Modification 2 is used, then we replace R2 in (18.1.1) by Q2.
Proof We substitute the values of IX; from Lemma 14.1.1:
_
A1 .
_
R4
A 2 ..
HNQ2V
R2
03 x
HN
R2
HQ
04 x
,
Q
H
The choice (18.1.2) of U is valid in (16.1.6) of Lemma 16.1.3 if H2N3 << MR4
(18.1.18)
(which implies (8.1.5)) and if the constant B in Lemma 16.1.3 is sufficiently large.
The last term in (16.1.7) of Lemma 16.1.3 dominates the other terms with V= U3"2, if 1
(18.1.19)
O2 04/3PQ x MR4Q513/H1113N3,
1 << 0 1 m in (P2 , Q 2)
« H Q mi n ( N
l
Z
R4
,
it .
(18 . 1 . 20)
The choice (18.1.3) is extremal in (18.1.19), and it satisfies (18.1.18). The conditions (18.1.5) imply (18.1.20). With V chosen so that O1Q2 x 1, the last term in (16.1.9) of Lemma 16.1.3
dominates the first term if
1«L1202/3P/QxMQ2"3/H513N2.
(18.1.21)
The extremal choice of N and R is (18.1.7), satisfying V >_ 1 when (18.1.6) holds. When (18.1.6) is false, then we choose V= 1, which puts N and R into the ranges (18.1.9), satisfying (18.1.21), and also satisfying (18.1.18) in the range (18.1.11). With V chosen so that A1P2 x 1, the last term in (16.1.10) of Lemma 16.1.3
dominates the first term if 1 « L)2 24 3Q/P X Q213R4/H513M.
(18.1.22)
The extremal choice of N and R is (18.1.13), and V>_ I when (18.1.12) holds.
Exponential sums
375
When (18.1.12) is false, then we choose V= 1, which puts N and R into ranges (18.1.15), satisfying (18.1.22), and also (18.1.18) in the range (18.1.17). We have to satisfy (8.1.4)-(8.1.7), so that
R :s- H5 N/64C,
(18.1.23)
R >- 2C H _+1
(18.1.24)
,
N<M/10.
(18.1.25)
The requirement (18.1.23) gives the conditions (18.1.4), (18.1.8), (18.1.10), (18.1.14), and (18.1.16). We find that, up to a constant factor, (18.1.24) follows from (18.1.23) and the condition A1Q2 >> 1 which our construction always satisfies. The constant B2 is chosen large enough for (18.1.24) to hold with
the right constant of proportionality. Also (18.1.25) follows easily from (18.1.18) if M and T are sufficiently large. The most difficult verification is (18.1.18) for the choice (18.1.7), where we use both lower bounds (18.1.6) and (18.1.10) in different ranges. Similarly, both (18.1.12) and (18.1.14) are used to verify (18.1.18) for the range (18.1.13). 0 Theorem 18.1.2 Let F(x) be a real function three times continuously differen-
tiable for 1 <x5 2, and let g(x), G(x) be bounded functions of bounded variation on 1 5 x S 2. Let C0,. .. , C6 be real numbers z 1. Let H, M, and T be large parameters. Suppose that
IF(')(x)I < C,
(18.1.26)
forr= 1,2,3, IF(')(x)I
1/C,
(18.1.27)
for r = 1, 2, and that either Case 1 or Case 2 holds:
Case l M5 C0T 1I2 and (18.1.27) holds for r = 3 also, Case 2 M >- Co 'T 1/2 and IF'F(3) - 3F" 21 Z 1/C4.
Let S denote the sum
S= E g( H
H)
m=M
G(M)e(M
F(M)).
Then there are positive constants B1 and B2, depending on C0,. .. , C6, and on the functions g(x) and G(x), such that for M in the range CS 1T49/114 SM5 C5T651114
and H satisfying
H
Lattice points and area
376
and also
HZC61T4/M9 for M
H z C61M11/T6
for M Z C5T9116,
we have
H
3/70
T23/70(log T)9"4.
<
S
(18.1.28)
In the range
CS 1T1135M
H<
min(B1M1/2T-1/12
,
B1M3/2T- 1/12, C6T 4M-9),
we have S
(18.1.29)
+B2H17/1sM-1/6T7/18(log T)9/4.
In the range CS 1T9116 <MS C5T213, min(B1M312T-7/12, B1M1/2,C6M11T-6),
H:!-,
we have S < B2H 15114M-5114 T 1/2 (log T)9/4
+ B2H17/1sM5"8T1/6(log
T)9/4.
(18.1.30)
Proof We substitute into Lemma 8.4.4 and perform the summation over Q. In Chapter 13 we have 71 x R2/HN, 77K x Q/H, so Theorem 13.2.4 gives A << K2L2(1 + 77K)log K << K2L2 log K.
When Modification 2 applies, then Lemma 18.1.1 gives A2 04'3P2Q2V <<
BV
Q 2/3
M2Q2
2 (T:) KLNR 2
2
,
2
and Lemma 8.4.4 gives
E (E F, e(hg,(n)) j
h
n
QW <<
R
(HN) log N + Q
Q
HM
R
(HN)
<< -
1/4
R2 ( ABKL4NR2W 2
log N+
)
Q3
Q )1/3 HM(log N)9"4 (HN3R2)1
6
log2 N,
Integer points and rounding error
377
The sum over j counts each minor arc with Q 5 q < 2Q several (>> Q2/R2) times. When we allow for this, then the sum over ranges Q = 2" converges, and the range Q R dominates. The major arcs contribution from Lemma 8.2.1 is smaller, so
HM(lo N)9"4 S <<
g
(HN3R2
R
1/6
2/3 V
H)
)914,
(log M
(18.1.31)
and we save a factor (R/H)2"3 (up to a logarithm power) over the easy estimate of Lemma 8.2.2.
There is a useful methodological comment here. If all the conditions are satisfied except R5 H, then the bound (18.1.31) is weaker than Lemma 8.2.2. However, for small H we can dispense with the whole apparatus of Chapter 8. Using Corollary 1 to Lemma 5.4.3 for each value of h gives
S << (H3T/M)112(1 +M2/HT),
(18.1.32)
which is better than (18.1.31) when
M2/T << H<< R.
If H «M2/T, then we use Lemma 5.4.1 to approximate the sum over m by an integral, which is estimated by the First Derivative Test (Lemma 5.1.2). This gives
S <<M2/T,
(18.1.33)
which is much better than (18.1.31). Hence we can omit from the hypotheses of Theorem 18.1.2 the lower on H in (18.1.4), (18.1.8), (18.1.10), (18.1.14), and (18.1.16) of Lemma 18.1.1. In the statement of the theorem we have collected cases, and worked out
the ranges for M in which each bound is used. We have renumbered the
constants B. and C;. The conditions on the function F(x) derive from Lemma 14.3.1. We have introduced the weight functions g(x) and G(x), which do not appear in Chapter 8. Because the ranges of summation for h and m are independent, we can remove the weight functions by applying Lemma 5.1.1 to the outer summation, interchanging orders of summation, and applying Lemma 5.1.1 again.
O
18.2 INTEGER POINTS AND ROUNDING ERROR We apply Theorem 18.1.2 to the problems of lattice points and area consid-
ered in Chapter 2, and of integer points close to a curve considered in Chapter 3. The second problem is technically simpler. Theorem 18.2.1
Let F(x) be a real function with three continuous derivatives
for 1 <x 5 2. Let M (an integer) and T (a real number) be large, and let 5
Lattice points and area
378
satisfy 0 5 6< Z. Let C z 1 be a constant. Suppose that for 1 <x S 2 either Case 1 or Case 2 holds: Case 1
The derivatives F" (x) and F(3)(x) are non-zero and M< CT 1/2.
Case 2
The derivatives F'(x), F" (x) and the expression
F'(x)F(3)(x) - 3F" (x)2 are non zero, and M > C-1 T 1/2. Let N(S) denote the number of solutions of T
(18.2.1)
m IIM F(M )IIS8
with M:5 m 5 2M - 1. Then in both cases/ (18.2.2)
N(S) << SM + T23/73(log T)315/146
for T63/146(log T)63/292 S M < T83/146(log
T)-63/292'
(18.2.3)
and in Case 1
N(S) «SM+M4/15T1/5(logT)21/10 +M-4/17T7/17(log T)81/34 (18.2.4)
for T 1/3 5 M < T 63/146 (log
T)63/292
and in Case 2 N(S) «SM+M-4/1sT7/15(log T)21/10+M4/17T3/17(log T)81/34 (18.2.5)
for T83/146(log
T)-63/292
< M < T 2/3.
The implied constants are constructed from C and the range of values taken by the derivatives of the function F(x).
Proof We suppose first that S is not too small, so that when we use Lemma 5.3.2 in the form
N(S)<-
D1 2M 4 +D-Re D
dM
dT
m
(1-D)Ee(MF(M)),
(18.2.6)
then
D=[1/881+1 is so small that Theorem 18.1.2 can be applied to each range H< h < min(D,2H), where H runs through the powers of 2. The bounds of Theorem
Integer points and rounding error
379
18.1.2 are increasing functions of H, and the order of magnitude does not change when we pass from the case of H large in (18.1.28) to the case of H
small in (18.1.29) or (18.1.30). The estimate for the sum in (18.2.6) is therefore dominated by the largest range. We want the term 2M/D to dominate in (18.2.6). In the case of H large we require D 3/70
D(M)
T23/70(log T)9/4 <<M,
which is true for S Z So given by So = M-1 T 23/73 (log T )315/146.
This choice of D is valid since 23/73 = 0.3151... > 17/57 = 0.2982... . For S 5 So we use the estimate for N(SO) instead. The largest value of H remains in the range where (18.1.28) is valid when M lies in the range (18.2.3). For H small and M small we use (18.1.29) in place of (18.1.28). The term 2M/D dominates for S >_ 51 given by 7,3 81
= min
M11
1/ 15
7.
(lo8T)21/10,
7/ 17
(M3)
(logT)81/34
For 5S 51 we use the bound for N(6) instead. This case gives the bound (18.2.4). The bound (18.2.5) follows similarly from (18.1.29).
Theorem 18.2.2 Let F(x), M, and T satisfy the conditions of Theorem 18.2.1. Let R be the sum m. T
R=Ep(MF(M))'
formed with the row-of-teeth function p(t), where M2 is an integer in the range M < M2 < 2M. Then in both cases T)315/146
R << T23/73(log
(18.2.7)
for
T 63/146 (log T) 63/292 S M < T 83/146 (log T) -63/292
(18.2.8)
and in Case 1 R << M4/15T1/5(log T
)21/10
+M-115T 2/5 (log T )81/40
(18.2.9)
for
T 1/3 < M < T63/146(log T
)63/292,
(18.2.10)
and in Case 2 R << M-4/15T7/15(log T )21/10 +
M1/5T2/5(log T )81/40
(18.2.11)
for
T83/146(log 7.)-63/292 <M < T2/3.
(18.2.12)
Lattice points and area
380
The constants implied in the << symbol depend on C in the definition of Cases 1 and 2, and on the range of values taken by the derivatives of the function F(x).
Corollary If, in addition, F(x) is defined for 1 - 112M5 x< 2 + 1/2M, and has two continuous derivatives in this wider range, then MZ
T
m
M+1/2 T
x
l
L(i)J-IM-1/2(MF(M)-zldx satisfies the order-of-magnitude bounds for the sum R given in Theorem 18.2.2.
Proof We use the expansion for p(t) with tapered partial sums in Lemma 5.3.1:
(1 --+0 + min 1, D h=2-2D k(0) 27r ih 2D-2
p(t) =
k(h) e(ht)
1
D311t11311
(18.2.13)
ho0
instead of the simpler sum in Lemma 5.3.2 which we used in the last theorem; we have replaced H in Lemma 5.3.1 by D to show the analogy. We divide the sum over h into blocks of the form M2
H2
k(h)
1
hT
m
R(H)= F, E k(0) 27r ih e(M F(M)), m=M h=H I
(18.2.14)
where H is a power of 2, and H2 = min(2 H - 1, 2 D - 2). The result (18.1.28) of Theorem 18.1.2 gives R(H) << (H/M)3/70T23/70(log T)9/a
The sum over values of H is a sum over powers of 2 in which the term with H largest dominates. We have R(H) << M/H provided that H << MT-23/73(log T)-315/146
(18.2.15)
We give D the largest value consistent with (18.2.15), provided that the largest block is one for which (18.1.28) is valid, which happens in the range C1T63/146(log T)63/292 <M < CI 1T83/146(log T) -63/292
for some constant C1, which we discuss below. This estimate holds for all blocks if C2T
7/16 5M < C2 1 T91 16
for some constant C2. For M :!g C2T7/16 we must consider blocks R(H) with H << T4/M9 separately. For these (18.1.29) gives R(H) << H1/14M3/14T3/14(log T )9/4 +H- 1118M- 1/6T7/18(log T
)9/4.
(18.2.16)
Integer points and rounding error
381
We can also use the Poisson summation bound (18.1.32), which gives
R(H) << (HT/M)112
(18.2.17)
We change from (18.2.16) to (18.2.17) when H << (M3/T )1/5(log T)81/2o Summing H through powers of 2, we have
E
min(H-1/18M-1/6TH/18(log T)9/4,
(HT/M)112`
H )81/40
<<M-1/5T2/5(log T This expression is smaller than (18.2.7) in the range (18.2.8). For T)63/292, (18.2.18) M< C1T63/146(log all blocks fall into the range of validity of (18.1.29). The block with H largest
dominates. It contributes O(M/H), provided that -21/10, (M3/T)1/2). C3 min((M11/T3)1/15(log T)
H<_
(18.2.19)
We give D the largest value consistent with (18.2.19). The first choice in the minimum gives the first term in (18.2.9). The second term gives the minimum only when the first term in (18.2.9) has ceased to dominate. As M passes through the value on the right of (18.2.18), then the order of magnitude of the estimate for the block R(H) changes smoothly. By adjust-
ing the constants in the upper bounds, we may change from the estimate (18.2.7) to (18.2.9) at T63/146(log T)63/292, so that C1 does not appear in the statement of the theorem. At the lower end of the range (18.2.10), the trivial estimate M is better, and we may drop the constant C2 similarly.
For M;_> C2 1 T9/16 we must consider blocks R(H) with H <<M11/T6 separately, using the estimates (18.1.30), (18.1.32), or (18.1.33). The discussion
runs entirely parallel to the case H << T 4/M9 above. Alternatively, Case 2 can be reduced to Case 1 by taking y as the independent variable on the curve y = (T/M)F(x/M), and counting lattice points in the y-direction; we find that O(T/M) is a trivial bound for the error when M5 T 2/3. The error term in (18.2.13) contributes M + r N(2'/D)
«
231
in the notation of Theorem 18.2.1, where the sum is over integers r with r >_ 0, 2'+ 1 D. The terms of the sum corresponds to the block sums R(H), and the corresponding estimates are the same in the range (18.2.8), and equal or smaller in the ranges (18.2.10) and (18.2.12). The corollary follows when we expand out the row-of-teeth function in the
sum R, and use fm+1/2 m- 1/2
f(x)dx=f(m)+O(maxIf"(x)I)
(18.2.20)
Lattice points and area
382
term by term with f(x) = (T/M)F(x/M). The error term in (18.2.20) sums to O(T/M2), which can be absorbed by the bounds of the theorem for M>- T1/3
The second term in (18.2.9) and (18.2.12) of Theorem (18.2.2) comes from
block sums R(H) with H smaller than D, in ranges for which the saving given by the double exponential sum is not so great. We can treat these sums as a family of single exponential sums, provided that we have more information about higher derivatives of F(x). We can extend the validity of (18.2.7)
this way. Our next theorem states a uniform result which is useful for applications like Dirichlet divisor problem, where the order of magnitude of the integer variable m is not fixed. Theorem 18.2.3 Let F(x) be a real function with four continuous derivatives for 1 <-x <_ 2. Suppose that the derivatives F" (x) and F(3)(x) and the expression
F" (x)F(4)(x) - 3(F(')(
X))2
(18.2.21) are non-zero. Let C z 1 be a constant. Then for M < CT1/2 and M2 < 2M we have M2
TM( m)) p(FM 4 T23/73(log T)315"146 1
M=M
If in addition, F(x) is defined for 1 - 112M5 x <- 2 + 1/2M, and has two continuous derivatives in this wider range, then Mz
T
M
F(m)11M+1/2 _
MM-1/2 (MF(M)
2)dx
+ O(T/M2 + T 23/73 (log T)315"146 ) The constant implied in the upper bounds depends on C and on the range of values of the derivatives of the function F(x).
Corollary
In Dirichlet's divisor problem we have T
L d(t) = T log T + (2y - 1)T + O(T 13/71 (log
T)461/146)
t=1
Proof We modify the proof of Theorem 18.2.2 by treating blocks R(H), with H small, more elaborately. We choose D to be the largest value consistent with (18.2.15). The block R(H) is split into (at most) H sums over m in which h is fixed, which we treat as single exponential sums with parameter
T' = hT/M. For example, when Mx T2/5, then T' x M7"4. When Mx T 1/2
x M1/4,
hx
MT-3120
hx
MT-3110 X M2/5
Integer points and rounding error
383
then we have T' x M 7/5. Hence M is rather larger than T, so we begin with the inversion step (Lemma 5.5.3). This gives an exponential sum formed
with the complementary function F(y). For best results we should apply Theorem 17.4.2 with R = 1, that is, a differencing step, then Theorem 17.1.4. However, for the accuracy that we require, we can replace Theorem 17.1.4 by the simpler Lemma 7.3.4. In its turn, Lemma 7.3.4 can be replaced by another differencing step, followed by Corollary 1 to Lemma 5.4.3. The only, property
required of f(y) is that F(4)(y) = (F"F(4) -
be non-zero, as in the statement of the theorem. The non-vanishing of (18.2.21) corresponds to the condition in Theorem 17.1.4 that F(3) is non-zero, which allows the construction of the Farey arcs. We have a four-step van der Corput iteration (Graham and Kolesnik 1991), of type BARB, where A denotes a differencing step, and B denotes Poisson
summation, that gives the exponent pair (2/7,4/7) in the sense of (17.4.6). The bound for the block R(H) simplifies to R(H) <<
1
H
T' 2/7
E(M)
M4/7<< (HT) 2/ 7 ,
(18.2.22)
h
which is acceptable for H << T 15/146 (log T
)2205/292'
and thus for the entire range (18.2.15) when M << T61/146(log T
)2835/292
(18.2.23)
For slightly larger values of M we use (18.2.22) when H is small, and (18.2.16) when H is large. We change from (18.2.16) or (18.2.22) when
H << M- 21/43 T 13/43(log T )567/86 Summing H through powers of 2, we have
E
min(H-1/18M- 1/6T7/18(log T)9/4, (HT)2/7) << M- 6/43T16/43(log T)81/43
H
This term is acceptable for M >> T 179/438 (log T)
-573/292
(18.2.24)
Since 61/146 =183/438 > 179/438, the ranges (18.2.23) and (18.2.24) overlap, and the estimate (18.2.7) of Theorem 18.2.2 is valid for M smaller than
the range (18.2.8). The corollary on Dirichlet's divisor problem follows immediately from (11.1.14) of Lemma 11.1.3.
0
Lattice points and area
384 18.3
LATTICE POINTS INSIDE A CLOSED CURVE
We are now technically equipped to return to the lattice point problem of Chapter 2. Accordingly C is a simple closed curve made up from pieces C, on which the radius of curvature p exists, is bounded and non-zero, and has at least one continuous derivative with respect to the tangent angle 41. In Chapter 2 we approximated the enlarged curve MC by a polygon whose sides
had rational gradients. In Chapter 8 the enlarged piece MC, has equation y = (T/M)F(x/M) in local coordinates, replacing the straight sides of the polygon by quadratic curves (splines).
For each piece C, of the curve, we set up a local coordinate system on which the hypotheses of Theorem 18.2.2 hold, and we use the automorphisms of the integer lattice to express the discrepancy in the new coordinates. We
may have to subdivide the piece C, into smaller pieces, each with its own local coordinate system. This subdivision is finite, and independent of the enlargement factor M. First we subdivide the discrepancy. In this section we use the name lattice to mean a group of points in the plane (under vector addition) with two specified generators, which are linearly independent. Lattice lines mean the straight lines through the points of the lattice in the direction of either of the two generators. The lattice lines divide the plane into parallelograms whose area is the determinant of the two generating vectors, called the determinant of the lattice. We call the centres of these parallelograms the midpoints of the lattice, and we call lines through the midpoints in the direction of either of the generators the midlines of the lattice. The midlines divide the plane into the second set of parallelograms whose centres are the lattice points. The integer lattice has the unit vectors (1, 0) and (0, 1) for generators.
The enlarged curve MC is made of pieces MC,, i = I,-, J. Let the endpoints of MC, be P; and P;+1. Since the curve is closed, we have Pr+ 1 = P1. Let Q. be the midpoint of the lattice square of the integer lattice which contains Pi (any of the possible squares if P, lies on a lattice line). We suppose that the tangent angle 41 changes by at most -7r/6 on MCj. Case 1 All tangents to the arc MC, make an angle at least nr/6 with the y-axis. Choose a point Si on the midline y = y(Q,) as follows. The arc MC, meets this line at most once. Let Si be the point of intersection if it exists. Otherwise let Si be the point of intersection of the midline y = y(Q,) with the tangent at P, (which cannot be parallel to the y-axis). Similarly choose T, on the midline y = y(Q,+1) to lie on the arc MC, if possible, otherwise on the tangent at Pi, 1 (Fig. 18.1).
Case 2 All tangents to the arc MC, make an angle at least a/6 with the x-axis. Choose Si and T, similarly on the midlines x =x(Q,), x =x(Q,+1).
In both cases the distances Q,S, and Q,+1S, are at most (1 + v)/2. Let U
Lattice points inside a closed curve
385
xT1 x Qi+1
Qi x
Ui
FIG. 18.1
be the intersection of the midlines x=x(Q,),y=y(Qi+1) The polygon Q1U1Q2U2 ... QJUJ is a union of unit squares formed by midlines of the integer lattice, so that the area of the polygon equals the number of lattice points inside. The polygon may intersect itself; in this case some unit squares are counted negatively. In Case 1 we compare the number of lattice points and the area within the region E. bounded by the arc S;T, and the midlines Qiu,,, y = y(Q1) and y =y(Qi+1). Again, lattice points and area below the base line Q;U are counted negatively. In Case 2 the region E. is bounded by the
arc S;T and the midlines UQr+1, x=x(Qi) and x=x(Qi+1). Note that when we fit consecutive arcs C, and Cr+1 together, then Si+1 is
not the same point as T,. A bounded region about P;+1 and Qi+1
is
overlapped or omitted. This causes a bounded error in the discrepancy at each point of subdivision of the curve. When we write ., for the discrepancy
of the set E,, 0 for the discrepancy of the interior of MC, then A
1, + O(J).
To avoid bad orientations of the curve MCi (such as horizontal or vertical), we rotate coordinates by an angle 6, with a rational tangent a/q. The angle 13 must be chosen from some interval I of length o(< it/4). If I contains nir/4 for some integer n, then we choose /3 = nir/4, with a2 + q2 < 2. If not, then some interval It = mir/2 ± I is a subinterval of (0, it/4). The values of tan a for a in It form an interval 12 of length at least 8. Let d = [1/S] + 1. The interval d12 has length greater than one, so it obtains some integer c. We take /31 = tan-1 c/d, /3 = ±(/31 - mir/2). Then tan /3 = a/q with
a2 + q2 < c2 + d2 < 2d2 < 8/S2;
(18.3.1)
this inequality also holds in the case /3 = nir/4.
Lemma 18.3.1 (Rotating rounding error sums) Let the curve y = g(x) be continuously differentiable for b 5 x < c, and let a/q be a rational number in its
Lattice points and area
386
lowest terms (or a = 1, q = 0), and let 6 be a number such that no angle in the a 6 6 a is that of a normal to the curve. Define - 2 , tan-1 + interval tan 1
2 q q the function v = h(u) implicitly by
au + qv = rg((qu - av)/r), where r = a2 + q2. Then there are integers e, f, and s with
f-e << q(c - b) + alg(c) -g(b)I + 1/6, and C
Y, P(g(m)) _
m-b
f
h(n) - sn r p n=e+1
r
1
+OI r+ s + J
(f - e) r
I
1
We can take s
a
a
q
q
r
qr
q
a
ar
(18.3.2)
where as + qq = 1.
Proof The rounding error sum can be regarded as the discrepancy of a region E bounded by lines SQ: x = b - Z, TU: x = c + ? , QU: y = d + Z , and the part of the curve y =g(x) with b - 15x 5 c + if necessary we extend
the definition of g(x) to be linear for x < b and 2;x > c, with continuous derivatives at b and c. Figure 18.1 shows one possible configuration. Integer points and area below the line y = d + 'z are counted negatively. The three straight sides of E are midlines of the integer lattice, which we shall call A. We divide the lattice points (m, n) of A into r = a2 + q2 different classes,
according to the residue of m + in mod q + is in the Gaussian integers. Points of A in the same class form the vertices of a system of squares with sides of length Vr-, in the directions of the vectors (q, a) and (-a, q), which are the directions of the new coordinate axes. The sides of these squares from all the r classes of integer points form the lines of a square lattice Al of side
1/%. We pick points V and W to be midpoints of the lattice Al as close as possible to S and T, respectively. There are now two cases. Either SQ or TU has length at most one. (Fig. 18.2). SQ is the short side. We take U' to be the intersection of the midline through V in the direction (q, a) with the midline through W in the direction (-a, q).
Case 1
Case 2 Otherwise TU is short. We take U' to be the intersection of the midline through W in the direction (q, a) with the midline through V in the
direction (-a, q).
Lattice points inside a closed curve
387
14G. 18.2
In both cases we define S' and T' to be the intersections of the curve (extended linearly past S and T if necessary), with the midlines of Al through
V and W in the direction (-a, q). The intersections are unique because the vector (-a, q) makes an angle at least 5/2 with any tangent. We now have a
region E' bounded by the arc S'T' and the lines S'V, VU', and U'T' (or S'U', U'W, and WT' in Case 2), which are midlines of A1. As usual, if the
curve crosses the line VU', then some of the area and lattice points are counted negatively. The new region E' differs from E by the addition and subtraction of a bounded number of regions of three types. Type 1 Regions about S and T which can be covered by rectangles with one side of length at most 1/ W, the other side of length at most (1 + cot 8/2)/ V. The discrepancy of a region of this type is trivially
<< 1+1/Sr. Right-angled triangles formed by two midlines of A and a midline of A1. The discrepancy is given by a sum of the form Type 2
(a(n q1/2)1 P
1
(18.3.3)
n
By the addition formula (Lemma 2.3.1) the summand in (18.3.3) sums to zero over any set of q consecutive integers, so the whole sum is O(q) = WT).
Right-angled triangles formed by two midlines of A1, and a midline of A. The discrepancy is given by two sums of the form Type 3
a(nP n
2)-b
9
(18.3.4)
for some b, which are again O(q). Since discrepancy is additive on disjoint sets, the discrepancies of E and E'
differ by 0(r+ 1/8r). Our last task is to express the discrepancy of E' as a rounding error sum.
388
Lattice points and area
We take new coordinates u = qx + ay, v = qy - ax. Integer values of x and y give the lattice whose (u, v)-coordinates are the multiples of q + is in the Gaussian integers. Since a/q is assumed to be in its lowest terms, we have (a,r) = (q, r) = 1, and the points of A can be characterized by qu + av = 0, (mod r) (18.3.5) and the determinant of A in (u, v)-coordinates is r. We note that (18.3.5) is equivalent to v = su (mod r), where s is a solution of as + q = 0 (mod r); the formula (18.3.2) defines an integer satisfying this congruence. Suppose that
the curve MC; is v = h(u), and the other boundaries of El are midlines u = e + 2, u = f + 2, and v = d + 2. The number of points of A in El is
f ([h(n)-snl - [d+ 1/2-snll r
11
(p(h(n)-11 sn1 -p(d+1/2-sn n=e+1
r
r
h(n)-d-1/21 /I
+
r
/I
The first term is the required rounding error sum, and the second is analogous to (18.3.4), and so is O(r). The third term is estimated as 1
fef+ +
1/2(h(u)-d-
2)du+O f -e)
max I
(18.3.6)
r J 1/2 r by (18.2.20). The integral in (18.3.6) is the area of E1 in (u, v)-coordinates, which is r times the area in (x, y)-coordinates.
o
Theorem 18.3.2 Let D be a Euclidean plane domain with area A, bounded by a simple closed curve C composed of finitely many pieces C, which are three times continuously differentiable in the following sense. The radius of curvature p is continuous and non-zero on each piece Ci, and p is continuously differentiable with respect to the tangent angle 41. Let E be a plane set obtained by expanding D linearly by a factor M > 2, followed by a rigid motion. Then, for any embedding of E in the Euclidean plane, the number of integer points in E is M)3151146),
AM2 + O(IM46/73(log where I is a number depending on the curve C, but not on Moron the position or orientation of E.
Theorem 18.3.3 In Theorem 18.3.2, suppose that the pieces C, are four times differentiable, in the sense that p is twice continuously differentiable with respect
to the tangent angle 41. Suppose also that p is either constant or has a finite number of maxima and minima on each piece Ci, and that d log p/d 4' has a bounded number of maxima and minima between any two consecutive extreme values of p on Ci. Then we may take
I= E r' pk/73' i
k
Lattice points inside a closed curve
389
where the inner sum is over local maxima Pik (including endpoints, and counting constant values only once) of the radius of curvature on Ci, provided that M is so large that the bounds M > 1 / p and M40
63 Z 1+
(log M)"
1
P
2
dP
2 103
d/,)
1 1
hold on each curve C1.
Proof of Theorems 18.3.2 and 18.3.3 We have to choose local coordinates on each piece Ci so that the conditions of Theorems 18.2.1 and 18.2.2 apply. The new coordinates are u and v in Lemma 18.3.1, with
u = r (x cos /3 +y sin f3 ),
v=W(ycos/3-xsin so, for (x, y) on the curve MCi, we have dv
- = tan(4,- /3),
(18.3.7)
du
d2v
due d3v
du3
1
r1/epcos3(f-/3) 3sin(4r-/3) 1 dp rp2cos5(ap-p) rp3cos4(q-/3) dq, 3sin(4i- f3-A) rp2
ip - /3 )cos A '
(18.3.8)
(18.3.9)
where the angle A is defined by
dp 3p d+/ 1
tan A =
We also have
dv d3v - 3 (d2v 2 du du3 due
3cos(4,-/3 -A) rp2 cos5(r/l - /3 )cos A '
(18.3.10)
We prove Theorem 18.3.2 by a convexity argument. The values of 4 on a
particular piece C. of the curve form a closed interval K. For any 4 in K there is an open interval of values of f3 for which the expressions in (18.3.8) and (18.3.9) are bounded away from zero and infinity. We pick a(41) to be the midpoint of this interval. For fixed 4 = 4i0, we take f3 = a(410) and
consider (18.3.8) and (18.3.9) for this P. Since p and A are continuous functions of +/, there is an interval of values of 41 about 1//0, open relative to the interval K, on which neither expression (18.3.8) nor (18.3.9) is zero or infinite. We call this interval I( /3 ). Each 41 inK lies in its own open interval
Lattice points and area
390
I(a(t/i)). There is a finite subcollection I1,...,IR of these intervals which covers K; we suppose that the subcollection is minimal, and numbered in order, so that I, nIr+2 is empty, but I, fl Ir+ is non-empty. We form closed intervals Jr by taking the midpoint of I, fl I,+ as the common endpoint of J, and Jr+ 1. Then J1,..., JR are closed intervals, disjoint except for endpoints,
that cover K, each corresponding to some angle /3 = ar. Let Sr be the smallest distance from an endpoint of Jr to the corresponding endpoint of Ir. The derivatives in (18.3.8) and (18.3.9) are finite and non-zero for 0rEJr+[-6r/4,6r/41. 1 /3-arl <6,./2, (18.3.11) Since the derivatives are continuous, their moduli lie within closed intervals not containing zero when the inequalities (18.3.11) hold. We have found a subdivision of K for which we can apply Lemma 18.3.1 to find a suitable angle /3 with rational tangent, then Case 1 of Theorem 18.2.2 with T = M2, perhaps with a shift of origin in the variable u. The enlargement factor M enters only into the radius p, not into the angles i/i and A. The implied
constant is constructed from the maxima and minima of the expressions (18.3.8) and (18.3.9) under the conditions (18.3.11), and from the angle /3 chosen, by way of the integer a2 + q2. Since a2 + q2 can be bounded in terms
of the length of the interval for /3 in (18.3.11), independently of a the bound in Theorem 18.3.2 is uniform under rotation as well as translation. To prove Theorem 18.3.3 we estimate all the constants which depend on the shape of the curve, and apply Theorem 18.2.2 with order-of-magnitude parameters M' and T' corresponding to the lengths of the ranges for u and v/r, satisfying rT
d2v
d3v
rT'
due Mi3' du3 and if Case 2 of Theorem 18.2.2 is used,
(_ )___3 dv
d3v
s
r d2v I
Mi4' 2
(18 . 3 . 12)
r2T'2 (18 . 3 . 13)
M'6 To simplify the expression in (18.3.13) we note that a/q has a neighbour b/d du2 )
in the Farey sequence Aq) for which ad - bq = 1, and we can take s = -(ab+dq) in (18.3.2). Let y= tan '01d). Then ad - bq 1 tan(/3-y)= ab - dq s By (18.3.9) and (18.3.10), for s non-zero the left-hand side of (18.3.13) is
3sin(if-y-A) rp2 sin( /3 - y)coss(gr- 13)cos A In Theorem 18.2.2 we require order-of-magnitude relations cos A x lsin(tr - /3)sin(4r - /3 - A)I, and, if Case 2 is used, Isin( /3 - y)cos Al x Icos(/r- /3)sin(gi- y- A)I.
(18.3.14) (18.3.15) (18.3.16)
Lattice points inside a closed curve
391
We subdivide the arc C, into smaller arcs C.1 on which:
(1) The angle changes by at most 1r/12; (2) The angle A changes by at most ir/12; (3) The quantity log p changes by at most log21/20; (4) The quantity log cos A changes by at most log 21/20. There are two cases, depending on the angle A. Case 1
JAI <- 11ir/24 on C,1. We want to choose /3 so that
I4-/3-kar/21 Z Ir/24,
141 -/3- i/i-1Ir/2IZ ir/24 (18.3.17)
for all integers k and 1. Modulo it/2, the set of (3 for which (18.3.17) holds
forms one or two intervals of total length at least Ir/12. Hence there is a suitable interval for /3 of length at least -7r/24. In Lemma 18.3.1 we take 6 = 7r/24, and (18.3.1) gives
r = a2 + q2 5 8/32 = 4608/ar2 < 500.
We take M' to be an integer, T' to be real, with M' = r112Po + 0(1),
T' = Po
for some value po of the radius of curvature p on the arc MC,1. When we write the Cartesian equation of MC;1 as u v -su T' r M' then F" and F(3) are bounded away from zero and infinity, and Case 1 of
F(M(18.3.18)
Theorem 18.2.2 applies. Case 2 For some iii on C,1 we have 0 < cos A 5 sin it/24. By construction cos A lies in some range -q ScosA52177/20,
so A is close to ±,7r/2. We treat the case when A is positive. We have
dt
IT
775
where
2
21,q
1177
(1-t2) 5-20 cosh/24 < 10
= min(21-7/20, sin a/24). Also
- f drlr53f tan AdA5[log p]c,,5log-, 10
7q c,j
21
20
c,1
and 4i changes by at most
-log-2120 5 200 771
10
777
Lattice points and area
392
Let i/ o, po refer to a particular point on the arc MCij. If 8 lies in the interval IT
2
-
37)
-<'o-13<
4
IT
7)
-
2
4
then IT
2
15717
200 -
-
a
5
IT
4371
2
200
Thus 437
- - a -A 5 200
1777)
200
and (18.3.15) holds with
cosAxcos(O -13)xsin(fi-/3-A)x17.
sin(/i-/3)xcos(+y-/3-A)x1. We take 6 = 77/2 in Lemma 18.3.1, and (18.3.1) gives
r = a 2 + q 2 <8/52
<32/7)2.
We take M' to be an integer, T' to be real with
M' = 7)2r1/2p0 + 0(1),
T=
7)3po,
for some value po of the radius of curvature on the arc MC;1, with
T'/M'2 x 1/pr. For r << 1/7) we can use Case 1 of Theorem 18.2.2. For r >> 1/7) we should use Case 2. However, the requirement (18.3.16) involves the angle y with
tan y = b/d, which depends non-uniformly on the angle /3, and so on the orientation of the arc. We choose a modulus k, and divide the sum over n into residue classes modulo k, putting n = km + c for c = 0,..., k - 1. This gives a congruence family of sums with the same T', but with M' now given by
M' = 712r1/2p0/k + 0(1),
and ((ns/r)) modulo one replaced by ((mks/r)). We choose
k x (711) << 115 and use Case 1. The rounding error sum in Lemma 18.3.1 is
123/73 (log T r)315/146 <<
po)315/146'
o
(18.3.19)
provided that n1o3p02o
>> (log
p0)63/2
The expression (18.3.19) has fixed order of magnitude on C.
(18.3.20) .
A family of lattice point problems
393
Summing over the different subdivisions C11 of C, gives
<< min(cos A)65/146p46/73(log p)315/146 cr
+fC(COS A)65/146p46/73(log p)315/146(I d l'
d
d
p
o
- logcos Oil
dir.
If p and A are monotone on C,, then each term is << max p46/73(log p)315/146 c;
In general we must subdivide C. into pieces on which p and A are both monotone; the number of such subdivisions has the same order of magnitude as the number of maxima of p on C,. To complete the proof, we replace p by Mp, and bring the scale factors in M outside the sum; the remainder terms in Lemma 18.3.1 are negligible. 0 Theorem 18.3.3 would be more elegant if the error term was a power of the area A, not of the maximum radius of curvature. However, area in intrinsic coordinates is given by the functional A
_1 f2 f2 4
9)dao r d0,
0
0
where k(0) is the kernel function defined by
k(0)=(1-0/ir)sin0 for 0 50 <7T, k(-0) =k(0), k(0+27r) =k(0). If p(0) is allowed to be negative, as for the four-cusped hypocycloid with
x= - 3 cos 41 - cos 3o,
y=3sino-sin3o,
p= -6sin2af,
then the area A need not be positive. Hence the area A is not a norm on periodic functions p(0), and it cannot appear on the upper bound side of an inequality.
18.4
A FAMILY OF LATTICE POINT PROBLEMS
Theorems 18.1.2, 18.2.1, 18.2.2 can be extended to a family of functions F,(x) in the same way as Theorem 17.1.4. Theorem 18.4.1 extends Theorem 18.1.2.
It is complicated, because there are five independent parameters, and we have chosen to consider several cases rather than to impose more conditions on the parameters H and M. Weaker results can be obtained for larger H by the method of Lemma 17.1.5, making the product hm divisible by a suitable prime number p (Huxley 1990, Section 7).
Lattice points and area
394
Theorem 18.4.1 Let F(x, y) be a real function defined for 1 5 x5 2,0:5y< 1, for which the partial derivatives are continuous and satisfy I ai a2F(x, y)I <- C,
for r55, s52, r+sS5, and 1/C1 S I a,F(x, y)I
(18.4.1)
for r = 1, 2, and al-1 a2F(x,Y)
a1F,(x y)
a1
z C2
(18.4.2)
for r = 2, where C1 and C2 are positive constants. Suppose that either Case 1 or Case 2 holds: Case 1
(18.4.1) and (18.4.2) hold for r = 3 also.
Case 2 For some positive constants C31 C4
I3F2 -F1F111I2C3,
IAIzC4,
where A is the determinant 3F11 + 4F1F111
3F1F11
F1
F1111
F111
F11
F1112
F112
F12
Let y1, ..., y, be points in [0, 1] with yi+1 - y; z 1/J, where J5 T. Let g(x) and
G.(x) be bounded functions of bounded variation on [0, 1] (uniformly in i in the case of G,(x)). Let S. denote the sum Si
ig(H)
mM
G'(M)e(M
F`M'Y,)),
where H and M are positive integers, and T is a positive real number. Then there are constants C5 and C6 constructed from C11 ... , C4 such that if
C5T1/35MSC51T2/3
(18.4.3)
H< C6 min( M3/2/T1/2, M1/2, MT-3/11),
(18.4.4)
and then we have bounds of the form E 1S;I2 << EH2IT logs T,
(18.4.5)
where the constant in the upper bound is constructed from C1,. .. , C4, from the
bounds for the functions g(x) and G;(x), and from the extra constant C7 in Cases (b) and (c):
A family of lattice point problems
395
Case (a) In Cases 1 and 2, for
HT
J
M2 r HT2/5
T
(7) +I>_C61min M2+ T ,I
M
)
,
(18.4.6)
we have (18.4.5) with
E =I-4135(HIM)3135T- 12/35 + (J/I)4/21(H/M)1/21T- 8/21 +1-1/2 (HIM )11/4T3/4 + (J/I )1 /2(H/M)'/4T1/12.
(18.4.7)
Case (b) In Case 1 for M< CST 1/2, where C7 is a constant which may be chosen arbitrarily, we have (18.4.5) with
E
=1-2/7H1/7M3/7T-4/7(1
+JM2/IT)3/14
+(H/M)1/21T-s/21(1 +JM2/IT)1/2
+H-1/9M-1/3T-2/9(1 +JM2/IT)1/2 +1-1/2HT-1/2. (18.4.8) Case (c) In Case 2 for M >_ C7 1 T 1/2 we have (18.4.5) with /7M-5/7(1
E = I-2/7H1 +(HIM)112'T- 8/21(l
+JT/IM2)3/14
+JT/IM2)1/2
+H-1/9M5/9T-2/3(1 +JT/IM2)1
12
+I-1/2HM-2T1/2.
(18.4.9)
Proof We use Lemma 16.1.4 (with 8 = 1). The conditions on F(x) are those of Lemmas 14.3.1, 14.3.5, and 14.4.1. The choice (18.1.2) of U is valid, provided that O2A4PR2>> Q
holds, so that H2N2 << MR3,
(18.4.10) arising from the fifth term in the minimum in (16.1.11) of Lemma 16.1.4. The
third term in the same minimum is larger than the corresponding term in (16.1.6) of Lemma 16.1.3, and we may relax (18.1.18) to H4N6 << M3R7,
(18.4.11)
which is a consequence of (18.4.10) since R > 1,
OiQ2 >> 1.
(18.4.12)
The conditions (18.4.11) and (18.4.12), when combined with the other inequalities involving the parameters, entail the conditions (18.4.3) and (18.4.4) of the theorem.
Cases (a), (b), and (c) correspond to the three results (16.1.12), (16.1.14)
Lattice points and area
396
and (16.1.15) of Lemma 16.1.14. We summarize the choices of N which give
the various terms in the upper bound. If T is very large, then (18.4.12) is limiting, and we take NX
M2/(HT2)1/3
in Case (b). If M is very large, then (18.4.12) is limiting, and we take
N X (M2/H)1/3 in Case (c). If I is very large, then (18.4.11) is limiting, and we take
Nx (M11/H4T3)1/7 in Case (b) or Case (c). If H is very large, but I is small, then we take N x H in all three cases. If no parameter is too large, then we choose N so that the term in P2Q2 is
the same order of magnitude as the next largest term. In Case (a) this requires N x (I6m57/H22T17)1/35
for J/I small, Nx (I/J)2/7(M11/H4T3)1/7 for J/I large. In Case (b) this requires N x (I3M6/HST )1/7
for J/I small, and in Case (c) Nx (I3M18/HST 7)1/7
for J/I small. For J/I large we take Nx (J3M12/H5 T4)1/7 in Case (b) or Case (c).
O
We deduce an analogue of Theorem 18.2.2. To reduce the complications, we express the upper bound in terms of J, using 15 J + 1 to eliminate I, and we do not attempt to optimize the logarithm power. Theorem 18.4.2 Let F(x, y), M, T, and the points y, and functions G.(x) satisfy the conditions of Theorem 18.4.1. Let R. be the sum 2M-1 R`
m=M
m
T M
G`(M)P(MF(MYj
Rounding error and integration
397
If either Case 1 holds with C5T113 < M< C7T112,
or Case 2 holds with
CVT
M.-5 CS 'T2/3
then we have the bound 57, R; << J(T26/43 +
T415M-2/5 +M2/5T2/5)log6 T
+ min( J65/73T46/73, J'1115M8115T2/5, J11115M-8/15T14115)log6 T.
The constant C7 may be chosen arbitrarily. The constant` in the upper bound is constructed from C1, ... , C4 and C7.
18.5 ROUNDING ERROR AND INTEGRATION The theorems of this chapter have been directed to approximating sums by integrals. Even the purest of mathematicians wants to know the approximate value of the resulting integrals. Some functions can be integrated in closed form to a function which is easily calculated, or uniformly approximated by a sequence of such functions. Usually the integral has to be approximated by a sum. The simplest method is given by the trapezium rule (Lemma 5.4.1): we
put f(x) = F(x/M), a = M, b = 2M in Lemma 5.4.1 to get 2
1
f F(x)dx=-F(1)+ 2M
1
2M-1
1m -I -FI m=M+1 M M
+ 2MF(2) -
1
f 2a'(Mx)F"(x)dx.
(18.5.1)
M2
The remaining integral is O(max IF"(x)I/M2). Writing down 4/3 times (18.5.1) for M minus 1/3 times (18.5.1) for 2M gives Simpson's rule: 4M
f2F(x)dx= m=2M
a m F(2M) + M
f
o (2Mx))F"(x)dx,
where am is the finite sequence (1,4,2,4,2, ... , 2, 4,1), and the remaining integral is O(max IF(4)(x)I/M4). Further accelerations are possible. Some formulae have unequally spaced blocks of sampling points, such as
X-+-+. M 2M-6M m
1
If the function F(x) has enough continuous derivatives, then we can find a formula of this type in which the remainder contains a high power of the step
length 1/M. However, the function values are not calculated to infinite precision, but to some accuracy 1/N. For hand calculation N = 10" for some
Lattice points and area
398
integer n. We round down to n decimal digits by replacing all decimal digits after the nth digit following the decimal point by zero. This corresponds to replacing NF(x) by its integer part. A similar rounding process is used for binary digits. Rounding down gives a systematic error. To avoid it, when adding a series of function values, we insert digits 5 (decimal) or 01 (binary) after the last digit retained unchanged. Computer arithmetic routines (see,
for example, Logan and O'Hara 1983) usually take the integer part of NF(x) + 2, using the instructions `shift right, jump if no carry, increase'. We call p(t) the rounding error function because 1
N([NF(x)] +
2)
=F(x) +
1
p(NF(x)),
N[NF(x) + 2] =F(x) + N p(NF(x) + D. The sampling nodes of the integration formulae form a finite number of sets of M points each of the form (m + a)/M, for m = M, ... , 2M - 1, so there are correction terms
zEi N
p(NF(mMa)),
N Z1 p(NF(mMa)+Z).
These sums are certainly O(M/N). If the rounding error functions were replaced by uniformly distributed random variables, then the sums would be
O(Vi/N) in root mean square. If F(x) is a polynomial of degree d with integer coefficients, and N = Md, a= 0, then all the p-functions are evaluated at integers, and the error is systematic. On the other hand, F(x) = x F2, and N = M, a = 0, then, since % has the continued fraction 1 + 1/(2 + 1/(2 + )), the explicit discrepancy formula of Lemma 2.3.2 gives O((log N)/N). We take the basic rounding error sum to be
R = R(M, M', N; F(x)) = F_ pI NF(M)1,
(18.5.2)
M
where M' < 2M. We expect to find cancellation in the sum R except possibly
when certain derivatives or determinants of derivatives are zero. In the vanishing case, there may be a systematic bias when F(x) has a good approximation by an algebraic curve with small integer coefficients. By analogy with the linear case given above, the bias is expected to disappear
when F(x) is replaced by /3F(x) for some number G3 whose continued fraction has bounded partial quotients. The numbers 1, v, and 2V - 1 are independent in the sense that the ratio of any pair of them has an essentially different continued fraction. Near any point y, at most one of the functions
Rounding error and integration
399
F(x), (I)F(x), and (2VI - 1)F(x) has a good algebraic approximation. So if a relevant derivative vanishes at x = y, then we should calculate fy+8RF(x)dx y-8
with 0 = 1, F, 2v - 1. We expect at most one of the calculations to be affected by a bias in the rounding error. Here 6 is chosen so small that the only critical point in the interval is y itself. Theorems 18.2.2 and 18.2.3 with T = MN give such bounds when N is close
to M in order of magnitude. We use single exponential sums to get bounds for large N. Numerical analysts will protest rightly that these bounds are only theoretical: the construction of the constant in the upper bound from the upper and lower bounds for various derivatives is not given explicitly. There are certain excuses for not giving the construction. 1. The exponents of M and N are not final; they are just numbers thrown up by the method in its present form. 2. The exponent of log M has not been optimized, and if Lemma 11.1.4 is used to optimize the exponent, then there is a function CO Xlog M)8 to minimize, where the `constant' C(S) is implicit in Hall and Tenenbaum (1986).
3. There are many contributions to the error estimate with different orders of magnitude, involving different determinants in the derivatives of F(x). 4. A complicated argument of n steps usually gives a numerical constant like 2", from repeated use of `a + b < 2 max(a, b)'. 5. The formulae would become very much more complicated.
First we note some consequences of the addition formula for p(t), Lemma 2.3.1.
Lemma 18.5.1 (Rounding error identities)
For any integer q > 1 we have
q-1 (M-a M'-a
R(M,M',N;F)= L RI1 a=0
q
q
,N;F x+
a
i1 (18.5.3)
q - i
R(M,M',N;F)= F, a=0
1
N;F(x)+ NJ, q
(18.5.4)
and for p a prime number n-i ` aMxl pR(M, M', N; F) = F, R It pM, pM, N, F(x) + N /I
a=0
- R(pM, pM', pN; F) + R(M, M', pN; F ). (18.5.5)
Lattice points and area
400
Proof We obtain (18.5.3) by writing m = nq + a, 0 < a < q, and (18.5.4) follows at once from Lemma 2.3.1. For (18.5.5) we have
P-1
EP a=0
an _ (pp(t)
if p In,
P)
if not,
t+-
St
p(pt)
so that n ant -1 + PM p
pM'
L n=pM
PEP(NF(m))_pE1
M
M
-
pin
P-1 PM _I 7
n
pM
n=pM
)_
M PM
p(pNF(PM))
n
'
+ E p pNF
n
P
PM
n = pM
0
+
( PM)
pin
We use the Fourier expansion of p(t) to deduce bounds for rounding error sums from bounds for simple exponential sums. Lemma 18.5.2 (Rounding error by simple exponential sums) Let F(x) be a real function, and let M and M2 be integers with M2 < 2M. Suppose that we have MZ
S=
T
K
e(TF(M)) «(M) M'(logT)µ,
(18.5.6)
where K > 0, uniformly for MN)")1"(K+1)
N< T < MN(MANK(log
(18.5.7)
Then M2
R=
.
M
m
p(NF( M)) <<
(MANK(log K
If (18.5.6) holds for M" < T < Ms, then (18.5.8) holds for M" <_ N << M(P-'XK+1)+A(log M)"'.
Proof We pick a positive integer H with
2H<M(M'NK(logMN)µ)
1),
and use Lemmas 5.3.1 and 5.3.2 to obtain
R« M
2HE-2
+
h1
Mz
h
e(hNF(M m=M
We apply the bound (18.5.6) term by term with T = hN.
(18.5.9)
Rounding error and integration
401
The parameter h in (18.5.9) is not used in the proof of Lemma 18.5.2. However, we can fit the sum over h into the iteration of Theorem 17.4.2. Let E1 be the average size of the upper bound for the sums with h * 0. Then
R << M/H+E1. We difference repeatedly to get
El << M/ D2 + (E2M) , where E2
is the average size of exponential sums formed with A(d2)hNF(x/M) in the notation of Section 17.4, and so on. We write D1 for H, and we choose parameters so that
D,., 1 = Dr
Dr = M/Er ,
which is (17.4.1). The smoothing-the-difference argument of Lemma 17.4.1
goes through with f'(x) replaced by (N/2)F(x/M), and the parameter choices are the same, when we interpret T as MN/2. Theorem 18.5.3 Let q be a positive integer. Let F(x) be a real function q + 5 times continuously differentiable for 1 < x 5 2, with
F(r)(x)
(18.5.10)
0
for r = q + 1, q + 2, d2
dx2
log
F(r) * 0
(18 . 5 . 11)
forr=q+1, 3(F(q+2))2 - F(q+1)F(q+3) 00' and
3(F(q+2))2 + 4F (q + 1)F(q + 3)
F(q+3) F(q+4)
3F(q+l)F(q+2) F(q+2) F(q+3)
(F(q+1))2 F(q+1) F(q+2)
0.
Let R be the sum in (18.5.8), with M sufficiently large. Define Q by Q = 2q. In
the range N z N6 we also require (18.5.10) for r = q + 3 and (18.5.11) for r = q + 2. Then, for any e > 0 we have R <
for N1 >- N >- max(N2, N3), and also for N >_ N1 when we replace the power of log N by NE, R <<
M(214Q-49q-165)/(214Q-49)N49/(214Q-49)(log N)33(5g2+6)/(214Q-49)
Lattice points and area
402
Table 18.1 Steps
D
D, E
E
C(1)
C(1)
C(1)
Rounding error bounds
To a=
Exponent /3
From a =
23(1 + a)
63
83
73
83
63 =
7+3a
83
4447
15
63
3252
355 + 496 a
4447
837
1396
3252
608 =
197 + 344a
837
643
902
608
466 =
68 + 43a
643
3001
171
466
2068 -
43 + 4a
3001
19
64
2068
12 =
62 + 29a
19
374
140
12
227 =
356 + 89a
374
872
641
227
423 -
80 + 11 a
872
122
423
87 = 2.1609.. .
34 + 13a
188
3323
73
87
1424 -
107 + 4a
3323
299
132
1424
120 =
593 + 49a
299
2461
807
120
967 =
193 + 29a
2461
300
967
979 + 89a
3619
2255
1353
1393
779
193 + 11 a
2255
485
246
779
161 =
107 + 13a
485
9325
159
161
2848
1.3175 ...
=1.3675 ...
1.3767... 1.3798...
1.4512...
1.5833... 1.6476...
2'0615...
188
2'3326...
C(2)
C(2)
C(2)
3619
1393 =
2.4917... 2.5450...
2'5980...
= 2.8947.. .
=
3.0124.. .
3 .2742.. .
Rounding error and integration Table 18.1 Steps
C(3)
C(3)
C(3)
(Cont.)
Exponent 13
From a =
239 + 4a
9325
2473
268
2848
720 - 3.437.. .
1400 + 49a
2473
7012
1663
720
1983 =
484 + 29a
7012
3356
620
1983
929
2314 + 89a
3356
5666
To a =
3.5361.. .
= 3.5740... a
3.8001.. .
1491 =
2777
929
430 + 11 a
5666
1214
494
1491
309 =
266 + 13a
1214
24177
331
309
5696 =
507 + 4a
24177
141
540
5696
32
3063 + 49a
141
3639
3375
32
803
1095 + 29a
3639
5169
1260
803
1133 =
5073 + 89a
5169
2769
5625
1133
583 =
915 + 11a
2769
581
990
583
121 =
597 + 13a
581
59577
675
121
11392 =
C(4)
C(4)
C(4)
403
3.9288... 4.2446 ...
= 4.4062...
= 4.5318... 4.5622.. .
4.7496...
4.8843...
5.2297...
for N2 >_ N >_ N4, and for N8 >_ N >_ Nlo if either of these ranges is non-empty, M(62Q-llq-33)/(62Q-2)N11/(62Q-2)(log N)6(5g2+6)/(31Q-1)
R <<
for min (N3,N4)zNzN5, R <<
M(356Q-89q-267)/(356Q-71)N89/(356Q-71)(log N)57(5g2+6)/(712Q-142)
for N5>_N>_N7, R <<
M(8oQ-
29q- 69)/20(4Q-1)N29/2o(4Q-1)(log
N)3(5g2+6)/5(4Q- 1)
Lattice points and area
404
for N7 >_ N max(N8, N9), M(34Q-4q-21)/(34Q-4)N2/(17Q-2)(log
R <<
N)3(5q'+6)/(17Q-2)
for min(N9, N10) z N Z N11, and also for N5 N11 when we replace the logarithm power by N. The ranges are defined by N1
=M9+2-6E
N 2 5Q = M75gQ + 64Q + 26 (log M)15(5g 2 + 6xQ -1)
N37Q+13 = M37gQ+31Q+13q+39(log
M)16(5g2+6)(Q-1)
N76Q+49 = M76gQ+62Q+49q+ 143(log M)S8(5g2+6)(Q -1) 4
0
N7867 =M178gQ+124Q+67q+201(log
M)82(5g2+6xQ-1)-356Q+71
N6267Q+18 = M267gQ+160Q+18q+125(log
M)57(5g2+6xQ-1)
N356Q - 31 = M356gQ+ 196Q- 31q+49(log M)32(5g2+6XQ - 1) - 356Q+71
N254Q-49 = M254gQ+134Q-49q-9(log
N37Q-2 = M37gQ+17Q-2q+8(log
N Q =M90gQ+34Q+41(log
o
M)8(5g2+6)(Q- 1)
M)4(5g2+6)(Q- 1)
M)18(5g2+6xQ-1)
N 1Q =M2gQ+1+4QE
The implied constants in the upper bounds are constructed from q and the derivatives of the function F(x), and, where appropriate, from e.
We can now set up a table of the best bounds for the rounding error sum R, when all the appropriate derivatives and determinants of derivatives are non-zero. We put N=M, a z 1. The bound R << MO (log M)5 (for some bounded 8) is valid for a in the range indicated. We have not optimized the logarithm power; this enables us to change from one estimate to the next at a number of the form MT rather than MT(log M)8. The suppressed order-ofmagnitude constants are constructed from the range of values of the deriva-
tives of the dimensionless function F(x). If the rounding error bound behaved like a random variable, then we would have the upper bound with /3 = 1 `almost surely'. We have shown that there is enough regularity in the rounding error to ensure some cancellation. In the first column of Table 18.1, C(q) denotes a case of Theorem 18.5.3, D denotes the double sum of Theorem 18.1.2, and E denotes that Lemma 18.5.2 has been used with some bound for the simple exponential sum. In some ranges we use double sum bounds for H large and simple sum bounds for H small. The bracketing of rows follows the same rules as in Table 17.1. When these bounds are used to estimate the number of integer points close
Rounding error and integration
405
to a curve, then small ranges of H cannot dominate; Theorem 18.2.1 can be used on the whole range a < 643/466. In certain ranges of a, corresponding to corners of the graph of /3 against a, the iterative ideas sketched in Section 19.2 give a small improvement. There is a rather different application to integrals in two dimensions. Let g(x, y) be a smooth function of two variables, and let D be a region bounded
by a piecewise smooth curve C. The simplest grid method of numerical integration is to take a square lattice of side 1/M, sum the values of g(x, y) at the lattice points within D, and divide by M2 to approximate the integral of g(x, y) over D. By the Riesz interchange, this estimate corresponds to
M1 I toN(to) + f "N(t) dt l
to
where to and tl are the infimum and the supremum of g(x, y) on D, and N(t) is the number of lattice points inside the subset D(t) of D on which g(x, y) >_ t. Let C(t) be the boundary of D(t), made up piecewise of arcs of C
and contour lines g(x, y) = t. We can expect further cancellation using Theorem 18.4.2 on each component of D(t), if the curves C(t) change smoothly with t. In general, part of C(t) consists of an arc of the curve C. However, if g(x, y) vanishes on the boundary of D (the Dirichlet condition), then each arc of the curve C(t) changes with t. Calculations of this type were made by Huxley (1993b) for the case when the surface z = g(x, y) is a convex cone. The mean square rounding error is given by the term in T26/43 in Theorem 18.4.2 whenever C has a radius of curvature which is non-zero, piecewise continuous, and twice continuously differentiable. However, a cone can be treated directly by Theorem 6.3.1, in which the exponent is 1, not 26/43. The Fourier series method seems to be the appropriate one in three dimensions and higher (Hlawka 1950).
19 Further results 19.1
EXPONENTIAL SUMS WITH A DIFFERENCE
The three sections of this chapter are unified only by the idea of iteration. In the course of treating an exponential sum, we come to another exponential sum, or a problem of integer points close to a curve. The first idea is to use the differencing step of Lemma 5.6.2, which produces a double sum with a central difference as in Chapter 9. The end result is slightly weaker
than Theorem 17.1.4, but still better than any bound obtained by the van der Corput iteration in one variable. The result would remain weaker than Theorem 17.1.4 even if we had the best possible result in the First Spacing Problem, with Lemma 13.3.1 as sharp as Lemma 13.1.1. These results
improve those of Heath-Brown and Huxley (1990), and were partly announced in Huxley (1994b). Theorem 19.1.1
Let F(x) be a real function, four times continuously differen-
tiable for 1 <x < 2, and let g(x), G(x) be bounded functions of bounded variation on 1 5 x < 2. Let CO, ... , C7 be real numbers >- 1. Let H, M, and T be large parameters. Suppose that IF(r)(x)I < Cr
(19.1.1)
IF(r)(x)I z 1/Cr
(19.1.2)
for r = 2,3,4,
for r = 2, 3, and that either Case 1 or Case 2 holds:
Case l M5 CoT 112, and (19.1.2) holds for r = 4 also.
Case 2 M> Co'T'/2, and 3(F(3))2I > 1/C5.
Let S denote the sum
S=
1g(H)m=M
2H-iG(M
M
)e(TF(mh)-TFI
Exponential sums with a difference
407
Then there are positive constants B1 and B2, depending on CO,..., C7, and on the functions g(x) and G(x), such that, for M in the range C6 'T 41/111 S
MS C6T 65/114
and H satisfying
H < B1MT 43/138,
(19.1.3)
and also
H>_C;1VIM' forM5C61T7/16, and also
H> C71M11/T6 for M> C6T9/11, we have H 87/140
H 13/70
<
S
I
T23/7o +
(M)
T18/35l (log T)9/4. (19.1.4)
In the range C61 T1/3 5 M 5 C6T7/16, min(B1M13/19T-10/57 BjM9/7T-3/7,C7T4M-9),
HS
(19.1.5)
we have S:5 B2(H 15/14M3/14 T 3/14 +H17/18M-1/6T7/18
+H93/56M-15/56T5/14
+ H 107/72M- 19/24 T 43/72 )(log T)9/4. (19.1.6)
In the range C61T9/16 <M 5 C6T2/3,
H5
min(B1M25/19T-28/57,B1M5/7T-1/7,C7M11T-6),
(19.1.7)
we have S
+H17/18M5/18T1/6 + H 107/72M- 13/72 T 7/24 )(log T)914. (19.1.8)
Proof We use Lemma 18.1.1 as in the proof of Theorem 18.1.2, making the same choices of N and R. There is a new condition H3 << N2R from (9.3.4) of Lemma 9.3.3, which makes the upper bounds for H in (19.1.3), (19.1.5),
and (19.1.7) smaller than in Lemma 18.1.1. The bound for A in Lemma 13.3.1 is A << (K 2L2 + KL4 )log K,
which gives an extra factor (1 +H2/RN)1/4 in the bounds for S, with extra terms in (19.1.4), (19.1.6), and (19.1.8) which do not appear in Theorem 18.1.1.
Further results
408
O
In order to state our next theorem in generality, we need one of the two-variable forms of Lemma 5.2.1.
Lemma 19.1.2 (Partial summation in a modulus squared integral) Let g(x) be a real function of bounded variation V on the closed interval a S x 5 b, where a and b are integers. Let fn(t) be continuous functions for a 5 t 5 /3, a 5 n 5 b. Then 2
b
2
n
R
L afm(t) dt.
max f E g(n)fn(t) dt 5 2(Ig(a)I + V)2 n5b f'3 a a a
Proof We write g(x) as the difference of two positive monotone functions on a S x5 b, g(x) = h(x) - k(x), with k(a) = 0 if g(a) > 0, and h(a) = 0 if g(a) < 0. We put n-+
Fn(t) = L.fn(t) a
for a 5 n 5 b, and hl(n) = h(n + 1) - h(n), for n = a + 1, ... , b, and
hl(a) = h(a). Then b
b
2
h(n)fn(t)) =
I
b
2
E hl(n)F,(t)I 5
hi(n))
a
a
(F',
a
a
hl(n) IFn(t)12) .
Integrating, we have 2
b
f13 a
b
E h(n).fn(t) dt
a
a
Now h(b) + k(b) = Ig(a)I + V, so the result follows from Cauchy's inequality, b
b
2
2
b
2IFlh(n)fn(t) +2 Ek(n)fn(t)I a
a
2 .
O
a
We apply Theorem 19.1.1 to the mean square of a simple exponential sum when the parameter T is regarded as a continuous variable. Theorem 19.1.3
Let M, T, and the functions F(x) and G(x) be as in Theorem
19.1.1, with (19.1.1) and (19.1.2) also holding for r = 1. Let S(t) be the exponential sum 2M-1
S(t)=
M
(;(M)e(tF(M))'
Exponential sums with a difference
409
and let A be any real number with A >> T72/227(log T)315"227
Then for T187/454(log T)279"454 << M << T267/454(log T)-279 454,
(19.1.9)
we have the mean square bound
fT+AI T-A
S(t)12 dt << AM.
(19.1.10)
Corollary Under the hypotheses of the theorem we haven S(t) << M1/2T36/227(log T)315/45a
19.1.11)
Proof After Lemma 19.1.2 we only need consider the special case when G(x) is 1 for 1 <x:5 M2/M (for some M2 < 2M), otherwise zero. To obtain central differences, we let S1(t) and S2(t) denote the terms in S(t) with m odd and m even, respectively. Then, since S(t) = S1(t) + S2(t), we have IS(t)I2 .2 IS1(t)I2 + 2 IS2(t)I2.
We apply Lemma 5.6.4 to the short interval means of S1(t) and S2(t) with sm = F(m/M), and t as the variable of integration. We have fT+AIS;(t)I2 dt
T-A
e((fl)
-TF(M)(fl)
TFM
<< A
-2AF(k\\
l
n k n =k =i
where n and k are summed from M to M2. With the substitution k = m + h, n = m - h, the terms for which h = 0 are seen to be O(AM), whilst the other
terms fall into ranges H :!g h < 2H, or - 2H < h 5 -H, where H is some power of 2. The congruence condition is m = h + i(mod 2), which disappears when we sum the contributions of Sl and S2. We consider M2-1
eI TF1 mM h
h=H m M+h X
`
l
-TFI mM h 1
A2AF(mMh) -2AF1 mMh
(19.1.12)
The expression (19.1.12) is of the type estimated in Theorem 19.1.1, except
that the variables are not separated in the weight function A(x). On the major arcs we used Poisson summation over m and an upper bound for the resulting integrals. Any weight which has bounded variation in m for fixed h makes no difference to the upper bounds on the major arcs.
Further results
410
On the minor arcs we approximate the weight as a function of h only. Let n be the centre of the minor arc containing m. Then
2A(F( mMh )_F( mMh ))_F( nMh )+F( nMh 4Ah(m - n)F" () M2
1 I
AHN
« M2
for some . After dividing the sum (19.1.12) into Farey arcs, we can evaluate the A-function at the centre of the minor arc, so that it becomes a function of h only, with total error
O(AH2N/M) << AM.
We now have a weight which is monotone in h, but depends on the minor arc. By Lemma 5.1.1 applied to the sum over h, we must estimate sums in which h runs through a subinterval of H5 h < 2H, depending on the minor arc. The next step in the argument, Lemma 9.3.1, involves taking a maximum over subintervals anyway, so the argument continues as before.
The bounds of Theorem 18.1.2 are increasing functions of H, and are dominated by the top range, which is H x M/A, since the A-function in (19.1.12) is A(4AhF'(e)/M) for some , and F'(x) does not vanish. The bound of Theorem 19.1.3 is linear in A, and the choice A x T72/227(log T)315"227
just gives S << M in (19.1.4) of Theorem 19.1.1. The bound (19.1.6) is smaller than (19.1.4) for H31 >> T19/M39,
and the bound (19.1.8) is smaller than (19.1.4) for H31 >> M101/T51
Thus we get (19.1.10) in part of the region where (19.1.4) or (19.1.6) is the appropriate bound when H x M/A. The range (19.1.9) where (19.1.10) is valid is longer than the range in which (19.1.4) in Theorem 19.1.1 is valid. The corollary follows from the mean-to-max argument of Lemma 5.6.3; we apply Theorem 19.1.3 with G(x) replaced by G(x)exp(crF(x/M)) for some o in -1 < o-< 1. Since A > 2, most of the range of integration in (19.1.10) is not used; in fact we can replace the integral in (19.1.10) by a sum over I S(t;)l2 at a sequence of well-spaced points t; in the interval
T-A +25 t;5T+A-2, with t,.,.,-t;>-4. We also have forms of Theorems 19.1.1 and 19.1.3 for a family of exponential sums.
411
Exponential sums with a difference
Theorem 19.1.4 Let F(x, y) be a real function defined for 1 :5x:5 2, 05 y5 1, for which the partial derivatives are continuous and satisfy t9 c9 F(x,Y)I S C1
for 25r56, s52, r+s56, and (19.1.13)
1/C1 5 I Br1F(x, y)I
for r = 2, 3, and
a;-' a2F(x, y) a' d,F(x, y)
(19.1.14)
- C2 2
for r = 3, where C1 and C2 are constants. Suppose that either Case 1 or Case 2 holds:
Case 1
(19.1.13) and (19.1.14) hold for r = 4 also.
Case 2 For some positive constants C31 C4 13F111
Io1
C31
- F11F1111
C4
where A is the determinant 3F1i1 + 4F11F1111
0=
3F11F111
F1i
F11111
F1111
F111
F11112
F1112
F112
Let Y1, ... , y, be points in [0,1] with y;+ 1- y; Z 1/J, where J:5 T. Let g(x) and G;(x) be bounded functions of bounded variation on [0,1] (uniformly in i in the case of G;(x)). Let Si denote the sum
2H-1 /h
S;= E gl -) H h=H
M-2H+1
E
G,
m=M+2H-1
m1 M
M)e(TF(I
m+h
m-h
,Y;) -TF( M 1M
,Y;
where H and M are positive integers, and T is a real number. Then there are constants Cs and C6 constructed from C1, ... , C4 such that if C5T1/35MSCS 1T2/3 and
H5 C6 min(
M3/2/T'/2,
M1/2, MT-8/27),
(19.1.15)
then we have bounds of the form
(H IS;12 << D1/2E4/21H2I
121
IT13/21 logs T
M)
(H 25/21 IT41/42 logs T, +D1/2El1/42H2I H) M
(19.1.16)
provided that
E 5 C6M9/H9T8/3,
(19.1.17)
Further results
412
where the constants in the upper bounds are constructed from C1, ... , Ca1 from the bounds for the functions g(x) and G.(x), and from the extra constant C7 in Cases (b) and (c): Case (a)
We have (19.1.16) with
J
(HT)1/5
+I,
E=(I3M
D=1,
provided that Case 1 or Case 2 holds for F(x), and that T
E z C6 1 min M2 +
Case (b)
M2 HT 2/5 T ,(
5
.
M
We have (19.1.16) with
D=1+
/ HM5 11/2
JM2
IT
,
E=1+I D3I3T2)
TS
1/6
(H5M9)
provided that Case 1 holds and M < C7T1/2, Where C7 is a constant that may be chosen arbitrarily. Case (c)
We have (19.1.16) with
JT
D=1+IM2,
HTa
11/2
E=1+(D3I3M7)
( M19 \ 1/6
+I H5T9)
provided that Case 2 holds and M > C7T 1/2.
Proof We make the same parameter choices as in Theorem 18.4.1, but omitting the case N x H, and imposing the condition H3 << N2R from (9.3.4) of Lemma 9.3.3, which leads to the extra condition (19.1.17) and the third term in the minimum in (19.1.15). The extra condition is not essential. For H3 >> N2R, the extra term in Lemma 9.3.3 becomes larger as the gardening becomes less effective, eventually dominating the contribution from the large
sieve, so that the bound no longer has the form (19.1.16). The condition H3 << N2R, by eliminating the choice N x H, allows us to collect terms in the form (19.1.16).
Theorem 19.1.5 Let M, T, the points yl,..., y,, and the functions F(x) and
G;(x) be as in Theorem 19.1.4, with the bounds (19.1.1) and (19.1.2) of Theorem 19.1.1 also holding for r = 1. Let S(t) be the exponential sum of Theorem 19.1.3, and let 0 be any real number with A >> 1- 11/227 T 72/227 (log T)350/227
A major arc estimate
413
Then for
J
1/227
T31
I <<
I134(log T)7o
and 1107/454 T 187/454 (log T )155/227 << M « I-107/454 T 267/454 (log T) - 155/227
we have the mean square bound 2
1
T,
19.2
I
2
(S(t)I dt
«IO2M2,Y
= T(1 +y1).
A MAJOR ARC ESTIMATE
We consider the major arc sum of Lemma 7.6.1, with length parameter N' = max(N2, -N3), which may be longer than the usual range for N in Chapter 7. The sums over r have
r «µgNi2.
(19.2.1)
Since r does not have constant order of magnitude, we divide the sum into blocks with H :!g r5 2H - 1, where H is some power of 2. As in Chapter 7, we can write
G(a, b + r; q) = e
4a(b + r)2 q
G(a, 0; q)
(19.2.2)
for q odd, and similarly in the other cases. The weight function remaining is
G(a,0;q)
«
(12µg3(r - K)) 1/4
1 .
(19.2.3)
(tr,gH)1/4
As in Lemma 7.3.1, we have
h(gyr) - ryr
-2
(r- K)3 27µq3
>
H3
(µq3) .
(19.2.4)
We assume that this term dominates the term in r from (19.2.2). Since we can
replace 4a and 2ab by integers congruent to them modulo q in the range
-q/2 to q/2, we require µg3H < 1/C1 for some sufficiently large constant C1.
(19.2.5)
Further results
414
The exponent (19.2.4) is a monomial function with monomial part -2r 3/2 / (27µq3) The conditions of Theorem 17.4.2 hold for all R >- 1, but in Theorem 17.1.4 (which corresponds to the case R = 0) the condition .
(17.1.27) fails. In the special case F(x)=x312, coincident minor arcs I(a/q) and I(a'/q') have a' close to a (analogously, for F(x) = x3, coincident minor arcs have q' close to q). We must argue differently; one way is to choose the parameters so that there are no non-trivial coincidences. The condition (17.1.27) of Theorem 17.1.4 is needed only for
N' >> 1/µq2, a very long major arc. We can use any of the bounds in Table 17.1 when
a
2log H
(19.2.6)
= log(H3/µq3) 5 z)
which is a consequence of (19.2.5).
As the length H of the transformed sum increases, the exponent (19.2.4)
increases like H312, and the weight (19.2.3) decreases like H-'14. The bounds of Table 17.1 have the form 13 5 p + as with p + o, >_ Z, so, within
each range, the estimate is largest when H is largest. The exponents /3 change continuously from one range to the next, and the logarithm powers are bounded. The largest value of H in (19.2.1), H x µgNr2, corresponds to the largest value of a, a -,
log µgN'2
(19.2.7)
log µN'3 '
and a weight function
1/(
1/ (µqN')
.
The dominant range contributes <<
µP+Q-;qo-ZN?3P+2o-_.
(19.2.8)
Smaller values of H contribute something smaller. We can simplify the calculations by using a weaker bound for blocks with H smaller. Suppose that
a given by (19.2.7) is in a range where we find ARC in the first column of Table 17.1. We use Theorem 17.4.2 with this value of R (Theorem 17.1.4 if R = 0) even for small H. Again, within each range, the estimate is largest when H is largest. There is a case M < N, where an extra factor T E appears in Theorem 17.4.2. The corresponding blocks are inside ranges in which we would use AQC with some Q > R, and (19.2.6) gives values of a strictly less than either (19.2.7) or the next change-over point in Table 17.1. Without the factor `TE', these blocks would give something smaller than (19.2.8) by a positive power of N'. The factor `T E, becomes O(Ni3E), which does not
cancel out all the positive power of N. We summarize the result of this discussion as a lemma.
Exponential sums with a large second derivative
415
Lemma 19.2.1 (Long major arc estimate) Let R >- 0 be an integer, and let F(x) be a real function R + 6 times continuously differentiable (four times differentiable when R = 0), with F(3)(x) non-zero for 1 <x5 2. Let M (an
integer) and T (a real number) be large, and let [ m + N3, m + N2 ] be a subinterval of [M, 2M-1]. Write N' = max(N2, -N3). Suppose that
f" (m) = 2a/q + O(T/M3). Let
µ =f'()/6. Then there are absolute constants CO and C2, depending on the function F(x), such that if M 1 11 q5 , 1/3,
N'S-min 3,M C2 µq
COT
and if log µgN'2
a = log µN,3
has a < Z, and falls into a range ARC in Table 17.1, then m+N2
e(TF(m
+n))
M
<<
µP+o-
qv-+N,sp+2Q-
(logT)e+µ-1/3q-i/2 log T,
where p and o, are the coefficients of the linear bound a = p + o a in Table 17
valid for this value of a. The exponent B depends on R, and is uniformly bounded.
O
Lemma 19.2.1 is powerful when q is small. However, in order to treat all arcs as major, we must take N' NR/q for q/R. With these choices, r is bounded in (19.2.1) when q x R. Large q dominate in Lemma 7.3.4, so
the improved bound for small q does not lead to an improvement of Lemma 7.3.4.
19.3 EXPONENTIAL SUMS WITH A LARGE SECOND DERIVATIVE In Part IV we saw that the conditions for the coincidence of two minor arc vectors by a fixed magic matrix implied that there was an integer point close to a curve, the resonance curve. This curve is given by a complementary function construction, with a cusp corresponding to the point where the error in the Second Condition changes sign. The number of integer points that might be close to the resonance curve was estimated trivially, by the length of
Further results
416
the curve. There are two cases where the error in the Second Condition has constant sign, and the resonance curve has non-vanishing curvature. These are the triangular matrices (type 2) for a simple sum, and the identity matrix (type 1) for a family of sums with a parameter. The construction of Chapter 15 gives a sequence of short resonance curves depending on the reference fractions. The construction of Section 16.3 gives one resonance curve, but it
may be longer than the underlying curve y =f'(x) in the construction of Chapter 7 or y =f(x) in the construction of Chapter 8. We can use Theorem 3.6.3 which does not require extra conditions on the derivatives, or Lemma 18.5.2 together with a bound for exponential sums. The second choice may lead to an iteration, with the original function f(x) transformed by differentiation, differencing, and taking the complementary function. The class of those monomial functions (in the sense of Chapter 3) for which f'(x) becomes proportional to a negative power of x as M and T tend to infinity is closed under these operations, and satisfies all our conditions about the size and non-vanishing of derivatives. We obtain the results of Huxley and Kolesnik (1996). We consider a family of resonance curves z = k(y, t) constructed from the
upper triangular matrices where
(10
B ) or the lower triangular matrices ( u
B=A2Pt/Q,
J,
C=A2Qt/P.
Since the Coincidence Conditions are symmetric, we can suppose for convenience that B > 0, C > 0, and so t > 0. We see from Lemma 14.3.2 that t is bounded. We use Lemma 16.3.5 with rs «R2. If a coincidence extends over L z 1 consecutive minor arcs, then Lemma 13.4.2 allows us to replace A2 by O(A2/L2) in Lemma 14.3.2, so that t << 1/L2. Both estimates (14.3.15) and (14.3.16) for h"(a/q) give MQt
Ax02
P
At this point we must distinguish the two constructions. The double sum has
an extra parameter H; for the simple sum we replace H and N in the following estimates. We use the non-linear relation HR2
R2
2 A2x04NQ x04NQ2)
in which H appears naked. The result of Lemma 16.3.5 becomes R2
1
Ik(l)+ni<8<
K1 X A2Mt X
Q4
D4PQ2t.
(19.3.1)
Exponential sums with a large second derivative
417
Since ak/ay is an argument of h(x), z = k(y, t) runs through a range of length K2 x A2
PMt
Qx
R4
taP2Qt.
Q4
For upper triangular matrices we take y as the independent variable, M' = K1, N' = K2. For lower triangular matrices we take z as the independent variable, M' = K2, N' = K1. In both cases Lemma 14.3.2 gives M' <<
02N'. Lemma 19.3.1 (Points close to resonance curves) Suppose that the conditions of Lemma 14.3.5 hold, and that we have a bound of the form
R' << SM' + E M'AIN'"I(log M'N')Ai
(19.3.2)
for R', the number of integer points (1, n) satisfying (19.3.1). Then for M << FT and V chosen so that 1 << 401Q2 S 1, we have in the Second Spacing Problem BV<< L1L2A304P2R2V r
+0102
I
Ra
";+A; 1
Q4
pz";+A;+1Q";+zA;+1V(logT)A', (19.3.3)
Aa I
for a choice of N and R in the construction of Chapter 8 with N in the range
MM
M2
1/3
Nxmin (HT2)1/3' H(T)
M2"i+4Ai(logT)A'
+
H1+I(;+A;TA;
I
provided that the conditions (8.1.4)-(8.1.7) hold, and for a choice of N and R in the construction of Chapter 7 with N in the range M3
Nxmin T
1/2
M4 1/6
,(T
1/(2+3"i+3A;)
M2"i+4A;(log T)A1
+
TA
provided that the conditions (7.2.3)-(7.2.5) hold. 1 For M >> FT and V chosen so that 1 << B4 i1 P2 5 1, where B4 is the constant of Lemma 16.1.4, we have (19.3.3) with P and Q interchanged for a choice of N and R in the construction of Chapter 8 with N in the range M2 1/3 M8/3 M4+4"i+2Ai(logT)A' 1/(1+2"+2A;)
N x min
H) ' HT +
l
H1+";+A;T2+"i
I
provided that the conditions (8.1.4)-(8.1.7) hold, and for a choice of N and R in the construction of Chapter 7 with N in the range Ai 1/(2+3Ki+3Aj) M4/3 M4+4" +2A' T Uj 1/2, T
Nxmin M
1/2 +
z+ ";
provided that the conditions (7.2.3)-(7.2.5) hold.
Further results
418
Proof In the upper triangular case the number of coincidences of between K and 2K consecutive minor arcs given by a fixed magic matrix is <<
hiK2t
zR2
+ E M,a;N'K,(log MIN')A, M
+
K;+A;
R°
(-
We sum this over O(A2P/K2Q) magic matrices to
P << a2 K2Q 0304 P
+ 02 K2Q
R2
1
Q2 PQ
K2
K;+A;
(R4
p2K;+A;QK;+2A,(log T)A'.
I Q4 04
We sum K through powers of 2 to get P2R2 <<
2A3A4
Q2
R4
+A2
K;+A;
R A41 l
P2Ki+A;+1QK;+2a;-1(logT)A,
17C
(19.3.4)
coincidences. The identity matrix contributes O((R2/Q2)PQ) trivial coinci-
dences. We want to make the ratio R/N small, but not so small that the trivial coincidences dominate. Either H3N2 << MR2,
or, for some i, H1+K;+A;N1+2K;+A;
<< M2K;+A;R2Ai(log T)A'.
These inequalities give the second and third terms in the bounds for N. The first term comes from the requirement that V Z 1 in the First Condition. The result (19.3.3) follows when we multiply (19.3.4) by A1Q2V. We calculate similarly in the lower triangular case, using Lemma 3.1.1 to interchange the
0
variables.
Corollary 2 to Lemma 5.4.3 gives
E p(k(l) ± 8) << (M'N')1/3, which gives (19.3.2) with i = 1, K1 = Al =
3,
Al = 0. In this case (19.3.3)
becomes R
BV z A1A20304P2R2V+ 0102 «L 1L 2I /3P2R2V,
(Q
141
1/3
P2R2V
Exponential sums with a large second derivative
419
and we get (17.1.9) of Lemma 17.1.2, apart from the power of R/Q. To improve Lemmas 17.1.2 and 18.1.1, we must use bounds stronger than the corollaries to the truncated Poisson summation formula.
It is a nuisance to check the conditions of Chapter 7 and 8 in applying Lemma 19.3.1. The purpose of these conditions is to exclude unsuitable ranges of the parameters. In practice, a choice of N in Lemma 19.3.1 that leads to a good bound for exponential sums will fall into a suitable range. For this reason we have left unspecified the ranges for M in the iteration lemmas that follow.
Lemma 19.3.2 (Iteration step for the simple sum) The choices of N in Lemma 19.3.1 give bounds for the sum S of Chapter 7: S << M2/3T1/12 log T+M114T1/4 logT + 57, M(6+10"i+11Ai)/(10+15"i+15Ai)T(2+3"i+2Ai)/(20+30"i+3OAi)
i T)1+Ai/(20+3OKi+3OAi)
x (log
+E
M(9-5"i-Ai)/20T(3+3"i+Ai)/20(log
T)1+Ai/1o
for a range of M with M << FT, and S << M113T1/4 log T + M314 log T
+
M(4+5"i+4Ai)/(10+15"i+15Ai)T(4+8"1+9Ai)/(20+30"1+30Ai)
X (log
T)1+Ai/(20+30"i+30Ai)
+E
M(11+5"i+Ai)/20T(1-"i)/10(log
T)1+Ai/10
for a range of M with M >> FT V.
Corollary A bound of the form (19.3.2) valid for a certain class of functions and a certain size range for (log N')/(log M') implies two other bounds of the same type, valid for related classes of curves and certain size ranges. In the first bound, the exponents (Ki, A,) are (1/13, 9/13), (1/5, 2/5), and two terms
2+3K+2A 14+23K+24A (22+33K+32A' 22+33K+32A)'
12-2K (23+3K+A' 22+3K+A 3+3K+A
corresponding to each term M'AN" in (19.3.2). In the second bound, the exponents (K;, A.) are (1/5, 7/15), (0, 3/4), and two teens
4+8K+9A 12+18K+17A 24+38K+39A'24+38K+39A)' corresponding to each term in (19.3.2).
1-K 13+3K+A
(11-K' 22-2K
Further results
420
Lemma 19.3.3 (Iteration step for the double sum) The choices of N in Lemma 19.3.1 give bounds for the sum S of Chapter 8:
S << HM'/'T'/'(log T )9/4 + H2/3T 1/3 (log T )9/4
+ 5' H(4+9K,+9A;)/(4+8x,+8A,)M(K,+2A;)/(2+4x,+4A,)T(1+2x,+A,)/(4+8K,+8A,) T)9/4+A,/(4+sK,+8A,)
x (log
+ E H(12-xi-A,)/12M-x,/2T(3+4K,+A,)/12(logT)(9+A;)/4 for suitable ranges of H and M with M << FT, and S << HM-1/3T 1/2 (log T)9/4 + H2/3M2/3(log T )9/4 H(4+9x;+9A;)/(4+8x;+8A;)M-(2x;+3A;)/(2+4x;+4A;)T(1+5x;+6A;)/(4+8x;+8 A)
+ Y` i
x (log
+E
T)9/4+A,/(4+8K,+8A,)
H(12-x
A,)/12M(4xi+Ai)/6T(1-x,)/4(log T)(9 +A,)/4'
i
for a range of H and M with M >> fT .
Corollary A bound of the form (19.3.2) valid for a certain class of functions and a certain size range for (log N')/(log M') implies another bound of the same type, valid for related classes of curves and certain size ranges. The exponents (Ki, A.) are (1/6, 1/2), (1/2, 0), and two terms
1+2K+A 1+5K+6A (4+9K+9A' 4+9K+9A)'
3+4K+A
3-3K
(12-K-A' 12-K- A)
corresponding to each term M'AN" in (19.3.2).
The proofs of Lemmas 19.3.2 and 19.3.3 follow those of Theorems 17.1.4
and 18.1.2. The corollary to Lemma 19.3.2 uses Lemma 18.5.2, and the corollary to Lemma 19.3.3 follows the proof of Theorem 18.2.1. The two cases
of Lemma 19.3.3 give the same exponents in (19.3.2) with M' and N' interchanged.
The new exponents from the iteration are not very sensitive to the values of K and A, and are useful only for a certain range of (log N')/(log M'). In practice this range is a short interval of the longer range in which the bounds are valid. We start the iteration from the bounds of Table 18.1, except that we use Theorem 18.2.1 in place of Theorem 18.2.2 and Lemma 18.5.2 for the
range 83/635 a5 643/466. Moreover, we can use the analogue of Theorem 18.2.1 for a family of exponential sums, using t as the parameter. This is the only range in which we can use t as a parameter. The bounds lower down
Exponential sums with a large second derivative
421
Table 19.1 Bounds for integer points close to a curve Iterated
Steps
D DD DDDC
23
23
41
73
73
32 =
17
35
7
3
41
396348
79
79
17
17
32
309023
DCC
DC
69787
3118
6577
396348
49115058
146195
146195
14735
14735
309023
38213083 -
15603
40525
3143
8083
49115058
1590221
84191
84191
16837
16837
38213083
1229946 =
1251
3306
115
342
1590221
3799577
6829
6829
679
679
1229946
2859937 =
3118
6577
598
333
3799577
8366056
14735
14735
1589
1589
2859337
6185661 =
3143
8083
675
1571
8366056
269897
16837
16837
3363
3363
6185661
197842
342
4
43
269897
27
679
679
64
64
197842
19 =
598
333
29
62
27
1375774
1589
1589
140
140
19
939695 =
675
1571
4
43
1375774
44151
3365
3365
64
64
939695
29740 =
4
43
44151
19
64
64
29740
12
29
62
19
374
140
140
12
227
89
356
374
872
641
641
227
423 =
C
1.2812... 1 . 2826
...
1 .2853
...
1 .2929
...
1.32 88...
1
.3525
...
=1.3573.. .
115
CC
C
=
27548
DDCC
DDC
from
Exponents K, A
Range for (log N)/(log M) from to
1
.
4211 ...
1
.
4641 ...
1 . 4846
...
=1.5833...
=
1
.
6476 ...
2 0615 .
...
Table 18.1 already use Theorem 17.2.2, with a parameter from the differencing step or from the Fourier series for the row-of-teeth function. Tables 19.1 and 19.2 give some further bounds for integer points close to a curve and for exponential sums. In the first column D means the double sum, C the simple sum, and A the differencing step, so that DCC means Lemma 19.3.3 applied to the result of Lemma 19.3.2 applied to one of the bounds that uses Theorem 17.2.2, labelled C(1) in Table 18.1. As usual, rows bracketed together correspond to an upper bound with two or more terms. Estimates for exponential sums in other ranges can be obtained by the A (differencing)
Further results
422
Bounds for exponential sums
Table 19.2
Iterated
Steps
Exponent (3
C
from
Range for a = (log M)/(log T) from to
89 + 285a
106822
139817
570
246639
246639
= 0.5669...
2387 + 17972 a
115
342
1033325
106822
27290
679
679
2642746
246639 =
2819 + 19177«
598
CDC
0.4331...
333
699371
1033325
2642746 =
29855
1589
1589
1647930
11897 + 88442 a
675
1571
156527
699371
134680
3363
3363
370694
1647930
CCC
= 0.4244...
113 + 897a
4
43
263
156527
1345
64
64
638
370694
CC
CC
0.4288...
= 0.4222...
491 + 3624«
29
62
143
263
5530
140
140
349
638 -
569 + 1053a
29
62
307
143
2800
140
140
641
349
1273 + 2484«
89
356
68682
307
6410
641
641
171139
761
AC 1
0.4122...
= 0.4097.. . = 0.4034.. .
4 + 103a
12
68682
128
31
171139 =
29 + 173a
227
12
280
601
31 =
0.4011...
0.3871.. .
or B (reflection) steps as in Table 17.1. We get some improvement near the values of a for which the steps in Table 17.1 change from A'C to A'+ 1C. For most values of a, the saving obtained from the iteration is less than the saving from using Theorem 17.2.2 instead of Theorem 17.1.4 after a sequence
of differencing steps. The iteration works best for exponential sums with a = (log M)/(log T) near 5/12, where the third derivative is still small, but the second derivative has order of magnitude larger than one. This range of a provides the title for this section.
20 Sums with modular form coefficients 20.1
EXPONENTIAL SUMS
We obtain the main results of Jutila (1987a et seq.) for the sum M2
S = E b(m)e(f (m))
(20.1.1)
M
of Chapter 10, where E b(m)e(mz) is a cusp form of even weight k for the full modular group. Lemma 20.1.1 (Choosing parameter sizes) Suppose that the conditions of Lemma 14.3.1 hold. Then, in the notation of Lemma 10.5.1, we have R 13/2
B(01(V ), 02(V ), 03(V ), 04(V )) << -
P2Q2,
Q1
V
where V is summed through positive powers of 2 satisfying
V 5 L1 /(LM)1"4,
(20.1.2)
for a choice of N and R in the ranges
NxMT-1/3,
RxM112T-1/3.
(20.1.3)
The implied constants are constructed from the derivatives of the function F(x).
Proof First we note that
0(V)<< 1
1
I1
M LQ2)
1
12
1
+ <<7 +PQ.
Type 3 matrices have B and C non-zero, and B << O1 P2, C << L 1 Q2. If AI(V) << 1/PQ, and type 3 matrices occur, then P x Q and B,C are bounded. We enlarge the definition of type 1 matrices to cover this case. Hence we can suppose that 01(V) « 1/V2 when type 3 matrices occur. Next
11
Q. 0z « 01(V) V!9M « 172+_
Sums with modular form coefficients
424
Type 2 matrices have B «O2P/Q, C z i 2Q/P. We can suppose similarly that 02(V) << 1/V2 when type 2 matrices occur. By Lemma 14.3.5, the sum is
K F, VI PQ+ y2 (P2 +Q2)+ VQ P2Q2) <
`
If
We have L3Q2 1/2 M
V2 N2R4 P2Q2 << M2Q4
R3
N2
« Q MR2 '
)
0
which motivates the choice (20.1.3)`.
Theorem 20.1.2 Let F(x) be a real function four times continuously differentiable on the interval 1 < x < 2, whose derivatives satisfy the following conditions IFV'>(x)I
(20.1.4)
I F(')(x)I z 1/C1
(20.1.5)
for r = 2,3,4, for r = 2, where C1 is a positive constant. Let b(m) be the coefficients of a modular form of even weight k for the modular group, and let MZ
m
b(m)e(TFI M )),
S=
m=M
1
where M and M2 are positive integers, T is a large real number, and M< M2 < 2M. Suppose that, for some C2 Z 1, either Case 1 or Case 2 holds:
Case 1 M 5 C2 T; Case 2 M z C2 'T and (20.1.4) and (20.1.5) hold for r = 1 also and IF'(x) + 2 xF" (x) I _ C3 for some positive constant C3. Then there is a constant C4 constructed from C1, ... , C3 such that, for C4T 2/3 5 M:!-. C4 'T 4/3,
(20.1.6)
we have S << BMk12T1/3(log M)1/2 + BM(k+1)/2T-1/6(log M)112, (20.1.7)
where B is the constant of Lemma 10.2.1. The implied constant is constructed
from C1,...,C4. Remarks The second term in (20.1.7) appears when we pass from a smoothed sum in Lemma 10.3.1 (with N x MT-1/3) to the sum (20.1.1). This term may
Exponential sums
425
be reduced by appealing to a stronger bound than Lemma 10.2.1, such as the
full result of Rankin (1939) or the results of Moreno and Shahidi (1983) (Lemma 10.3.3) or Deligne (1974) (Lemma 10.3.2).
Proof The conditions on F(x) are those of Lemma 14.3.1. In Case 1 we can subtract a linear term from F(x) which changes TF(m/M) by an integer, and so does not affect the sum S. Having done this, we suppose that (20.1.4) holds for r = 1 in both cases. In Lemma 10.5.2 the number of rationals satisfies J(Q) x PQ, and Lemma 20.1.1 bounds the block sum by 2
min(2L-1,K) (Q)
E 1=L
F, b(l)hj(l)e(gj(l)) i
(R
L3
5/2
«B2Mk 1NRI Q) P2Q2
Mk+1Q
L3
1/4
(Q )
log M
lMR2)1/4
log M.
N
Summing over the blocks L:!-. 1 < 2L, Q5 qj < 2Q, we have in Lemma 10.4.1
E E wj(m)b(m)e(f(m)) j
m M2R2
M314
(NR)
1/2
<< BMk12T1/6(log
+ N3 +
M 1/6
(-)
(log M)1/2
M)112,
by the choice (20.1.3) of N and R. The lower bound in (20.1.6) ensures that
R Z 1, and the upper bound ensures (10.3.5). This gives the first term in (20.1.7) of the theorem. The second term in (20.1.7) is the smoothing error in Lemma 10.3.1.
When we consider a family of sums, then we do not use Lemma 14.4.3 because the assumption 02 PQ >> 1 may fail. We consider types of magic matrices separately. We take M << FT, so at most a bounded number of lower triangular matrices occur. We reclassify them as type 1. We define a set of Farey arcs to be a transversal set if, for any pair of indices i, j,
a')'M(a,) q1 qj for any type 1 matrix M except the identity. The Farey arcs can be divided into a bounded number of transversal sets.
Sums with modular form coefficients
426
Lemma 20.1.3 (Parameter sizes for a family of sums) Suppose that the conditions of Lemmas 14.3.1, 14.4.1, and 14.4.2 hold, and that the sum over j in Lemma 10.5.2 is restricted to a transversal set of Farey arcs for the sums of a family. Then the bound (10.5.32) of Lemma 10.5.3 is
JR2 (LM)V4
O
I2P2Q2 log IPQ
(20.1.8)
QZ
for (20.1.9)
T2/3 << M<< T,
and suitable choices of N and R in the following ranges:
R 1-1/3M112T-1/3.
Nx1213MT-1/3,
(20.1.10)
for T1/3 12/3
J2 T »IZT+M
(20.1.11)
and
Nx (J/I)-2 MT,
R 4 JrM- /IT
(20.1.12)
for J2
T
T 1/3
+
T1/3 >> 12T » 12-3
M
(20.1.13)
1
(20.1.14)
and
N x M2/T,
R
for J2 T 1/3 T + 12T . M >> 12/3
(20.1.15)
The implied constants are constructed from the derivatives of the function F(x), and from the implied constants in the ranges (20.1.9), (20.1.11), (20.1.13), and (20.1.15).
Proof Since the Farey arc indices i and j are limited to the same transversal set, only coincidences given by type 3 and upper triangular type 2 matrices
are counted in B'(O 1, ... A4). As in Lemma 20.1.1, type 3 matrices contribute
12p2Q2 LQ2 «01(01 + A2)I2P2Q2 <<
V4
By Lemma 14.4.2, an upper triangular matrix I o
M
.
i) gives coincidences
only when the first derivative a/q lies in a `short interval of length
427
Exponential sums
O(O2P2/I bI Q2), which contains at most O(i,2P2/I bl) + 1 rationals by Lemma 1.2.3. There are two cases. As in Lemma 14.2.2, triangular matrices with
L Pz LQ2 1 5 Ibl 5 O2P2 5'117 M +A1 M
(for some constant A1) contribute
« O2I2P2 log PQ. Since Ibi z 1 and L5 18M/R2, if R is sufficiently large, then R Z 6 A1, and
P2 LQ2 A252A1Vz M , so this case contributes
12P2 LQ 2
« -F F M
log PQ.
The second case is that of triangular matrices with Ibl z 02P2, which give a bounded number of coincidences each. We use the conditions involving i3 and 04, with
03(V) «
1
(LQ2)31'4 1 M + PQ
V!L
1
PQ
LQ z
M
provided that N2>> MR2, and z
P
04(V) X VT
M LQ2
+ PQ
( M) >> i,3(V).
To avoid double suffices, we write yi for the value of the parameter on the Farey arc with index i. We have U.
-=1+O(03), ui and
f11(u3,Yi)
= 1 +O(D4).
f1i(uj ,Yy)
Since F12 and F112 are supposed bounded and non-zero, f11(u?,Yi)
f11(ui3, Yj)
1 XIYi -y,IX
f1(ui,Yi) z fl(u; ,Y;)
-1
Sums with modular form coefficients
428
Now
f1(ul,Y1) =fl(u),Yj) +b, so that bQ
i,Yi) P - fl(u),yj) .f1(uz
-1 «lyi-Yjl+L3
<< A4 + 03 << 04
and we have
b «A 4
P
2
Q2
l
2
M 1« V2
Q2
QV2
vM
we can drop the second term since b is a non-zero integer. This case contributes
of
I2P2
VT
v2
Hence
B'(A1(V).... ,04(V)) «
I2V2Q2 2
IogPQ (
M2 )
and in Lemma 10.5.3
E v
r
(LM)114V
B'(A1(V ), ..., A4(V )) +J(Q)L + Q2
<< IPQL +IJPQ log l (
QJ(q)log I min lyi -yjl
+I2P2Q2 log PQ )
(
Q2)
.
(20.1.16)
In the first two cases of the lemma, the third term on the right of (20.1.16)
has the same order of magnitude as one of the other two terms in the extreme ranges Q x R, L x M/R2. Theorem 20.1.4 Let F(x, y) be a real function defined for 1< x < 2, 0 S y < 1, for which the partial derivatives are continuous and satisfy I a1 2'F(x, y)I < Cl
forr54, s<2, r+s<4, and i/C1 < I a1F(x, y)I
(20.1.17)
I a; a2F(x, y)I > C2
(20.1.18)
for r = 1, 2, and
Mean value theorems
429
for r = 1, 2, where C1 and C2 are positive constants. Let yi,... , y, be points in [0,1] with yi+1 -yi Z 1/J. Let G;(x) be bounded functions of bounded variation on [0,1] (uniformlly in i). Let Si denote the sum analogous to (20.1.1): 2M-1
m
m
T
Si = E b(m)G,(M)e(M
F(j,Yj)),
m=M
where T is a positive real number, and M is a positive integer with T2/3 << M<< T.
(20.1.19)
Then we have
I ISi1 <<
BIMk/2T1/4(
T1/3
IZT
i=1
T "3 +AIMtk+ 1>/2
J2
I2/3
+
+
J2
T
-
1/4
T M)
log IM -1
log M,
I ZT
where A is the sum of the moduli of the coefficients when the modular form E b(m)e(mz) is expressed as a linear combination of newforms, and B is the Rankin constant of Lemma 10.2.1. The terms in A can be omitted if the functions G,(x), when defined as zero for x < 1 and x >_ 2, have G;:> continuous with
IG;S)(x)I <(CON/M)5
(20.1.20)
for i = 0, ... 4, for N as chosen in Lemma 20.1.3, and some constant Co. The implied constant is constructed from C0,. .. , C4, from the bounds for the functions G,(x), and from the implied constants in the ranges for M. Proof We modify the proof of Theorem 20.1.2, using Lemma 20.1.3 in place
of Lemma 20.1.1 on the smoothing error, and Lemma 10.3.2 in place of Lemma 10.3.1. The conditions on F(x) are those of Lemmas 14.3.1, 14.3.5, and 14.4.1.
O
The smoothed version of Theorem 20.1.4 can be generalized beyond classical cusp forms with a congruence level. There is a similar result for FT << M << T413, but the second and third terms are bigger, because the conditions involving 03 and 04 do not apply.
20.2 MEAN VALUE THEOREMS We use the notation of Section 10.6, so S,Q>(t) is a smoothed sum over a transversal set.
Lemma 20.2.1 (Parameter sizes for the mean square) Suppose that the
Sums with modular form coefficients
430
conditions of Lemmas 14.3.1, 14.4.1, and 14.4.2 hold, and that S;Qky) in Lemma 10.6.5 is formed with M << T and
T =0+yi)T, where y 1, ... , y, are points in 0 - 11J, and with weight functions gi(x) = Gi(x/M) satisfying (20.1.20) for i = 0,..., 4. Then, for
U x MQ/NR,
U >> T/J,
(20.2.1)
we have fu
UIsIQ)(t)I2 dt <
(20.2.2)
for a choice of N and R in the ranges (20.1.10), provided that
M>> (IT)2"3.
(20.2.3)
For a single sum, the logarithm power can be reduced to log4 M.
Corollary For a single smoothed sum we have f U IS(t)I2 dt << B2MkT2/3 log6 M -U for
UxT2/
M>> T2/3.
Proof We must estimate Bo(01,..., L4) in Lemma 10.6.5, the number of solutions within a fixed transversal set involving a fixed minor arc. As in Lemma 14.4.4, type 3 matrices contribute
<< t1Imin(P2,Q2)+A102IPQ<< i1IPQ, and, as in Lemma 20.1.1, we can take A1(V) << 11V2. When the parameter takes the same value on both Farey arcs, then there is
no contribution from upper triangular matrices. If we fix the parameter on the variable Farey arc, then a bounded number of upper triangular matrices occur, satisfying
pi
b«V2
F1 LQ 1 M 1
as in Lemma 20.1.3. Hence
Bo(i1(V),..., 04(V)) << IPQ/V2 +I, and the second term can be omitted if V>>P(LQ2/M)114.
Mean value theorems
431
Hence
VBo(O1(V),..., 04(V)) << IPQ +IP(LQ2/M)1'4 << IPQ. The expression in Lemma 10.6.5 is
B2Lklog2L L+
(L
1/4
IPQ+JIogI
We sum over blocks for L in Lemma 10.6.2 to get f UIS,Q>(t)12 dt
<<
M B2Mk(R2
IM3/2 Q 3/2
(R)
+ NR
Q
+ RJlogIlogoM.
The term in J may be omitted if I = 1. The Lemma follows at once, and the corollary follows since IS(t)12 << log R L IS,Q>(t)12 Q
in Section 10.6, the sum being over all transversal sets for each power of
o
2, Q.
Our next lemma is valid for exponential sums with arbitrary coefficients.
Let F(x) be a real function, twice continuously differentiable for 1 5 x5 2, with F'(x) and F" (x) Lemma 20.2.2 (Large values of exponential sums) bounded away from zero. Let 2M- 1
!
g(m)eI tF( M M )),
S(t) _ m=M
and define G by 2M-1
G2M= F Ig(m)12. m=M
Suppose that t(1),..., t(N) is a sequence of real numbers that satisfy IS(t)I Z GW,[M-
for all n, and t(n + 1) - t(n) >_ 1,
t(N) S t(1) + T.
Then N <<
MlogM W2
+min
(T log M MT log M W2
, yy6
The implied constants depend on the bounds for the derivatives of F(x).
Sums with modular form coefficients
432
Proof We use Lemma 5.6.1 with M1 = M, M2 = 2M - 1, N1(m) = 1, N2(m) = N, f(m, n) = t(n)F(m/M), and h(m, n) = rt(n), a complex number of unit modulus independent of m, chosen so that -q(n)S(t(n)) is real and positive. The inner sum in Lemma 5.6.1 is 2M-1
(
E e(f(m,nl)-f(m,n2))= F, el(t(n1)-t(n2))F( M))
m=M
m
This sum is M if n1 = n2. For n1' n2, but n1 - n2 <<M, the trapezium rule (Lemma 5.4.1) gives
m
el(t(nl)-t(n2))F'(M)) «1+ It(n1)Mt(n2)I `
« In
1
Mn2l
For I n1 - n2l >> M, Corollary 1 to Lemma 5.4.3 ives
7' e ( (t(n) m
1
-t(n 2 ))F( mM ) ) «M
+
M2
M
It(n1)-t(n2)I
<< +V 4 V . The double sum on the left of Lemma 5.6.1 is real and at least GNWVi, so Lemma 5.6.1 gives
(GIWX-)2<< G2MIEM+E ` n,
M
nt n2on, In1 - n2I
+EEVT ni n2
<< G2M(MN log M + N2FT).
The term in N2 can be omitted if T <M. Hence
NW25C(MlogM+NlT), where C is some constant constructed from the bounds for the derivatives of
F(x). If T< M, or if TS T0=W4/4C2, then
N S (2CM log M)/W2. If not, then we put T1 = max(M, To), and we divide the sequence t(n) into at o most 2T/T1 subsequences, in each of which t(n) 5 t(1) + T1. Lemma 20.2.2 was proved by Huxley (1972) for Dirichlet series, using the approximate functional equation for a(s) (Lemma 5.5.4). This simple proof, replacing the approximate functional equation by Poisson summation, was suggested by S. W. Graham; the more elaborate result in Huxley (1975), in which the sum is over Dirichlet characters as well as points t(n), still seems to
Mean value theorems
433
require special properties of Dirichlet series. The lemma turns on the Halasz coefficients rt(n). The full Halasz construction would be
h(m, n) = rt(n) w(m)
,
where -q(n) has modulus unity, and w(m) is a non-negative weight function which is one for M 5 m < 2M. If F(x) can be defined suitably for x < 1 and for x >_ 2, then we can use the weighted First Derivative Test and remove the logarithm factor. Theorem 20.2.3 Let F(x) be a real function four times continuously differentiable on the interval 1 5 x 5 2, whose derivatives satisfy the following conditions: IF(')(x)I 5 C1
for r = 1, ... , 4, IF(')(x)I >_ 1/C1
for r = 1, 2, where C1 is a positive constant. Let G(x) be four times continuously
differentiable for all x, and zero outside the range 1 5x 5 2. Let b(m) be the coefficients of a cusp form of even weight k for the modular group, and let 2M-1
/
M=M
`
S(t) = m b(m)GI M)el tF( )), `
/
where M is a positive integer. Let C2 be a positive constant, and let T be a real number with M5 C2T. Then f2TIS(t)I6
dt <
T
where B is the Rankin constant of Lemma 10.2.1. The implied constant is constructed from C1 and C2, and from the bounds for the derivatives of the weight function G(x).
Sketch proof We divide the range of integration into unit intervals, and for each power of 2, V, let I(V) be the number of unit intervals on which BMk12V 5 max IS(t)I 5 2BMkl2V.
(20.2.4)
Then
f2TIS(t)I6 T
dt 5 E I(V)(2V )6B6M3k.
(20.2.5)
V
We apply Lemma 20.2.2 with g(m) = b(m)G(m/M), so that G2 << B2Mk,
and we form the sequence of points t(n) by picking the values of t at which IS(t)I is maximum from alternate intervals satisfying (20.2.4). This ensures the
Sums with modular form coefficients
434
condition t(n + 1) >- t(n) + 1 of Lemma 20.2.2, with W>> V. For each power of 2, V, we have
I(V) <<
M log M MT log M V2
+
V6
(20 . 2 . 6)
If the second term in (20.2.6) dominates, then I(V)V' << MT log M << T 2 log M.
(20.2.7)
If the first term in (20.2.6) dominates, and if I(V) >> M312/T,
(20.2.8)
then
I(V)V6 <<
T2
M3
(I(V)V2)3
<<
T2 log 3 M.
(20 . 2 . 9)
The difficult case is when
I = I(V) << M3/2/T. Using the mean-to-max argument of Lemma 5.6.3, we have max
IS(t)12 <<
uSf5u+1
max fi+31S(o., t)12 dt,
-15051 u-2
where the sum S(o,, t) has an extra factor exp(27ro-F(m/M)) in each term. We can choose integers R and N by (20.1.10), and apply Lemma 20.2.1 with U Z 1. We write S(o-, t) as a sum of O(log R) sums S'Qk(o,, t) corresponding to fractions aj/q, with Q S qj < 2Q, with the Diophantine approximation taken for the value of t at the centre of the unit interval. The conditions on G(x) ensure that we can write
G(x/M)exp(27rcrF(x/M)) = Wj(x)g(x), with g(x) of bounded variation. There are two complications, one obvious, one less so. The intervals in (20.2.2) are essentially non-overlapping by (20.2.1), so that, if possible, we must cover several unit intervals counted in I(V) with one interval to - U :..g t < to + U. However, the definition of a,,(x) depends on the unit interval, in that N changes by O(QR) as t changes by U = O(MQ/NR). We can overcome this by partial summation, at the cost of a factor log M. So Lemma 20.2.1 leads to the bound IV2 << (IT)2"3 log9 M, or
I <<
T2 V6 log27 M,
(20.2.10)
with one logarithm power extra for each factor S(t) from the dissection into sums S(Q), and another from the partial summation. We use (20.2.7), (20.2.9), and (20.2.10) for O(log M) values of V, and I(V) << T when V is small.
The modular form L -function
435
The third form of Jutila's method leads to a mean square result (Jutila 1989, 1990b).
Lemma 20.2.4 (Short intervals mean square) Suppose that the conditions of Theorem 20.2.3 hold. Let U, t1,. .. , t1 be real numbers U S t, < T - U, t;+ 1 - t, > 2U. For any e > 0, however small, if T2/3+E<< M<< T,
T1+E/M1/2<< U<< T2/3,
then
tf
t,+U
-U
1S(t)12 dt << BZMkTE(IU+ (IT)2/3),
where B2 is the Moreno-Shahidi constant of Lemma 10.3.3, and the implied constant depends also on e.
Both Theorem 20.2.3 and Lemma 20.2.4 imply slightly weaker forms of Theorem 20.1.2 by the mean-to-max argument of Lemma 5.6.3.
20.3 THE MODULAR FORM L-FUNCTION Exponential sums with modular form coefficients appear in the modular form L-function W b(m)
cp(s) _ E
s+tk- 1W/2 ' M=1 m
(20.3.1)
which has been much studied lately. It is a Fourier transform of the modular
form F(z) = E b(m)e(mz) along the line from 0 to ico. We hope that, whenever F(z) corresponds to some algebraic or geometric object, then certain special values of cp(s) will correspond not only to integrals involving
the modular form, but to integrals on the geometric object in its intrinsic metric. Different structures can give the same L-function. The derivatives of
modular forms can be used to construct new modular forms. We have normalized (20.3.1) so that F(z) and F'(z) have the same L-function, up to a factor 2ir i. Changing the sampling line 0 to ico corresponds to twisting the
modular form (Atkin and Lehner 1970). When F(z) is a newform, the simplest of a family of twisted forms, then cp(s) factorizes as an Euler product of Dirichlet series, with each factor a sum over those m which are powers of
some fixed prime. From its construction cp(s) has a functional equation cp*(s) = cp*(1 - s), where
cp*(s) = (21.)-SF(s + (k - 1)/2)cp(s).
(20.3.2)
The corresponding Dirichlet series constructed from non-holomorphic modular forms include 2(s), the square of the Riemann zeta function. It makes sense to ask whether newform L-functions cp(s) are non-zero in the region
436
Sums with modular form coefficients
Re s > z and of size O(tE) for s = o- + it, with cr- 1 fixed, as t --> . Even a bound of the form tA(°) would enable us to deduce the Wilton formula from
(20.3.2). However, to bound cp(s) we need an approximate functional equation, obtained by the Wilton formula and analytic continuation (see Jutila 1985).
Lemma 20.3.1 (Approximate functional equation) x, y chosen with
For 0< o-< 1, t >- 2, and
xy=t2/47r2, we have
cp(s) = E
b(m)
(k- 1)12+s M 5X m
b(l)
+ G(s) l5y
l(k+ 1)/2-s
+O(Bt1"2-° log t), where
G(s) = (27r)2's-1r((k+ 1)/2-s)/F((k-1)/2+s), 0
and B is the constant of Lemma 10.2.1.
We deduce bounds for the modular form L-function. Theorem 20.3.2 For T large
cp(2 + iT) = O(BT1'3(log T)3"2), where B is the constant of Lemma 10.2.1.
Proof We split the sums over l and m in the approximate functional equation into blocks of the form M5 m::5 min(2M - 1, x). For M >- C4T 2/3 we use Lemma 5.1.1 to take out the factor 1/ m__, and then use Theorem
20.1.2. For small m we use Lemma 10.2.1. The factor G(4 + it) is the quotient of two complex conjugate numbers, and so it has unit modulus.
0
Theorem 20.3.2 was proved by Good (1982), with the same exponent 3, but
a better logarithm power, using a Fourier expansion into non-holomorphic modular forms. Jutila's triumph (1987a) was the sixth-power moment. Theorem 20.3.3 For T large
f2TI
cp(Z +it)I6dt=O(B6T2log34T),
where B is the Rankin constant of Lemma 10.2.1.
(20.3.3)
0
The modular form L -function
437
Remark The same bound holds for the integral over - T to T; we fit together ranges on which IZ + it) has fixed order of magnitude. Theorem 20.3.3 is deduced from Theorem 20.2.3. The sums in the approximate functional equation are smoothed by averaging over x in Lemma 20.3.1, and then subdivided into O(log T) smoothed sums over ranges of the type M to 2M - 1. We use Holder's inequality to sum over ranges for m. Similarly we deduce a mean square result from Lemma 20.2.4 (Jutila 1990a). Theorem 20.3.4 For any e > 0 and T sufficiently large we have fT+T2/31
T-T2/3
(p(z +lt)12dt=O(BZT2/3+e),
where B2 is the Moreno-Shahidi constant of Lemma 10.3.3, and the implied constant depends on e.
21
Applications to the Riemann zeta function 21.1 INTRODUCTION The Riemann zeta function, C(s) = E 1/ns, is a generating function for the positive integers. It has a single pole as a function of s, at s = 1 where the series first diverges, because it is the generating function of an infinite set of positive coefficients. It has a functional equation (stated in Section 5.5 of Chapter 5) that relates its values at s and 1 - s; this, in a sense, is equivalent to the Poisson summation formula (Lemma 5.4.2). What makes it extraordinary is that log C(s) is a generating function for the prime numbers. Euler's product over primes 1
1
expands out as a sum of 1/n3, where each integer n occurs as many times as it can be written as a product of primes, that is, exactly once. Euler's product is simply C(s). Taking logarithms and summing the geometric progressions, we have 1
log
p
1_ p-s
°°
-, 1
= p r-1 rprs
we use the convention that p is a variable running through the prime numbers. Differentiation gives the meromorphic function log p
s
(s)
-
P r=1
P
s
(21.1.1)
The right-hand side of (21.1.1) is usually written as - E A(n)/ns, where A(n)
is log p when n is a prime power pr (with r > 1), zero otherwise (A was chosen by von Mangoldt as the initial of `logarithm'; a clash of notation with the kernel function A(x) of Chapter 5 seldom occurs). The coefficients A(n) can be recovered using the Mellin transform integral ('Perron's formula'), of which Lemma 5.2.4 is a truncated version. Riemann conjectured, and von Mangoldt proved, that the integral is equal to the sum of the residues; both the integral and the sum are only conditionally convergent. The pole of
C(s) at s = 1 gives the main term, telling us that A(n) averages to one, so prime numbers of size N are about log N apart on average, and the kth
Introduction
439
prime number is about k log k: the prime number theorem of Hadamard and de la Vallee Poussin. Points where C(s) is zero also give poles of C'(s)/C(s), and there is an extra pole from the factor 1/s in Lemma 5.2.4. The task of Hadamard and de la Vallee Poussin was to show that these zeros had real parts less than one; otherwise the residues would give oscillating terms as large as the main term.
The reciprocal 1/C(s) is the generating function for the combinatorial Mobius function µ(n). Just as A(n) averages to one, so µ(n) averages to zero, with an accuracy determined by the residues at the point s = 0 and at the zeros of C(s). This is why Selberg's expression on the left of (5.6.10) is used both in the sieve and in the study of the zeta function. Riemann predicted with some confidence that all the zeros of C(s) (except those on the negative real axis), beyond the six or so that he had calculated, should have real part one-half, and so lie on the line of symmetry in the functional equation. This Riemann Hypothesis, if true, would imply that the Farey sequence is distributed as uniformly as possible along the real line, and that the prime numbers are distributed rather uniformly among the integers. Both these sequences are irregular over short intervals. In particular, the gap between consecutive prime numbers p and would be bounded by the square root of n times a power of log n if the Riemann Hypothesis were true. Probably the gaps are always much smaller; but to prove this would require further results on the small-scale distribution of the zeros of C(s) along the `critical line' Re s = i.
The Hardy-Littlewood generation of number theorists doubted the Riemann Hypothesis, not merely because they had failed to prove it, but because there was no apparent reason why it should be true. Even the computer evidence is covered by Littlewood's observation that the zeta function is unusually well-behaved near the real axis where log log IsI is small.
The widely held view in the age of algebra is that the numbers s(1 - s) formed from the zeros of t(s) will be eigenvalues, and thus real. Suggestions as to the self-adjoint operator to produce these eigenvalues range from an infinite matrix, through a differential operator, to a homomorphism of an infinite product of topological groups. All agree that it should be a natural construction in which the rational numbers are fixed, rather as the ground field is fixed by the Frobenius automorphism in a finite field. A suggestive approach is the Selberg Trace Formula. In the special case of the modular group (which fixes the rational numbers as a set), the prime numbers appear
as a small correction term, corresponding to the term in the Riemannvon Mangoldt formula from the residues of zeros on the real axis. The coefficient sum in the formula is over units in real quadratic number fields. The main sum on the other side is over the other zeros of the Selberg zeta
function-for which the Riemann Hypothesis is true. The related trace formulae for Fourier coefficients of modular forms have been worked out by
Kuznetsov (1980) for the expansions of Chapter 10, which involve polar coordinates in hyperbolic space, and Good (1983) for Cartesian coordinates
Applications to the Riemann zeta function
440
in hyperbolic space. They provide useful information for analytic number theory. The analogies with scattering operators in quantum mechanics (Lax and Phillips 1967) should only disconcert those who like their mathematics totally disconnected. But for prime number theorists, the Selberg Trace Formula resembles a glass of beer: the most attractive part turns out to be froth, disappearing when you digest it, and the body of the drink, however nourishing, has a bitter taste. For all the loving care devoted to C(s) (see Titchmarsh 1951, and Ivic 1985, first conceived as a supplement to Titchmarsh) it is only one function. There are reasons to suspect that C(s) is extremal in the class of ordinary Dirichlet
series with bounded coefficients, like Koebe's f(z) =z/(1- z)2 among the univalent functions. The Minakshisundaram-Pleijel zeta function, E 1/An, where A is a sequence of eigenvalues, is an interesting but remote generalization. There is a select class of functions (usually called zeta function if they have a single pole, L-function if they are entire) with three properties: D: E:
a definition as a Dirichlet series; an Euler product expansion;
F:
a functional equation of the form f *(s) = g*(1- s), where g(s) is another function of the same type, and the stars indicate that there is a normalizing factor involving phase shift factors, sth powers of constants, and gamma functions or the like.
The Riemann Hypothesis is not known to be false (or true!) for any function of this class. Selberg's zeta function has E and F, but not D, since it is only defined by the analogue of C'(s)/C(s). Any direct definition of it as a series would be over the centre of the group ring of the modular group. The French school has its own motives for explaining the D, E, F properties. The same zeta function can serve several different motives. The Wilton summa-
tion formula for the Fourier coefficients of modular forms in Chapter 10 corresponds to D and F; that E holds was noticed by Ramanujan and first proved by Mordell.
While these mighty matters are mooted, we have to make the most of imperfect information, moving the Mellin contour past some of the poles, and estimating the integrand on the new contour. Useful information is:
(1) bounds for C(s); (2) mean square results (or higher powers) on lines Re s = (3) zero-density results, that there are very few zeros in some region bounded away from the critical line Re s = Z. Zero-density results use bounds of types 1 and 2, and ingenious arguments
based on partial sums of the Dirichlet series for 1/C(s) or C'(s)/C(s). Present function-theoretic methods, alas, seem incapable of disproving the following null hypotheses: ItI'18-3
(1) I C(s)I gets as large as for every S> 0; (2) 1 C(s)I is zero in Re s > 1 - 8 for every 6 > 0;
The order of magnitude in the critical strip
441
(3) functions with the D, E, F properties can vanish on the positive real axis.
Littlewood (1912) observed that convexity arguments applied to log i(s) show that off the real axis, for any e > 0,
(s) << TE (21.1.2) T to the right of the critical line that contains no
inside any region with Itl zeros. Hence the Riemann Hypothesis implies the Lindelof Hypothesis, that
(21.1.2) holds for v> 1 (with a constant depending on o-). The Lindel6f Hypothesis in turn implies the density hypothesis, that the residues from zeros with Re s = 1 dominate the sum, and so Pn+1 -Pn << n
1/2+
21.2 THE ORDER OF MAGNITUDE IN THE CRITICAL STRIP In complex function theory, knowing the maximum size of a function in a region is very useful. The series for a(s) converges for v = Re s > 1. The
functional equation reflects the region v > 1 into the region v < 0. The remaining region is the critical strip 0 5 o,5 1. We use the approximate functional equation (Lemma 5.5.4) to translate bounds for exponential sums into bounds for I (o + it)I for a fixed, t tending to infinity. First we quote a standard estimate (Jeffreys and Jeffreys 1962, art. 15.05). Lemma 21.2.1 (Stirling's formula)
Let S > 0 be fixed. Then
logF(s)=(s- Z)logs-s+ilog27r+O
Is11
as I s I -* o, uniformly in - it + S < arg s 5 it - S. In particular, as t -3, + o with o fixed, we have
logI'(s)=(v-2)logt-
i2n
+Zlog2ir i
+it(logt-1)+-(v-Z)+O
(t)
There are corresponding approximations to the derivatives of log F(s).
O
Theorem 21.2.2 Let a- be fixed in z < Q < 1. Suppose that we have MZ
r
/ m
eI TFI M)) << MATµ(logT)",
(21.2.1)
M
with F(x) = (log x)/21r, for some non-negative A, µ and v with
1-0,
(21.2.2)
Applications to the Riemann zeta function
442
uniformly in M, M2, and T satisfying M2 < 2M and µ log M
1-a.
logT S
2
Then, as It I -, cc, we have
(o. + it) = o(ItI, (log It!) ' Corollary
)
We have
(Z + it) = 0( ItI89/57° log2 ItI)
Proof By Stirling's formula (Lemma 21.2.1) we have r 21r l s
I-I
['(1 - s) x Itl 1/2
(21.2.3)
for s = o-+ it, Iti large. In the approximate functional equation (Lemma 5.5.4) we have +
I (s)I <<
log T
T1/2-°
where T = [ItI] + 2, and
N' =
[iiL}.
2irN
We can choose the integer N so that N and N' are less than
ItI
.
The trivial
estimates K 1
K
1ns
«K1-°
1
n. K'
1
nl-s << K
<
"log K',
are allowable for
K<< TK' << T('L+o-1/2)/° If µ > (1 - 0/2, then we can use the trivial estimates over the full range. Otherwise we have Tµ/(1-°)
<<
T(µ+°- 1/2)/° << T1/2,
and T here differs only trivially from the notation T = ItI in (21.2.1). The rest of the range is split into O(log T) blocks of the form M 5 n <- M2. We write n 1 1
n = n°M
M).
The mean square
443
We take out the factor n° x M° using Lemma 5.1.1, and estimate the resulting inner sum as O(M"Itl'`(log Itl )°). Since A5 o-, this block contributes O(Itl "(log Itl)°). Similarly we have Itl1/2_Q
My
Itl1/2-o
1
M1_o M" Itl"(logltl)v
n1_S
ltl)"+
< Itl1/2- a(
11tI A (log ltl) ° «Itl" (log ltl )
where we have used both inequalities of (21.2.2). The result follows on putting together the bounds in different ranges.
From Table 17.3 we deduce the further bounds 23208
43860
+ it)
«Itl 6299/43860 log
2
Itl,
(21.2.4)
and, for any integer R Z 0 and Q = 2R, 169(712R + 1577) 712(1424Q-338)
+ it
«Itl 169/(1424Q-338) log 22
(21.2.5)
These are values of o for which two outstanding vertices of the upper bound polygon in Table 17.1 correspond to sizes of M that give contributions of the
same order of magnitude. For intermediate values of o,, one range of M dominates, and we have t(o + it) << Itl µ log2 Itl,
where the exponent p. is obtained by linear interpolation between the exponents at values of a- given in the corollary, (21.2.4), and (21.2.5).
21.3 THE MEAN SQUARE Theorem 19.1.3 gives a bound for the short interval mean square of the zeta function on the critical line Re s = 2 Theorem 21.3.1
Let T and A be large real numbers with T72/227(log T)315/227
S 0 S T.
Then
+it)12dt<< T
log2T.
Applications to the Riemann zeta function
444
Proof The upper bound is linear in A, so we need only prove it for A <- F. In (21.2.3) we can fix N' as
T -31 4'
N'
2orN
since the effect is to omit a bounded number of terms in the second sum, whose total contribution O(T-O/2) can be absorbed by the error are divided into blocks term in (21.2.3). The sums E 1/ns and E M:5; n 5 2M - 1 (the upper limit for n may be replaced by N or N'). In 1/nl_S
Theorem 19.1.3 we take F(x) = (log x)/21r, G(x) =1v (if the upper limit is N or N', not 2M - 1, then we take G(x) = 0 for x > N/M or N'/M). Then S(t) is larger by a factor VM_ than the sum we want, so when (19.1.9) of Theorem 19.1.3 holds, then the squared integral for each block is O(ff). Blocks with M small are estimated as in Theorem 18.2.3. There are O(log T) blocks, so Cauchy's inequality gives O(A log2 T) for the whole integral.
Much deeper arguments give the corresponding mean value theorem T
Jo
+(2y-1)T+E(T),
with an explicit estimate for the error term E(T). We use the machinery of Heath-Brown (1979), following Heath-Brown and Huxley (1990).
Lemma 21.3.2 (Truncated mean square expansion) Suppose that 15 A S T1/3. Let t - 21rm2
E
g(t) = 2
ms (t/2a)
(m/n)"
G(t) = 2 m
n
m 1
(mn) ilogm/n
exp
02 log2 m/n 4
the conditions of summation in G(t) being
0 < I log m/ni 5 (log T)/0, mn <- t/tar.
(21.3.1) (21.3.2)
Then
g(t) = t log t/27r + (2y - 1)t + 0(1), and
+it)I2dt=g(4T)-g(2T)+G(4T)-G(2T)+0(t log2T). 2T
The mean square
445
Theorem 21.3.3 For T large
T
+it)I2 dt = T log - + (2y- 1)T + E(T),
Io with
E(T) << T 72/227 (log T
)629/227
Proof We estimate G(T) in Lemma 21.3.2. We write m + n = k, m - n = h, 2m = k + h, 2n = k - h, noting that k
1/2
h2
which may be expanded as a power series in h/k. We take A = T72/227(log
T)175"227,
so that h is much smaller than k. The contribution of the first term in the power series will dominate the sum. We also have log
m
n = log
1+h/k 1- h/k
r (khlr) 1
=2
(21.3.3)
r odd
The weight A2log2 m/n
1
log m/n eXp(
4
and the condition (21.3.1) may be expressed by a condition 0 < I log m/nl 5 S,
(21.3.4)
and a weight w(S), and then by integrating the weighted sum over S. The condition (21.3.2) is
k2 - h2 5 2T/ir.
(21.3.5)
By (21.3.1) and (21.3.3), h2/k2 5 (log2 T)/402, so the upper bound for k in (21.3.5) is one of at most two consecutive integers. We may divide the double sum over h and k into two parts, and replace (21.3.5) on each by a constant upper bound for k. For fixed k, (21.3.4) is an upper bound for h of the form
h
dH dk
x S.
Having chosen the endpoints and centres of Farey arcs, we may find that the integer part of H(k) changes within a minor arc. If so, then the sum over a
Applications to the Riemann zeta function
446
minor arc must be divided into two or more parts which are estimated separately. The number of subdivisions is 0(1 + SN), which is bounded when
HN << M. We divide the sum into blocks of the form
H ::g h s min(H2, H(k)),
H2 < 2H,
M
M2<2M.
By (21.3.1) and (21.3.2) we can suppose that
M« FT,
H<<
M log T
0
A final preparation concerns the integer summands h and k. To obtain only terms in which k - h = 2n is an even integer, we divide the terms into two groups. In terms with h and k both even we take h/2 and k/2 as new integer variables, and (21.3.6) F(x) _ (log x)/21r as usual. To obtain terms with h and k both odd, we take two sums in which h and k are unrestricted. In the first sum F(x) is still given by (21.3.6), whilst in the second sum we take
F(x) =
log x
M2x2
8T '
21r
so that TF I
k Mh)_TF( k M h)=
T 2
log k + h
-
hk
When we subtract the second sum from the first, then we obtain e(hk/2) _ (-1)"k, and only terms in which h and k are both odd remain. We can now estimate the blocks of the sum as in Theorem 19.1.1. The weight function is
0(1 MeX M H p(
4M2 YH2
so the block bound corresponding to (19.1.4) of Theorem 19.1.1 is H 87/140 H 3/70 Q2H2
<<exp(-
4M2
)(()
T23/70+(M)
Tis/3sl(logT)9/4.
For fixed M, this expression sums over blocks for H to «(0-3/70T23/70+0-87/140T18/35)(logT)9/4 <<
J
log T,
and then over blocks for M to O(0 log2 T). The conditions on M and H are those in Theorem 19.1.1. For M large, H small we can also use the bound
Gaps between zeros
447
corresponding to (19.1.6) of Theorem 19.1.1, and for M small, the onevariable 'exponent-pair bound (2/7, 4/7)' as in Theorem 18.2.3. The parameter choices in Theorem 19.1.1 satisfy the condition HN << M easily. Thus we find that G(T) = O(0 log2 T). The theorem follows on summing the integral of I I2 over the ranges T/2 to T, T/4 to T/2,..., and a trivial range 0 to T/2', where r = [(log T)/(log2)].
21.4 GAPS BETWEEN ZEROS Even if the Riemann Hypothesis is false, then there area zeros of i(s) on the critical line Re s = .1. The symmetrized function C*(s) = r(s/2)C(s)/?rs/2 (21.4.1) is real on the real axis and on the critical line, which is also a symmetry axis. Selberg (1942) showed that the number of sign changes of *(s) along the critical line is comparable to the change of argument along the parallel line
Re s = 2, so that a positive proportion of the zeros lie on the critical line Re s = .1. As a first step towards the local distribution of the zeros and the gap between consecutive primes, we try to localize this result. The whole argument can be regarded as a divide-and-conquer proof that there are zeros of i(s) on the critical line.
We quote the first lemma (due to K. Ramachandra), which is purely function theory, from Balasubramanian (1978). The lemma does not use the properties E and F of the zeta function, and it applies to any Dirichlet series with leading term 1 and an analytic continuation for large t to the left of the line a-= 2, where it is bounded by a power of It1. Lemma 21.4.1 (Mean minimum modulus) Given e > 0, however small, there is a To depending on e, such that for T > To and H Z (log T)E, we have
fT+HI
(2 + it) I dt >> H.
T
Lemma 21.4.2 (Approximation to
*(s))
Let
N=N(t)=[ (ItI/21T)]. Then for s = i + it with t large, we have
I'(s/2)
*(2 + it) = 2Re 1s/2
N1
ns + O
log t \ 1 (
ti/a
with
arg
r(s/2)
t( t _ 1 11 s/2ns = 2 log 2arn2 11 + 0 t /I
Applications to the Riemann zeta function
448
Proof This lemma follows at once from the definition of *(s), the approximate functional equation and Stirling's formula (Lemmas 5.5.4 and 21.2.1).
Lemma 21.4.3 (Repeated First Derivative Test) Let f(x) be real and r + 2 times continuously differentiable on the closed interval [ - /3, /3 ], with
f'(x)>-K>0 on[-/3,/3], and for 25k5r+2 If (k)(x)I < /31-kIf'(x)I. Then
1
e(f(x))cosr
R R
-x
dx <<
20
0)r+1
OK
The implied constant depends on r.
Proof We iterate the relation [(x)e(f(x))cosk1Tx/2/31 I1
irx
a
f Rg(x)e(f(x))cosk
2/3
dx= 7rx
27rif'(x) d
g(x)
-f e(f(x))cosk 2/3 dx f'(x)
kir 2/3
sin
rx
20
arx
cosk-1 20
g(x)
f'(x)
dx
= f Rh(x)e(f(x))cosk-1 2a dx, with
h(x) =
k7r
20
-7rx
sin
g(X)
20 f'(x) -
cos
g(x)f"(x) f (x) 2
irx g'(x)
20 f'(x)
The iteration ends when the integral on the right has no cosine factor, and we estimate each term at its maximum. Each term consists of numerical factors, sines and cosines, and a product
`(f(r+ 2) ..
°,+ z
f'
)
with
as+a2+ +ar+2=r+l. Now suppose that *(2 + it) does not change sign between T and T + H, where H >- log T. Putting a positive weight function into Lemma 21.4.1, as in Lemma 5.1.1, we see that fT+H
T
+it)Icosr (Hr t-T-
- dt>> H. 2
))
Gaps between zeros
449
Let
g(t) =
I'(1/4 + it/2)
1
2ir
arg
.1/4+ii/2
Then
I (2 + it)I =
it),
and the ambiguous sign does not change between T and T + H. Hence
2 ))dt
>> H.
By Lemma 21.4.1, we may replace g(t) by
f1(t) = with an error
-
(log 27r -
HN
1
H
1
T3/a
1
and replace e(g(t))t2 + it) by
2Ree(f1(t))
N
log T
1
(log T
2
N
+0(71/4)
7;T/ 2+ir
71/4
n
=
where
fn(t) =
1)
41r (log
with t
1
N
1
f,(t) = 41r log 2arn2 z
27r log n
We pick M < N and use Lemma 21.4.3 for n :!g M - 1 with f3 = H/2. The condition on the derivatives is satisfied if T N
H1ogM>1.
Then T+H T
M-1
Re
r
1
M-1
<< E 1
N/2
(( t- T-
H
1
n (H log 1
IN- ( <<1+ Hr
H dt 2) )
N/n)'+1
M-1
1
H-,+
N'
(N-M)r
1
1
Nr+1
rNHr .(N-n)r+1 )
« BrH ,
450
Applications to the Riemann zeta function
provided that
-
N-M>>
1/r
,
H( H
(21.4.2)
where B is a large constant. With this choice of M we must have M
N
1
n1/2+ir
>1
(21.4.3)
for some t between T and T + H. This is a short exponential sum formed with the functions 1
- log x,
F(x)
1
G(x) =
27r
,
TX
and parameters M (or N) and t, so that t x f(x)=--logM
We check that
f"(N)
M 2= ++O1
t
4irM2(NJ
2
1.
FT
The integer N is the centre of a major arc. We use Lemma 19.2.1 to treat the whole sum as a long major arc. Let
N' =N-M=MB. We have
µ=f(3)(N)/6 x 1/M, so that in Lemma 19.2.1 we have / 20-1 30-1 +OI
1 M).
After a partial summation step using Lemma 5.1.1, Lemma 19.2.1 gives M
1
«M1/2-v-v+9(sp+2v 112)(logT)B.
n1/2+U
Comparison with (21.4.3) gives
9>
p+o
3p+2a-1/2
-0
log log M
logM
(21.4.4)
After some experimentation with parameter choices, we suppose that 423/1295 5 a:5 227/601, corresponding to 0.6601 ... 5 05 0.7179.... In this
The twelfth power moment
451
range p = 89/1282, o- = 908/1282, so the rational number in (21.4.4) is 997/1442 = 0.6914... . We can choose H = TO in (21.4.2) with 9 /3
1
+0
+
2
1
(21.4.5)
g
and (21.4.4) gives 445
< 2884
+O
1
log log T
r
logT
'
(21.4.6)
This is an upper bound for H. We conclude by discussing a logical trap. Could /3 be larger than (21.4.5), so that a and 0 are smaller, so small that a < 423/1295, and the values of p
and o- change? If /3 = 3/20, then 9 is approximately 7/10, a is approximately 4/11, still in the range in which /3 = (89 + 908 a)/1282 in Table 17.1. Lemma 19.2.1 tells us that (21.4.3) cannot hold for large T; there must be a zero on the critical line with height t in T5 t< T + T 3120. So there cannot be a gap so large that /3 corresponds to a value of a < 423/1295. We have now proved a version of Karatsuba's theorem (1981). Theorem 21.4.4 Let /3 > 445/2884 (= 0.1543...) be a real number. If T is sufficiently large (depending on /3) then there is at least one zero p = 1 + i y of the Riemann zeta function with
T
O
21.5 THE TWELFTH-POWER MOMENT Jutila's method for modular form L-functions applies equally to 2(s), although the corresponding modular form is not a cusp form, and has a logarithmic singularity at ico. The summation formula in the untwisted case is the original Voronoi formula (1904). There is an extra non-oscillating term
corresponding to the singularity, and the Rankin series has a fourth-order pole at s = 1. We must replace B log M by log4 M in the bounds of Chapter 10; we can see this directly from Lemmas 11.1.2 and 11.1.3. The proof of Theorem 20.3.3 can be modified to give a sixth-power moment for 2(s). Theorem 21.5.1
For T large
LT Theorem 21.5.1 was first proved by Heath-Brown (1978) (with a better logarithm power logs' T), using Atkinson's formula (1949), which is essentially Wilton's formula for a non-holomorphic modular form whose Dirichlet series is (Z + it)C(Z - it). Another proof by Iwaniec (1980) uses Fourier expansions into non-holomorphic modular forms.
22 An application to number theory: prime integer points 22.1 PREPARATION A favourite source of problems in number theory is to take two integer sequences, and to ask how many members they have in common. Popular sequences are the prime numbers, the Fibonacci numbers, the powers of 2 and the squares (or, more generally, the values of a suitable polynomial g(x)). Problems of this type involving prime numbers are tantalizingly difficult: even primes of the form n2 + 1 are well out of reach. Gel'fond suggested
the sequence of integer parts of the values g(n) of a real function g(y) growing faster than linearly. Thus the point ([g(n)], n) is an integer point lying close to the curve x = g(y). Piatecki-Shapiro (1953) was able to show that there are infinitely many prime numbers of the form [n"] for each a in the range 1 < a < 12/11. Deshouillers (1976) showed that the set of a > 1 for which [n"] takes only finitely many prime values has measure zero. It would
be very surprising if this exceptional set were not simply the integers. Piatecki-Shapiro's range for a has been extended many times, most recently by Liu and Rivat (1992) to 1 < a < 15/13. We prove the more general result suggested by Gel'fond. For convenience we work with the function f(x) inverse to x = g(y), where
f(x) =NF(x/M),
(22.1.1)
and we count the prime values of m in an interval M< m < 2M for which there is an integer n with
f(m) 5 n < f(m + 1).
(22.1.2)
The conditions on F(x) will be of the type seen in our previous theorems. We require f(x) and g(y) to be monotone increasing, with
0
f(m + 1) -f(M) + p(-f(m + 1)) - p(-f(m))
(22.1.3)
is one if some integer n satisfies (22.1.2), zero if not. There is a technical
Preparation
453
simplification in counting prime numbers using von Mangoldt's weight A(m), defined by log p if m= p°, a> 1, p prime, A(m) = 0 if m is not a prime or a prime power. The basic property of these weights is E A(d) = log m. (22.1.4) dim
We use the prime number theorem in a weak form (see Davenport 1967 and Hardy and Wright 1960 for the two standard proofs). Lemma 22.1.1 (Prime number theorem) For P > 2 we have
E A(m) =P+O msP
P log P
Let I be a subinterval of 1 <x < 2, chosen so that no combination of derivatives of F(x) occurring in the proof vanishes on I. We work with the integers in the interval MI, the set of points of I multiplied by M; let the endpoints of MI be M1 and M2. Let E(2) denote a sum over integers m for which (22.1.2) has a solution. Then (2)
E A(m) _ MI
A(m)(f(m + 1) -f(m)) MI
+ L A(m)(p(-f(m + 1)) - p(-f(m))).
(22.1.5)
MI
By the prime number theorem and partial summation, the first term in (22.1.5) becomes
f(M2)-f(M1)+0 logM
(22.1.6)
.
On the left of (22.1.5) (2)
(2)
E A(m)
F, log p + OI
MI
r
A(m)(22.1.7) /
m52M,mnotprime
P,-Ml
in this chapter p is a variable of summation restricted to prime numbers. The second sum in (22.1.7) is
E E log P< p
rz2
log 2 M
log p
log P
p'S 2M
E
1 <<M1"2 logM<<
ps (2M) for
M << N2/logo N.
N log M
An application to number theory: prime integer points
454
The second term on the right of (22.1.5) contains p(t) functions as in Chapter 18. It is still essentially a sum over prime numbers because we have
the coefficients A(m). We use a sieve (in the original sense): a counting argument which puts more weight on the prime numbers, using Mobius inversion to remove many numbers which are not prime. Linnik's large sieve (1941) introduced generating function inequalities; Lemma 5.6.6 is a modern
version. Vinogradov showed that sieve ideas could be used to estimate exponential sums over prime numbers. Linnik and Vaughan gave different combinatoric arguments of the same general type. We follow Vaughan (1977). Let 0' (m)
= ( p(-f(m + 1))
- p(-f(m)) form in MI, otherwise,
0
and let u be an integer with u + 1 a power of 2, and (22.1.8)
u3 < M.
Let So be the finite sum
so= E µ(d) F, A(m) F, cr(dmr). d5u
r
m
We write So = S + S', where S consists of the terms with dr < u, S' the terms with dr > u. By Mobius inversion we have
S = E A(m)v(m), m
the second sum on the right of (22.1.5). Next we write S' = S, + Si, where Sl consists of the terms with m > u, S', the terms with m< u. We have S,
F, µ(d) F, A(m) E v(dmr)
d5u
r>u/d
m>u
F, A(m) E K(n)v(mn), m>u
n>u
where
K(n) _
F,
µ(d),
K(n)I
dln,d5u
In Si we can drop the condition dr > u, since cr(dmr) is zero when dmr < u2. We write S' = S2 + S31 where S2 consists of the terms with dm 5 u, S3 the terms with dm > u. We put s = dm,
A(s) _ E F, µ(d)A(m), d5u m5u
IA(s)I 5 logs.
dm=s
Then
S2 = E F, A(s)o (rs), sSu r
S3 = E
F, k(s)Q(rs).
u<s5U2 r
Preparation
455
Next by (22.1.4) we write So as
So = E E p.(d)log a o, (de). d5u
e
In Montgomery and Vaughan's classification the sums So and S2 are type
I, pure in the inner variable, which runs through a long interval (the monotone weight in So can be removed using Lemma 5.1.1). The sums Sl and S3 are type II, and can be written using (22.1.8) as
a(s) E /3(r)o,(rs).
F,
(22.1.9)
r>u
u<sSuZ
We can regard the coefficients a(rs) as forming a lacdnary square matrix of size 2M - 1. For large M we elaborate the construction, picking z so that z + 1 is a power of 2 and u2Y2 5z < VIM-,
(22.1.10)
uz2z2M.
(22.1.11)
and
Then
w=2M/z2 lies in the range 2:5 w 5 u. We write S1 = S41 + S42 + S'4, where S41 is the
terms with m < z, S42 is the terms with n < z, and S4 is the terms with m, n > z. Thus
F,
S41 =
A(m) F, K(n)a(mn), n>u
u<m
F,
S42 =
K(n) F, A(m)a(mn).
u
m>u
These are type II sums in which one summand runs through a short range. We expand S4 again as
S'4 = F, µ(d) F, A(m) F, a(dmr). dsu
r>z/d
m>z
Let S5 be the terms in S4 with d z w, S6 the terms with d < w. In S5 and S6 we cannot combine d and m into one variable s, because the condition on r is r > z/d, depending on d. However S5 and S6 both have the more general form
F, F, a(d,m) F, a(dmr). dm
r> z/d
We treat S. as a type II sum, S6 as a short type I sum. In S5 we have
r>z/u>u,
s=dm>wz=2M/z.
In S6 we have
r > z/w,
s > Z.
An application to number theory: prime integer points
456
22.2 TYPE I DOUBLE SUMS Long type I sums have the form
E a(s)
/3(r)a(rs),
s5x
(22.2.1)
r
where /3(r) is a monotone weight function. For So we have x = u, a(s) = µ(s), (3(r) = log r, and for S2 we have a(s) = a(s), /3(r) = 1. For x > z, the part of
S6 with s 5x also has this form, with 6(r) =1, and
a(s) _ F, µ(d) E A(m),
I a(s)I < logs.
m>z
d<w
dm =s
The inner sum over r in (22.2.1) is P2(s)
F, /3(r)( p(-f(rs + 1)) - p(-f(rs))),
(22.2.2)
Pa(s)
taken over a range P1(s) <- r5 P2(s) with P1(s)
X P2(s)
X M/s. By Lemma 5.4.3 Corollary 2 and Lemma 5.1.1, the sum in (22.2.2) is << (P2(s)N)1I3 max I a(r)I, and so the whole expression in (22.2.1) is
MN "
s5X
(-)
max Ia(s)/3(r)I << (MNx2)1/31
s
in all three cases. Here and in later order-of-magnitude estimates we write 1 for log M. The sum (22.2.1) can be swallowed by the expression in (22.1.6) for X <<
N M1/2
(22.2.3)
M 13 The terms in S6 with dm > x form a short type I sum. We suppose that x + 1 is a power of 2. We divide the ranges for r and q = dm into blocks of the
form PS r 5 2P -1, Q S q < 2Q -1, where P and Q are powers of 2. A typical block sum has the form 2Q-1
E F, µ(d) E A(m)
q=Q d5u
m>z dm=q
P2(d, m)
F,
r(qr),
r=P1(d,m)
greMI
and is bounded above by 2Q-1
F, q=Q
P4
F, F, A(m) dm dm=q
max Is> Pa
gP3,gP4eMl
I
E a(qr) r=P3
457
Type I double sums
The sum over d and m is simply log q. To apply Lemma 5.3.3 we put
o-(qr) = p(-f(qr+ 1)) - p(-f(qr)) = p(zr) - p(yr). We take the weights wr to be 1 for P3 < r5 P4, 0 otherwise. Then Lemma 5.3.3 gives
p I Ewro(gr)I 4
H
H
wr(e(hy,.) - e(hzr))
h=1
2arih
+E
H H h=1
H H h=1
1
1
+- E I Ewre(hyr)I +- E I Ew,.e(hzr)I; (22.2.4) Y
we have gained some simplification by taking the sum over h outside the modulus sign. If
M l3, N then the contribution 4 PQl/H of the first term is small enough. In the second term we write
(22.2.5)
H >>
e(-hf(m)) - e(-hf(m + 1))
f(m+1)
2,irih
f(m)
e(-ht) dt
=e(-hf(m))
ff(m+1)-f(m)e(-ht)dt. 0
There is some constant c with
f(m + 1) -f(m)
E wr(e(-hf(qr)) - e(-hf(qr+ 1))) r
fcN/m
we(-hf(qr)-ht)dt.
F, r
0
f(rs+1)-f(rs)z1
We suppose that f" does not vanish on the interval MI, so that the condition f(rs + 1) -f(rs) > t holds on a subinterval of P3 < r:5 P4. Hence the second term in (22.2.4) is bounded by
H N
<< F, -
P6
max
h=1 M IPs>P6)cIP3rP41
E e(-hf(qr))
.
P5
We can remove the minus sign by complex conjugation. In fact
P I Ewro(gr)I << H
1) H +(N-M +H h=1 - gP5,gP6EMI max
By (22.2.5) we can drop the term 1/H.
P6
e(-hf(gr)) P5
.
(22.2.6)
An application to number theory: prime integer points
458
We treat the sums in (22.2.5) as exponential sums over r with a parameter
q; the range for h is too short for us to expect an extra saving. We apply Theorem 17.2.2 with P in place of M, Q in place of I, and F(xy) for F(x, y) (with y in the range 1 5y <- 2), and T x hN. The results are much simplified if
Q>> max T8/3>>
(HN)8133
M8/3316/1 1;
(22.2.7)
h
we have assumed that H is chosen as small as (22.2.5) allows. Since Q does not satisfy (22.2.3), we have (22.2.7) provided that M49 <<
N661-246
(22.2.8)
We note that
aiF(x, y) = y'F(')(xy), a1-la2F(x, y)
+xyr-1F(')(xy).
= (r-
The derivatives of F(u) which must not vanish are: F" ( u),
F (3)(u),
F(4)(u) 3F ( 3)( u) + uF (4)(u)
F(5)(u)
3F(3)(u)2 - F (u) F(4)(u), F(3)(u) F(4)(u) 3F (3)(u) 2F" (u) + uF(3) (u) 2F" (u) F(4)(u ), F(3) (u)
F(5)(u) 4F(4)(u)
F(4) (u) 3F (3)(U) + uF(4) (u)
4F (4 )( U) + uF( 5)(u)
3F (3)(U)2 + 4F" (u)F(4)(u)
3F " (u)F(3)(u)
F(5)(u) 4F (4)(U) + uF(5)(u)
F(4)(u) 3F (3) (U) + uF(4)(u)
1 3F (3)(U)2
0
-F " (u)2
F(5)(u)
F(4)(u)
F(3 )(u)
4F(4)(u)
3F (3)(U)
2F "(u)
F(4)(u) 3F (3)(U)
F" (u)2 F(3)(u) 2F" (u) + uF(3)(u)
The theor em gives 5
P6(q)
E e(hf(qr)) q
<< p5/2 QT 5/6 +eE 3/8,
r=PS(q)
with 1
E << T8/33+
T2/3 :i
P2
+74/3
when we combine the second and third cases. The limits of summation P5
Type II double sums
459
and P6 also depend on h and t, but these are outside the summation over q anyway. We want to show that P6(q)
Q4
q
E e(hf(gr))
5
( 1 M. N
<
H N l3
r=P5(q)
which holds when T5/6+EE3/s <<
P5/2/H5120.
This inequality is satisfied for
M P» (NMM49/165+e + (N) 2
20/13
20/7
M1/3+E + (NM),
M4/21+F
(22.2.9)
22.3 TYPE II DOUBLE SUMS The type II sums have the form
E a(s) F, p(r)cr(rs). r>u
u<s5z
We divide the sums into blocks of the form P < r< 2P - 1, Q < s5 2Q -1, where P and Q are powers of 2, with QM/z>> M1/2. (22.3.1) Let a and /3 satisfy 2Q-1
2P-1
I a(s)12 < a2Q,
l3(r)12
1
s /32P.
P
Q
In S3 we have a(s) = A(s), /3(r) = 1, so that we can take a =1= log M, /3 = 1. In S41 we have a(s) = A(s), /3(r) = K(r), so we can take a = 1, p2 << 13. In S42 we have a(s) = K(s), /3(r) = A(r), so that a2 << 13, /3 =
1.
We write S5 (with a change of notation) as
S5 = E µ(d) E A(m) w5d5u
m>z
F, or(dmq). q>z/d
We remove the condition dq > z using Lemma 5.2.4 on the sum over q, with K2 = 2Q - 1, K1 = (z + Z)/d. This lemma introduces factors 1/ds, 1/q5 into the block sum, and an error term << E I µ(d)I A(m) log Q «P12. w5d5u
m>z PSdm <2P
The coefficient /3(r) has
/3(r) _ L
w5d5u
µ(d) ds
E A(m), m>z
dm =r
10(01 < log r,
An application to number theory: prime integer points
460
so we can take a = 1, f3 << 1. There is also a factor I from the integration over
s at the end. We apply the method of Lemma 5.3.3 with
Z. = -f(m + 1),
Y. = -f(m),
and wm zero for m not in the interval MI, 2Q-1
2P-1
wm = F, a(q) E f3(r) r=P
q=Q
for m in MI. In the argument leading to the third and fourth terms in the upper bound we use 2Q-1
IWmI s F,
2P-1
Ia(q)I F,
q=Q
I f3(r)I
r=P
for m in the interval MI. We continue as in the treatment of short type I sums, but we pay more attention to the limits of summation. Let J denote the
interval MI, and J(t) the subinterval of J on which f(x + 1) - f(x) Following the treatment of type I sums, we have 2Q-1
1 2Q-12P-1
2P-1
L a(q) F,
r=P
q=Q
f3(r)o- (gr)<< - F, E Ia(s)f3(r)I H q=Q r=P 2Q-1
H
IcN/M
q=Q
h=1 t-O
2Q-1 1
H
q=Q
H h=1 H
dt
E If3(r)Ie(hf(gr))
r=P
grEJ
2Q-1
+-H h=1 1
F f3(r)e(hf(gr))
r=P 2P-1
E Ia(q)I
+- F,
2P-1
E a(q)
2P-1
E I a(q)I
L 1 /3(r)I e(hf(gr + 1)) r=P
q=Q
.
(22.3.2)
greJ
We want to show that the block sum on the left of (22.3.2) is << N/13. The first term on the right of (22.3.2) is M N 1
<< Ha/PQ<< a(3H<<,
provided that we choose
Hx af3 a choice similar to that in (22.2.5).
M 13, N
(22.3.3)
Type II double sums
461
The other three sums on the right of (22.3.2) have the same form. We take the second sum as an example. First we remove the coefficients /3(r) by a Cauchy inequality as in Lemma 5.6.1. 2Q-1
2
2P-1
E a(q)
F,
/3(r)e(hf(gr))
r=P
q=Q
2P-1
2Q-1 E a(q)e(hf(gr))
2P-1
E I (3(r)I
F,
P
P
(22.3.4)
.
Q
greJ(t)
The first factor on the right of (22.3.4) is 5 /3 2P. In the second factor we apply the differencing step (Lemma 5.6.2): 2Q-1 E a(q)e(hf(gr)) Q
2
2
Q+2D-1'2Q-1
Ia(q)I +2Re
D2
D-I (
d
1
D)
d=
Q
greJ(t)
2Q-d-1
x
E
a(q+d)a(q-d)e(hf(gr+dr)-hf(gr-dr))
.
(22.3.5)
q=Q+d gr±dreJ(t)
The first sum on the right of (22.3.5) contributes
Q+2D-1 D2
«2Q
to the right-hand side of (22.3.5), << a2/32P2Q2/D << a2/32M2/D
to the bound for the right-hand side of (22.3.4), and
« HN M a/3M VD _
r
a2/32M13
(22.3.6)
to the second sum in (22.3.2), provided that D << Q. We choose DxaaRa(N)2
112,
so that the bound in (22.3.6) is again << N/13.
(22.3.7)
An application to number theory: prime integer points
462
In order to apply Lemma 5.4.3, Corollary 1 to the sum over r, we note that a2 ar2(hf(gr + dr) - hf(gr - dr))
=h(q+d)2f"(qr+dr) - h(q -d)2f"(qr-dr) = 2dh(2(q + Od) f" (qr + Odr) + (q + 9d)2 f (3)(qr + Odr)), (22.3.8)
for some 0 in -1 < 0 < 1. If 2F" (u) + uF(3)(u)
is bounded away from zero, then the expression in (22.3.8) has order of magnitude
dh N M2 - Q P2
dhNQ
Corollary 1 to Lemma 5.4.3 gives
L e(hf(gr+dr) - hf(gr - dr)) <<
(
dhN 1/2
I`
r
Q
Since
2I a(q+d)a(q-d)I.-I a(q-d)I2+Ice (q+d)I2, the non-trivial terms on the right of (22.3.5) sum to
«
Q Q
aD2 2
Q(DhN1/2 Q
contributing a2/32M2 (DHNP) 1/2 << « 2Pa ZQZ(DHN11/2 Q P M J
to the right-hand side of (22.3.4), and
DHN
M
M
(22.3.9)
) 1/4
to the second term in (22.3.2). The expression (22.3.9) is << N/13 for P>> a4R4DH5N 112>> (ap)13(M)
6
139
(22.3.10)
We also require Q >> D, so
Q>> (aa)4(N
2112
(22.3.11)
Prime numbers in a smooth sequence 22.4
463
PRIME NUMBERS IN A SMOOTH SEQUENCE
We can now give a form of the Piatecki-Shapiro theorem. Theorem 22.4.1 Let F(x) be a real monotone increasing function, six times
continuously differentiable on a closed subinterval I of [1, 2] on which the expressions F'(x), F"(x), and 2F"(x) +xF(3)(x) are non-zero. Let e be positive, and let M be a large integer, N be a large real number with
NmaxF'(x)<M
(22.4.1)
I
with 0 = 12/11. Then, for some constant B depending on I, F(x), and on e, but not on M or N, there are at least BN/log N integers n such that [g(n)] is a prime number p with M5 p < 2M, where x = g(y) is the function inverse to
y/N = F(x/M). If in addition the interval I is chosen so that none of the expressions
F(4)(x),
F(3)(x), F(4)(x) 3F(3)(x)
F(3)(x)
F(4)(x)
F(3)(x)
F" (x)
3F (3)(x)
2F"(x)
3F(3)(X)2 F(5 (x) 4F(4'(x)
F(5 (x) 4F(4)(x)
0
-F" (x)2
F(4)(x) 3F(3)(x)
2F"(x)
F(4)(x) 3F (3)(X)
F(3)(x)
vanishes on the interval I, then the result holds with 0 = 3300/3019 in (22.4.1).
Proof We follow the convention of writing a for any exponent which can be made arbitrarily small, so that exponents e in different bounds need not be
the same as one another, or the same as the a in the statement of the theorem. For the result with 0 = 12/11 we do not actually need the further dissection of S3 into S411 S421 S5, and S6. All that we use about the type II sums is that P >> Q, so that p >> M112. This can be achieved by interchanging q and r within each block of the sum if necessary. With the combinatorics of Section 22.1 we take z close to M112, so that w is bounded, to achieve the same effect. For p >> M 1/2, the necessary condition (22.3.10) holds for M<<
N12"11-
(22.4.2)
and (22.3.11) holds for M /2
u >>
(N
ME,
(22.4.3)
which is consistent with (22.1.8) and (22.1.10). The sum S6 may be treated as
An application to number theory: prime integer points
464
a type II sum. For the other type I sums, the condition (22.2.3) with x = u follows from (22.1.8). In (22.1.5) we have
N
(2)
E loge=f(M2)-f(M1)+O log M
PEW
p prime
Since log p >- log M and f(M2) -f(MI) >> N, we deduce the result. For the result with 0 = 3300/3019 we must choose z so that in (22.3.10)
M >>(N)
6
ME, (22.4.4)
and in (22.2.9) z3
z
M w >>
20/7
(M 1 2 M49/165+e + M )20/13 M1/3+e ,+,
N
N
M4 /21 + E
N (M) (22.4.5)
For M larger than the range (22.4.2) we take
()6_
to satisfy (22.4.4). When we substitute into (22.4.5), then the strongest condition is M <<
N3300/3019-
(22.4.6)
and both (22.1.10) and (22.1.11) with
ux
Me
hold for M<
a weaker condition.
We used Vaughan's three-factor combinatorics to replace sums with the von Mangoldt function A(m) by more tractable exponential sums. Using a five-factor identity of Heath-Brown (1983) lets us take P >> z in the short type I sums, increasing 0 from 3300/3019 to 330/301.
Part VI Related work and further ideas
23 Related work 23.1
INTEGER POINTS CLOSE TO A CURVE
Bombieri and Pila (1989) have improved Swinnerton-Dyer's estimate for the
number of points on a convex curve when the curve is very smooth. The argument again takes a `divide and conquer' form.
D. Divide the curve into arcs. A. Show that for each arc there is some algebraic curve P(x, y) = 0 passing through all the integer points on the arc, where P(x, y) is a polynomial with integer coefficients. C. Count the integer points on the corresponding arc of the algebraic curve.
With L = M = T, S = 0 in the notation of Chapter 3, the results take the form R:5 BMO+ e,
(23.1.1)
where a is positive, and can be taken arbitrarily small. The constant B depends on e. Irreducible algebraic curves (defined by P(x, y) = 0 where P(x, y) is an irreducible polynomial) are classified by an integer called the genus. Curves
with genus zero can be parameterized by rational functions x =f(t)/h(t), y =g(t)/h(t), where f(t), g(t), and h(t) are polynomials with integer coefficients. Suppose that we have the simplest such parameterization. Then (23.1.1) holds with 0 = 0 unless h(t) is a constant. When h(t) is a constant, then (23.1.1) holds with 0 =1/d, where d = max(deg f, deg g). For genus 1, R is unbounded as M tends to infinity, but (23.1.1) holds with 0 = 0 and B depending on the curve as well as on e. Swinnerton-Dyer uses the determinants
0=
1
ml
nt
1
m2
n2
1
m3
n3
If (m;, n.) are integer points on or close to a very short arc of a curve, then A
Related work
468
is numerically small. Since 0 is an integer, then 0 must be zero, and the three points must lie on a straight line. Bombieri and Pila use 1
m1
nl mj
...
mind-1
...
mDnD 1
nl
0= 1
mD nD MD
nD
where D is the number of terms mini of degree at most d, which is the binomial coefficient (d + 1)(d + 2) D =d+2C2 =
(23.1.2)
2
The conclusion from A = 0 is that the integer points (ml, n1), ..., (MD, nD) lie
on an algebraic curve defined by an equation P(x, y) = 0 of degree d with integer coefficients.
Suppose that F(x) is infinitely differentiable and equal to the sum of its Taylor series (strictly, for each point a there is a neighbourhood of a on which the Taylor series at a converges and equals F(x)). Suppose that G(x) is an algebraic function (not necessarily defined over the rationals). If F(x) = G(x) for infinitely many values of x with 0 <x 5 1, then there is some
point at which all the Taylor coefficients of F(x) - G(x) are zero, so that F(x) is the same as G(x). Consider all implicit real equations d
d-1
0=P(x,y)= E E a;;x'y1 i=o i=o of degree d, normalized by
F, F, a,j=1. r
i
The number of intersections of Y = F(x) and P(x, y) = 0 in 0 S x 5 1 is a function N(a) of the coefficients a1, continuous except at sets of coefficients where the two curves touch. For each value at = 0 there is a neighbourhood of 0 in D-dimensional space on which N(a) S N((3). Since the unit sphere is compact, we see that there is a maximum number of intersections. This is an ineffective argument: we cannot find the maximum number of intersections explicitly. For given e > 0, choosing d about 3/e leads to (23.1.1) with 0 = 0, and the constant B depending heavily on the function F(x) as well as on E.
If y = F(x) is a solution of Q(x, y) = 0, where Q(x, y) is an irreducible polynomial of degree e > d, then the curves P(x, y) = 0, Q(x, y) = 0 intersect in at most de points. Bombieri and Pila show by induction that if 0 < F'(x) < 1 for 0 s x< 1, then (23.1.1) holds with 0 = 1/e, and B depending only on E. Finally they give a uniform result for an arbitrary function F(x). If d and
Integer points close to a curve
469
D are related by (23.1.2), and if F(x) is D times continuously differentiable, then (23.1.1) holds with 8
(23.1.3)
B=z+3(d+3)
and with the constant B depending on the function F(x) only by way of an upper bound for the derivatives of F(x). There is a similar result with B depending on the number of zeros of F(D)(x). In principle the method extends to integer points at most S from the curve. But 8 must be extremely small for us to conclude that the determinant 0 is numerically less than unity. If d = 24, D = 325 (the first case in which (23.1.3) gives 0 < 3/5), then we require 6 «M-52649
Sargos has another approach to the problem of integer points close to a curve, based on the interpolation polynomial used in Lemma 3.2.2. The curve
is y = f(x), with f (") non-zero on an interval of length L. For n = 2 we considered the integer points close to the curve as a polygon. For any n + 1 points, there is a curve y = g(x) passing through them, given by the interpolation polynomial, of degree at most n. Each set of n + 1 consecutive integer points close to the curve defines an interpolation polynomial curve, which we call the Sargos spline. A general integer point lies on n + 1 Sargos splines. A spline is minor if it has degree n, major if it has degree at most n - 1. The coefficients of a Sargos spline polynomial are rational, and the coefficient of xn-1 in a major spline is a rational approximation a/q to f (" -1)(x)/(n - 1)!.
In the case n = 2 several integer points can lie on the same side of the polygon. Similarly several consecutive integer points can lie on the same Sargos spline. If n + 1 or more integer points lie close to the curve in a short interval, then they must lie on the same major spline. Theorem 23.1.1 (Huxley and Sargos 1995) Let 6, 0, and L be positive real numbers, with S < z and L > 1, and let n >- 3. Let f(x) be a real function with n continuous derivatives on an interval I of length L, with
If(")(x)I x 0
(23.1.4)
on I. Let R be the number of pairs of integers (1, m) with m in I and
IZ-f(m)I s6. Let C be any positive real number. Then S 1/" R«S2/"("-1)L+02/n(n+t>L+()
+1.
The implied constant depends on C, n, and on the implied constants in (23.1.4).
470
Related work
Theorem 23.1.1 improves Theorem 3.3.1 for large S; we have suppressed T
and M from the notation. In an iteration of the type in Section 3.4 which ends with an appeal to Lemma 3.1.2, we should replace the last iteration step by Theorem 23.1.1. More complicated results, in which an extra derivative is assumed not to vanish, will be found in Filaseta and Trifonov (1996) and Huxley and Trifonov (1995); they correspond to Theorems 3.3.1 and 3.6.3 respectively.
23.2 THE HARDY-LITTLEWOOD METHOD Our guiding principle has been that in problems with both continuous and discrete aspects, behaviour at real numbers is related to the behaviour at nearby rational numbers. In the large sieve inequality the comparison goes the other way, that an average over the Farey sequence modulo one can be bounded in terms of averages over all real numbers modulo one.
The Hardy-Littlewood method is one of the great themes of number theory this century. In its simplest form we count the number r(n) of ways that an integer n is represented in a certain form. Classically we form the generating function f(z) = E r(n)z". Usually this series converges inside the unit circle and tends to infinity for z = re(a/q), where a/q is rational, and r tends to one from below. The unit circle is a natural boundary. We recover the coefficients from 27rir(n) = f f(z)zn-1 dz, where the integral goes around the origin along a circle of radius slightly less than one. The circle is divided
into Farey arcs I(a/q). When a/q has small height (major arcs), then I(a/q) contributes approximately ?Ti times the residue at e(a/q), defined as
the limit of (z - e(a/q))f(z). When a/q has large height (minor arcs), then we use an upper bound for If(z)J, and we count the contribution as an error term. The major arc contributions are evaluated approximately, so there may be cancellation between different arcs I(a/q), first between arcs with the same denominator q, then between the sets of Farey arcs with different denominators. After order-of-magnitude factors are taken out, these contributions form the so-called singular series, with terms indexed by the Farey sequence modulo one. The singular series converges in most applications; if r(n) is zero for congruence reasons, then the singular series is found to converge to zero. Often the series factorizes into factors associated with each prime p, which can be thought of as the measure of the set of solutions of a congruence.
A simplification is to truncate the generating function f(z) to a finite
sum, and work on the unit circle with S(a) = Er(n)e(na), so r(n) = f S(a)e(- n a) d a, integrated over the real numbers modulo one. When the generating function is a product, then the truncation takes place in each factor. Complex analysis is replaced by Fourier theory. Inequalities can be treated analogously. This is the form of the Hardy-Littlewood method used
Other Farey arc arguments
471
in Chapter 12. Many complications were avoided in that chapter, as only an upper bound was required. There is a discrete form of the method, where the integral from 0 to 1 is replaced by the Riemann sum with step length 1/N for N sufficiently large. It seems to offer no advantage in practice. When the generating function factorizes into several factors, then there is
an elegant treatment on the minor arcs. Instead of multiplying an upper bound for each factor, we take some factors out at their maximum, leaving the integral of the modulus of the other factors on the minor arcs. Holder's inequality estimates this integral in terms of one or more integrals of the modulus squared of exponential sums. The range of integration is extended from the minor arcs to the whole unit interval. The modulus-squared integral becomes the counting function for a simpler problem. In Waring's famous problem of expressing n as x, + +xs, this leads to Vinogradov's mean value theorem (see Vaughan 1981), an analogue of the First Spacing Problem in Part III.
In delicate investigations we must treat all arcs as major arcs, to get cancellation over different arcs I(a/q) with the same denominator q. The endpoints of I(a/q) depend on the neighbouring Farey fractions. When q is large, then for some values of a, the arc I(a/q) does not even appear: for a = 1 and q large, the rational 1/q lies in the arc 1(0/1). In this version of the Hardy-Littlewood method, each Farey arc gives a main term, contributing to the singular series, a remainder term from the approximation, and a correction term involving the adjacent Farey fractions. Cancellation in the correction terms is called the Kloosterman refinement. In the Hardy-Littlewood method for d simultaneous equations, there are
d integrations. An equation in an algebraic number field of degree d corresponds to d simultaneous equations involving the coefficients with respect to a basis for the integers. The role of the rational numbers a/q is given to the characters o ,(v) of the additive group modulo an ideal I. The norm of the ideal corresponds to the denominator q. The characters can be
expressed as v(v) = e(Trace av), for some number a in the field; the denominator of a may involve a different ideal.
23.3 OTHER FAREY ARC ARGUMENTS Iwaniec, lecturing in Gottingen in 1992, suggested a discrete form of the Hardy-Littlewood method with the Kloosterman corrections. He requires a weight function w(t) with w(t) = w(-t), w(0) = 0, and W
w(n) = 1.
Related work
472
If n is non-zero, then
E w(k) = E to
n
kink
kin
by symmetry, whilst if n = 0, then
,w(k)=1,
L(k)=0.
kin
kin
Using Kronecker's notation 8(n) for the function which is 1 for n = 0, 0 for all other integers, we have n
8(n)
w(k) - w(k kin
k=1
=
k E eI In ) I w(k) - w bmodk
l1
`
n
k
* e( anl(w(k)-w(k))
1
k=1 k qlk amodq (an l
L * el -1
amodk 1 q
I`
q
JJ
1
E k=O(modq)
n
-(w(k)-w(_)) k k
an a mod k
q
where the star indicates a sum over a with (a, q) = 1, and Wq(n) denotes the
sum over k for fixed n and q, regarded as a weight function. To count solutions of gi+...+g:
=hi+...+h;,
we put
n =gi + ... +g, - hi - ... -hr, and sum over g1, ... , gr, hl,..., hr. Although the exponential e(an/q) factorizes,
the weight function Wq(n) does not. The weight Wq(n) makes
the expression a convolution of generating functions rather than a simple product. Everest (1989) has discovered a major and minor arc phenomenon when
studying the distribution of units in an algebraic number field. Let S be a finite set of prime ideals. An S-unit is an algebraic integer with no prime factors outside S. If S is an empty set, then we get the units of the field. The S-units form a group generated by a root of unity and a finite number (r, say)
of generators of infinite order. Let e be the number of embeddings of the field into the real or complex numbers. For each S-unit a, we form the
Higher dimensions
473
vector with e entries log I o (a)I, where o runs through the embedding maps. The S-units are mapped to an additive group of vectors in an (e -1)-dimensional subspace. The roots of unity map to the zero vector, and it can be proved that no other S-unit maps to zero. These vectors form the projection into (e -1)-dimensional space of an r-dimensional lattice. The distribution is non-uniform: the r generators v1,. .. , v,. have particular lengths and directions, so the shorter vectors have more integer multiples near the origin, and directions are favoured which correspond to linear combinations alvl + +a,.v,. for which (al,..., a.) is a primitive integer vector of small height, the major directions. The major directions determine the approximate distribution of solutions of the S-unit equation 13,x]+...+F'nxn=0,
where the variables X1,..., x are S-units, and the constants /3,, ... , . 6 are integers of the field. A recent paper of Baker and Harman (1991) makes an unexpected use of dissection into Farey arcs.
Theorem 23.3.1 (Baker and Harman) Let a be irrational, 13 be real. There are infinitely many primes p with
Ilapk+/311<1/p°
when k=2, 0<3/20, orkZ3, 9<1/3X2k-' Sketch proof
1. Approximate a by A/Q using Dirichlet's theorem (Lemma 1.5.1). 2. Express the condition in terms of exponential sums using Lemma 5.3.2. 3. Replace a sum over primes by a sum over integers m with weights A(m), and expand A(M) by a method similar to that in Chapter 22. 4. Since xk is a mononomial, the factors can be grouped in various ways to give triple sums involving e(Ahxkyk/Q).
5. For each y, approximate Ayk/Q by a/q in a Farey sequence of smaller order.
6. Difference k - 1 times in x using Lemma 5.6.2, to get sums involving e(bx/q), which can be evaluated, and are small for most values of b modulo q. The integer parameter b comes from combining h and the differencing parameters as in Lemma 17.4.1 and Theorem 18.5.3.
0
23.4 HIGHER DIMENSIONS Much less is known of lattice point problems in three or more dimensions. It
is easy to approximate a plane curve by a polygon, because the gradients correspond to the real line, which is ordered. A polyhedron with rational
Related work
474
normals that approximates a surface is more difficult to construct. However
Kendall's Fourier series method used in Chapter 6 does extend to higher dimensions (Hlawka 1950; Herz 1962a, 1962b). The expected number of lattice points inside a smooth closed surface containing a volume V in n dimensions, expanded by a linear factor M, is VM". The easy estimate for the
discrepancy D(M), corresponding to Lemma 2.1.1, is O(M"-'), and the mean-to-max argument replaces the exponent n-1 by n - 2 + 2/(n + 1). The limitation result corresponding to Theorem 6.3.1 states that the exponent must be at least (n - 1)/2. Kratzel and Nowak (1991, 1992) have improved
the exponent by grouping the Fourier series terms to form suitable twodimensional exponential sums.
Theorem 23.4.1 (Kratzel and Nowak) Let S be a closed surface in n dimensions with non-zero Gaussian curvature, with an infinitely differentiable one-one correspondence between points on the surface and their outward normal vectors. Let D(M) be the discrepancy of the interior of the M times enlarged surface MS. Then D(M) << M8(log M)A with A 5 1 and 9
_
n - 2 + 8/(5n +2)
for n=3.... 6,
n-2+3/2n
fornz7.
Exponential sums with two or more variables present new difficulties. Some difficulties already appear in the corresponding exponential integrals
f f e(f(x, y))dxdy. After we integrate by parts in x, the exponent for the integral over y depends on the limits of integration for x, and so on the shape
of the region of integration. Even for rectangles, there is no saving in the y-integration if f(x, y) =g(x), a function of x alone. If f(x, y) =g(x +y), then we can use a repeated First Derivative Test as in Chapter 21, but if g' has a turning point, then the exponential integral can be evaluated using Lemma 5.5.2, and again there is no saving in the y-integration. The most favourable case is f(x, y) = g(x) + g(y), when the variables separate over a rectangle. If the integrand has a local maximum or minimum, then we should get an asymptotic formula like Lemma 5.5.2. Since f f e(x2 +y2) dx dy and its
analogues in several variables do not converge at infinity, non-vanishing conditions are required for the third or higher derivatives, and the error term is proportionately worse than in Lemma 5.5.2. The van der Corput iteration (see Graham and Kolesnik 1991) consists of
two types of step, followed by a trivial estimate or an appeal to Theorem 17.2.2, with one summand as m, and another corresponding to the parameter.
The differencing step involves f(x + h) -f(x - h), where h is a vector in a box with centre at the origin. The box may have sides of unequal length; if a side has length one, then the corresponding component of h is zero, and the corresponding variable is fixed. The reflection step is Poisson summation (Lemma 5.5.3) in a subset of the variables, and it introduces all the difficulties of estimating exponential integrals. Even for a rectangle, the reflected sum
Exponential sums with monomials
475
may be over a region of complicated shape, as we saw in Chapter 8, and the error terms and the truncation error in the Poisson summation formula may add up to more than the estimate for the reflected sum. Another difficulty is that the reflected sum, when treated as a family of exponential sums with a parameter, may be non-standard. A standard sum is one with length M and the rth derivative of order T/M' for all required values of r. A sum becomes non-standard if the length is much shorter than M, or if certain derivatives
are too small or too large. Kratzel (1988) explains this method for many classical problems.
23.5 EXPONENTIAL SUMS WITH MONOMIALS Many applications of exponential sums in number theory involve monomial
functions in the sense of Chapter 3. When expressions like o-(dmr) in Chapter 22 are expanded as Fourier series, then the exponent in each term can be factorized as a product of powers of d, m, and r. The integer variables can be separated. They can often be recombined in ingenious ways, as in the proof of Theorem 17.2.2. A lesser advantage of monomial functions is that inverse functions and derivatives are easily found. Fouvry and Iwaniec (1989) have shown how useful the large sieve (Lemma
5.6.6) can be for exponential sums with monomial functions in several variables. As an example we sketch their Theorem 3, which is used by Liu and Rivat (1992) in the Piatecki-Shapiro problem of prime integer points close to a curve, considered in Chapter 22, when the curve is a power function y = x. That Fouvry and Iwaniec's method actually gives a slightly stronger result was noticed by Liu and Rivat. We indicate such an improvement at the end of the proof.
Theorem 23.5.1 (Fouvry and Iwaniec) For K, L, M positive integers, not all unity, T large and positive, a, 63, y non-zero real numbers, let 2K-1 2L-1 2M-1 Tm'k Ply S = E F, E a(m)b(k, l)eI MaKPLY 1 k=K I=L m=M where a(m), b(k, 1) are any coefficients. Then, for a 96 1 we have M215
T114 +M112
S << (KLM)1 /2(
(
+M1"5 + 7
/
X max la(m)II
log2 KLM
Ib(k,l)I2)1/2.
(23.5.1)
m
`k l The implied constant depends on a, (3, and y. Sketch proof
Step one The triple sum S is type II, so we apply Lemma 5.6.1 with the
Related work
476
double sum over k and I as the outer sum, followed by Lemma 5.6.2 applied to the inner sum over m. This produces a four fold sum involving
e(x(d, m)y(k, l)), with
x(d, m) = T((m + d)'- (m - d)a)/Ma, y(k, 1) = k PIY/KsLY.
Step Two We apply the large sieve (Lemma 5.6.6) in one dimension to blocks of terms H< d5 min(2H - 1, D). Step Three The Second Spacing Problem is to count the number of sets of integers k1, 11, k2, 12 with
k2
O(OKRLY),
(23.5.2)
where 0 x M/HT. Let k = (k1, k2), 1 = (11,12), k1/k2 = r/s, 12/l1= r'/s' in lowest terms. For fixed l there are O(L2/12) possibilities for r'/s', so r/s lies in an interval of length O(0) by (23.5.2). For k fixed there are now 0(1 + AK2/k2) possibilities for r/s by Lemma 1.2.3, so we get L2
O l2 +
02K2L2
k2l2
(23.5.3) )
possibilities for r/s and r'/s'. By symmetry, we can replace (23.5.3) by O I`
OZK2L2 ( k212 + Min l
which sums over k and 1 to give
K2 L2 2 , IZ
`
B «OK2L2 + KL log KLM, in the notation of Lemma 5.6.6.
(23.5.4)
Step Four The First Spacing Problem is to count the number of solutions of
(m1 + d1)a - (m1- dl)a = (m2 + d2)a - (m2 - d2)a +
O(OHMor_ 1).
(23.5.5)
For small H we write
(m + d)"- (m - d)a = 2 adma-1 + O(H3Mar-3), and the argument used for step 3 gives
A << HM logKLM+OH2M2 +H^. For large H we write
(m+d)a-(m-d)a=2adma-1+
(23.5.6)
a(a - 1)(a - 2) d3ma_3+O(H5Ma_5) 6
Exponential sums with monomials
477
We fix d1 and d2, and put A =
mi M
-
(d1/d2)1/t"- 1>. Then (23.5.5) implies
a-2 a; ( A
1
6
m
H44
111-
za
+0 0+ M J
(23.5.7)
)
Fouvry and Iwaniec study (23.5.7) by exponential sums. We can also use Farey sequence methods. For fixed A, the fraction m2/m1 lies in a short
interval. We take e/r to be the rational number of least height in this interval, and let f/s, f'/s' be its neighbours in the Farey sequence .9(r). Now m2/m1 lies in one of two reference intervals. For m2/m1 between e/r
and f/s, we have m1= ru + st, m2 = eu + ft for some integers t, u as in Lemma 1.2.2, but the common factor (t, u) = (m1, m2)'may be greater than one. We can also write
A = (e +f9)/(r+s9), where 0 may be negative. Substituting in (23.5.7), we find that (t - Ouxru + st) lies in an interval whose length is proportional to A. This is an integer point
close to a curve problem. The trivial estimate O((6 + 1XL + 1)) (in the notation of Chapter 3) gives
O(OM2 +H4/M2 +M/r), and by Lemma 1.2.3, we have r << R for O(HR) pairs d1, d2, and so A << HM log KLM + OH2M2 + H6/M2.
(23.5.8)
Fouvry and Iwaniec impose a condition H << M3/5 to remove the last term in
(23.5.8). If we retain this term, and optimize the choice of D, the upper bound for H, then the terms M1/5 +M2/5/T1/4 in (23.5.1) may be replaced by M1/7T1/14
M 1/6 +
M112
(KL)1/14 + T1/5 +
M1/2 (KL)1/24T1/6.
o
24 Further ideas 24.1 COMMENTS ON THE METHOD Mathematics delights in general principles and specific examples. The ideas of Bombieri and Iwaniec (1986a) for a specific problem have grown to a general method. A part of mathematics may not die, but it can hibernate,
either because all the main results appear to be known, or because the subject has become too complicated. The Bombieri-Iwaniec method is not yet in its final form: we end this book with some suggestions for improving or extending it. Complications
There are two main complications: short sums (over a subinterval of M to 2M) and a family of sums indexed by values of a parameter. We have not tried to combine these in all possible ways, because of the large number of
cases. In applications the sizes of the parameters S, I, J, M, and T are known approximately, so most cases do not occur. We have said nothing about congruence families of double sums. Two particular questions arise:
1. What happens when different values of the parameter correspond to different short subintervals of M to 2M?
2. Is there a saving over two independent parameters which cannot be combined into one as in the proof of Lemma 17.4.1.? Problems of technique Neither Spacing Problem is completely solved:
1. Find a good estimate for N12(81 0) in Chapter 12. 2. Can one handle the perturbed inequalities in Section 13.3? 3. The condition R2 >> N in Chapter 7 can probably be dropped, at the cost of writing f" (m)/2 = a/q + A, and carrying A through the Coincidence Conditions and all subsequent calculation. 4. Kolesnik's formulation of the Third and Fourth Conditions in Chapter 16
partly explains why the arguments of Chapter 15 work, but it loses accuracy in the Third Condition. Must accuracy be lost? The Kolesnik conditions are more elegant. Sometimes a method is first simplified, then extended in a way which was always possible, but looked impossibly difficult before.
Subdivision without absolute values
479
5. Can one show that most magic matrices do not give a coincidence?
6. Can one show that most minor arcs do not give a coincidence, using a divide-and-conquer argument that is not based on magic matrices? Extensions
It would be very interesting to find other sums which can be broken into Farey arcs this way. One such is
E F, E e(hf(m +k) -hf(m -k)), h
km
which corresponds to the mean square of a lattice point discrepancy. But estimates strong enough to improve Theorem 6.3.4 seem unlikely by this method.
24.2 SUBDIVISION WITHOUT ABSOLUTE VALUES Dividing a sum of length M into about M/N short sums of length N or less, which are estimated in modulus, cannot give an upper bound better than
(N M OE)=0
(
M NJ.
By Theorem 6.1.1, the sum is almost always O(MI/2+E), so there must be
cancellation between the sums over different Farey arcs. The HardyLittlewood method obtains this cancellation on the major arcs. With the Kloosterman cancellation, all Farey arcs can be treated as major arcs. The difficulty is transferred to estimating the Kloosterman corrections. To apply the same programme to the Bombieri-Iwaniec method for single sums, we need to know f(m) modulo one at the centres of the major arcs. To describe how they change, we will need more terms of the Taylor expansion of f(x). The double sum of Chapter 8 may be easier, as f(m) already occurs in the calculation.
In any argument analogous to the Kloosterman refinement in the Hardy-Littlewood method, the division into Farey arcs must be made arithmetically, corresponding to some Farey sequence AR). First method
If y lies between the consecutive Farey fractions cp,_ I = a,_ cpr = a,/q,, then qr- 1(ar - qrY) + gr(gr- 1Y - ar-1) = 1
1/q,_ 1
and
Further ideas
480
The Farey are sum on Ir = I(ar/qr) is now
E
gr(Rr-1f"(m) - ar- 1)e(f(m))
Wr-ISf"(m)S cVr
MSm5M2
Rr-1(ar+1 -Rr+1f" (m))e(f(m))
+ p,sf"(m)5 Wr+1
Msm5M2
_
E
kr(f"(m) - (pr)e(f(m)),
M5rn5M2
where kr(x) is a piecewise linear function with Fourier transform _s
Sr
Rr
kr(s)
Rr+1e(R q 4ar2s2 r
+1
)
+ Rr-1e(Rr
Rr
-1)
Rr+1 - Rr-1
When Mr is the centre of the Farey arc Ir, and m = Mr + n, then
kr(f" (m) - (pr) = f kr(s)e(-sf" (m) + s(Pr) ds. The integral is approximately
f kr(s)e(-µrns)ds, in the notation of Chapter 7. Second method
Jutila (1992) has proposed taking overlapping Farey arcs I(a/q), with length N(q) depending on q, for all rationals a/q with q S Q. For each integer m in the sum, let A(m) be the number of Farey arcs which contain m. We write A(m) = B(m) + D(m), where B(m) is the expected number, varying smoothly with m since f (3)(X) is non-zero, and D(m) is the discrepancy. If m is near the centre of a major arc, then A(m) = 1, and D(m) is large. On the minor arcs the discrepancy D(m) is smaller than B(m) in root mean square. Thus M2
Q
E B(m)e(f(m)) _ F,
F,
q=R M
(a, a
E e(f(m)) +E1 +E2, I(a/q)
where E1 is the correction for m on major arcs, which we estimate as before, and E2 is the correction for minor arcs. Using this idea, we can add together contributions from minor arcs with the same denominator and sums of the same length. A modification would be to take the Farey arcs so that the transformed sums had the same length H. In either method, we must sum over a for fixed q to obtain a Kloosterman refinement. For fixed q we have a double sum over a and h. The limits of
summation, and the numbers b, µ, and K, depend on a. We also have
Subdivision without absolute values
481
expressions like e(- 4ah2/q). When we use a finite Fourier transform to take them outside the sum, then we meet Kloosterman sums
K(a, b; q) = F,*
e
(an +bn) q
n mod q
where the star indicates a condition (n, q) = 1, so that integers n and q exist with nn + qq = 1. Lemma 24.2.1 (Factorizing the Kloosterman sum) If q factorizes as
q = np;', and a/q, b/q have partial fractions a q
i
a,
b
pi'
q
b, i
pi`
then
K(a,b;q)= fK(ai,bi;pi`) i
The factorization in Lemma 24.2.1 is twisted, in the sense that a is (usually) not congruent to a, modulo pi.
Lemma 24.2.2 (Weil's Kloosterman sum bound) The Kloosterman sum to a prime power modulus p` satisfies I K(a, b, p`)I 5 2p`l'2, unless both a and b are multiples of p.
The cases c >_ 2 of Lemma 24.2.2 are elementary, but the case c = 1 depends on counting the number of points on an algebraic curve over finite fields.
The terms of the Kloosterman sum correspond to matrices
of the modular group, and Kloosterman sums occur in the Fourier coefficients of modular forms. Kuznietsov (1980) followed ideas of Selberg to express sums of Kloosterman sums explicitly in terms of the Fourier coefficients of modular forms.
Further ideas
482
Lemma 24.2.3 (Kuznietsov's trace formula) For a, b positive integers, g(x) a continuous piecewise differentiable function of compact support q 2
(ab))
g(2
(ab)
a )kk-1/2
1)k
°°
+k=1 E
21T
b (-)
K(a, b; q) + Sab f J.
x
g(x) dx x
2
(cb(G2k a) - Sab)(2k -1) f J2k-1
4 (ab) bt(a)bt(b)g(K,)
c
1
+
cosh in.1
j=1
(2)
x)
g(x) dx
T;t(a)Tj,(b)g(t)
?r J- (1 + 2it)( 1 - 2it)
dt,
where G2k,a is a modular form of weight 2k, the ath Poincare series, and c is its nth Fourier coefficient (suitably normalized), b,(n) is the nth Fourier coefficient of the jth non-holomorphic Maass wave form (suitably normalized), and a + Kj is the corresponding eigenvalue, and Ta(n) is a normalized divisor function
F, E
T.(n)
d
()a,
e
de=n
Sab is 1 for a = b, 0 otherwise, and (x2
1T i
g(t)= 2sinhirt f (J2it(2)
where JS(y) is the Bessel function. In the same notation q 2 vV-Wab)
-
g
2
K(-a, b; q)
(ab)
4 (ab) b,(a)b,(b)g(Kt) j=1
cosh?rK,
T,t(a)T;t(b)g(t)
1
+ 1rL
dt,
with
g(t)= 2 cosh1rt f Kh2it(x)g(x)dx, where
Khf(y) _
2
W
fo exp(-y cosh t)cosh st dt
IT
1
=
1
I
exp(-2(' +u))u-s
du
is a less familiar Bessel function, the Hankel function of pure imaginary argument (Jeffreys and Jeffreys 1962, cap. 21).
Subdivision without absolute values
483
The two transforms g(t) and g(t) are analogous, since we can express Khs(y) in terms of 13(y) - I_S(y). The Kloosterman sum is a discrete analogue of the definition of Bessel functions of integral order as integrals round the unit circle. We call Lemma 24.2.3 a trace formula, as the right-hand side can be regarded as the normalized trace of an intertwining operator in representation theory. The lemma is proved using the Fourier theory of functions on the hyperbolic plane invariant under the action of the modular group PSL(2, 7L)
as rigid motions, and the Poincare series coefficients enter as residues at trivial poles.
Bombieri and Iwaniec (1986a) suggest that a full solution of the Second Spacing Problem will involve Lemma 24.2.3. We have calculated directly with
the matrices rather than with their Fourier theory. There are three integers a, b, and q in the Coincidence Conditions of Part IV. Perhaps 3 X 3 matrices will be needed. The representation theory of SL(3, 71) is still being investigated.
Are we on the right road to showing that the discrepancy of a smooth closed curve of linear dimensions M is O(MI/z+E)? Subdividing the curve according to rational gradients must be right. Perhaps the number of Farey
arcs should be ME, and we take about 1/e terms of the Taylor series. Alternatively, the y vectors may be so uniformly distributed that the contri-
butions of different minor arcs cancel. The Kuznietsov trace formula is appropriate here because SL(2, 7L) is the automorphism group of the lattice
points, a natural part of the problem. Should the representation theory of SL(n,7L) for n >_ 3 appear in a two-dimensional problem? In all events, we have reached the final formula: `If you want any more, you must sing it yourself.
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2
Chandrasekharan, K. and Narasimhan, R. (1967). On lattice-points on a random sphere. Bull. Amer. Math. Soc., 73, 68-71. 6 Colin de Verdiere, Y. (1977). Nombre de points entiers dans une famille homothetique de domaines de R. Ann. Sci. Ec. Norm. Sup. (4)10, 559-76. 6 Corput, J. G. van der (1920). Uber Gitterpunkte in der Ebene. Math. Annalen, 81, 1-20. 2,5 Corput, J. G. van der (1921). Zahlentheoretische Abschatzungen. Math. Annalen, 84, 53-79. 5 Corput, J. G. van der (1922). Verscharfung der Abschatzung beim Teilerproblem. Math. Annalen, 87, 39-65. 3,5 Corput, J. G. van der (1923). Neue zahlentheoretische Abschatzungen. Math. Annalen, 89, 215-54.
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23 6.3
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18
Huxley, M. N. (1994a). The rational points close to a curve. Ann. Scuola Norm. Sup. Pisa Cl. Sci, (4) 21, 357-75. 4 Huxley, M. N. (1994b). A note on exponential sums with a difference. Bull. London Math. Soc., 26, 325-7. 13.3,19.1, 21.3
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1.1
Index
analytic coincidence conditions 315-20, 327, 339-46 approximate functional equation 111-13, 432,
436,441-3,447-8 arc of coincidence 313, 321, 327-8, 341-2 area in intrinsic coordinates 393 Atkin-Lehner newforms 210, 435 Atkinson's summation formula 451
degenerate cases 81 Deligne, P. 210, 425 Deshouillers, J. M. `452 diagonal solutions 236 terms 363 differencing 45-6, 65 step 115-19, 135, 185, 257, 259-60, 363-5, 383, 406, 416, 421-2, 461, 473, 474 Dirichlet approximation theorem 16, 83, 229, 259, 473
Baker, R. C. 473 Balasubramanian, R. 447 Bombieri, E. 2, 3, 4, 125, 158, 160, 289, 467-9, 478, 483
Bombieri-Iwaniec method 142-196, 235-361, 372-383, 479 Borel, E. 7 Brun-Titchmarsh theorem 122-5 Burgess D. A. 135
boundary condition 405 characters 100-101, 135, 236, 362, 432 divisor problem 2, 28, 239-40, 382-83 pigeonhole principle 16, 42-3, 48, 56, 335 series 2, 237, 432-3, 435, 438-441, 447 discrepancy of lattice points 33-41,128-141, 372, 384-8, 398, 474, 479, 483 of sequences 97-8
divide-and-conquer 2, 42-3, 48, 56, 81, 83, 115, 143, 158, 236, 240, 255, 263-4, 307, 311, 447,
Cassels, J. W. S. 116, 242 Cauchy, A. 8 inequality discussed 114-16, 121-2, 125 centre of Farey arc 145, 169, 186, 339 Chaix, H. 41 circle method 143, 152, 255-71, 470-2 see also Hardy, G. H. problem 2, 31, 203 coincidence conditions 156-7, 216, 231, 235, 272, 283-4, 287-346, 349, 416, 426-9, 478,
479
divided differences 46, 65 divisor concentration 240-1, 367 function 29, 203, 237-41, 367 problem 2, 28, 239-40, 382-83 duality of points and lines 73-4, 137-8 of bilinear forms 122, 224
483
detection 119-20, 255 commensurable subgroups 198
Erdos-Turan theorem 96-7 Euler constant 239-40
complementary function 72, 110, 321, 341, 345, 383, 415
product 113, 435, 438-41 Everest, G. 472 exponent pairs 151-2, 369-70 exponential integrals defined 87 evaluated 104-14 exponential sums bounded 99,353-61,365-71,415-22 defined 87 non-standard 475
congruence family 336-9,349,357,361-3,392, 478
continued fraction 16-19, 398 Corput, J. G. van der 4, 41, 43, 93, 116, 125 iteration 43, 62, 116-17, 135, 152, 383, 406, 474
lemmas 87-120, 221-2, 225, 448 counting squares 1-2, 25-31, 41, 131, 167 cusp forms 200 Davenport, H. 4, 101, 453
Faltings, G. 235
Index
492
families of solutions 241-48, 249-51, 272-83 of sums 179, 194, 219-20, 300-306, 333-4, 353, 357-61, 393-7, 410-13, 420, 424-9, 458, 478
Farey arcs 2, 143, 167-70, 185-7, 208-9, 255, 336-7, 372, 425-30, 445, 470, 473, 479-80, 483 sequence 8-10, 19-23, 208, 346, 439, 470, 477
see also major arcs, minor arcs
Fejer kernel 93-4 Filaseta, M. 45, 470 First Derivative Test stated 88, 104, 113-14, 448
First Spacing Problem 3, 56, 158, 235-85, 307,
372,406,471,476-7 flanks of major arcs 151 Ford circles 10 Fortean function 312, 324, 327, 342, 343 Fouvey, E. 475, 477
gardening 152,166,183-4,195-6,225,228-30, 412
Gauss circle problem 2, 31, 203 sums 100-4, 149, 162 Gel'fond, A. 0. 41, 135, 452 generating function 198, 438, 470 Gershgorin's eigenvalue bound 224 Good, A. 436, 439 Graham, R. L. 8 Graham, S. W. 4, 432 with Kolesnik, G. 2, 43, 116, 125, 152, 297, 383, 474
Halasz, G. 433 Halberstam, H. 122 Hall, R. R. 240, 399 Hardy, G. H. 2, 31, 122, 143, 203, 256, 439, 453
Hardy-Littlewood circle method, see circle method Harman, G. 473 Heath-Brown, D. R. 2, 4, 125, 185, 406, 444, 451, 464
height 7 Herz, C. S. 128, 474 Hlawka, E. 128, 405, 474 Hobson, E. W. 130 Hooley, C. 4 divisor concentration 241-2, 367
intrinsic equation of curve 24-5, 384, 388-393 inversion step 108-10, 116-17, 149, 176, 191-2, 193, 383, 422, 474-5 Ireland, K. 210
iteration between rational points and lines 76-80 by major arc method 166, 413-15 of points close to curves 43, 48-53 of resonance curves 370, 415-22 van der Corput's 43, 62, 116-17, 137, 154, 383, 406, 474 Ivic, A. 4, 440 Iwaniec, H. 2, 3, 4, 125, 167, 372, 451, 471, 475, 477; see also
Bombieri-Iwaniec
Jarnik, W. 30-33, 57 Jeffreys, H. and B. S. 150, 177, 203, 441, 482 Jordan curve theorem 25 Jutila, M. 2, 4, 208, 225, 284, 423, 435, 436, 437, 451
Karatsuba, A. A. 125, 160, 451 Kendall, D. G. 135, 474 Kloosterman refinement 471, 479
sums 481-3 Koebe's function 440 Kolesnik, G. 2, 4, 255, 297, 339, 416, 474, 478, see also Graham S. W. Kratzel, E. 4, 43, 474, 475 Kuznietsov, N. V. 290, 439, 481-83 L-function 435-7, 440-1, 451 Landau, E. 31 large sieve 3, 120-125, 126, 166, 214-20, 224-5,372,470 applied 152-8, 178-84, 193-6, 231, 329, 475-6
lemmas 120-25, 224-5 large values results 126, 431-2 Lax, P. D. 440 limitation results 126, 248-9, 334-5, 474 Lindelof hypothesis 441 Linnik, Yu. V. 41, 122, 124, 135, 454 Liu, H.-Q. 452, 475 Logan, I. 398 magic matrix 289-308, 320-1, 329-335, 415-18, 423-8, 430 major arcs in Bombieri-Iwaniec method 148-51, 160-66, 168-72, 186-8, 377, 409, 480 flanks of 151 in Hardy-Littlewood method 143, 255, 256-7, 262-68, 270 long 160-66, 356, 413-15, 450 for S-suits 473 Sargos splines 469 as sides of polygon 55, 67-72 Mangoldt, H. von 438, 453, 464 Matthews, K. R. 125 mean-to-max 117-18, 122, 135, 136, 185, 410
493
Index
Mellin transform 91, 117-18, 181-2, 195, 438-40 Minkowski, H. 242 minor arcs in Bombieri-Iwaniec method 143-9, 151, 168-70, 172-8, 186-7, 188-93, 336, 377, 410, 414, 480
in Hardy-Littlewood method 143, 255, 256-62, 270 Sargos splines 469
as sides of polygon 55, 67 MSbius function 13-14, 63-4, 123-5, 439, 454 transformation 199 Modification One 147, 149, 168, 172, 315 Two 147, 149, 156, 169, 181, 186, 195, 351, 352, 374, 376 Three 169
reflection step, see inversion step resonance curves 320-21, 345, 415-19 Riemann hypothesis 15, 210, 435-6, 439-40, 447
zeta function 2, 3, 64, 111-13, 160, 237, 432, 435, 438-451
Riesz interchange 218, 246, 267, 330-2, 405, 457
Rivat, J. 452, 475 Rosen, M. 210 row-of-glasses function 95 row-of-teeth function 2, 33, 93, 131, 167, 379-81 rounding error function 2, 33, 93, 131, 167, 379-81 sums 93-8, 131, 167, 363, 372, 379-83, 385-8,397-405
modular group 11-12, 197, 289, 439-40, 481, 483 forms 2, 3, 197-9, 256, 290, 435, 451, 481-83
Sargos splines 469
monomial function 50-2, 117, 165, 369, 414, 416,473,475-7 Montgomery, H. L. 115, 213, 455 Mordell, L. J. 440 Moreno, C. J. 210, 425, 437 Motohashi, Y. 4 Mozzochi, C. J. 2, 4, 167, 372
Second Spacing Problem 3, 158, 287-346, 363, 372, 476, 483 Selberg, A. 4, 481 sieve 122, 125, 439
Nowak, W. G. 4, 135, 474
Sierpinski, W. 41, 167, 372 sieve
trace formula 439-40 zeta function 439-40 Shahidi, F. 210, 425, 437 short interval means 118-19, 221-4, 226-7,
408-10,435,443-7 general 454
O'Hara, F. 398 partial summation lemmas 89-93, 181-2, 225, 408
Phillips, R. S. 440 Piatecki-Shapiro theorem 452, 463, 475 Pila, J. 467-9 Pisot, Ch. 116 Poisson summation formulae 93-104 used 2,103,109,110-12,149-50,173-8,
large, see large sieve Selberg's 122, 125, 439 simple asymptotic 63 simple asymptotic sieve 63 Simpson's rule 397 stakanchik 95 stationary phase integrals 105-8, 133-4, 150, 177, 212, 263
Swinnerton-Dyer, H. P. F. 4, 30, 43, 45-6, 49, 50, 54-62, 197, 467
189-93, 257, 260, 263-4, 320, 363, 381,
383,409,419,432,438,474-5 polynomial approximation 2, 142-6, 161, 173,
188-9,384,469 prime numbers 1, 15, 122-5, 438-9, 447, 452-64, 473
Tenenbaum, G. 240, 399 trapezium rule 31, 98, 128-9, 397 Trifonov, O. 4, 470 Type I double sums 115, 455-60 Type II double sums 115, 213, 455, 459-62, 475
Rademacher's binary decomposition 180 Ramachandra, K. 446 Ramanujan's hypothesis: bounds 210 hypothesis: multiplicativity 440 sum 13-14, 123, 125 work on partitions 143, 256 Rankin R. A. 4, 200, 210, 425, 433, 436, 451
reference fractions 19-20, 308, 312, 329-30, 332
Uchiyama, S. 180 uniform Diphantine approximation 19-23, 221 uniform distribution 12-15
van der Corput, see Corput, J. G. van der Vaughan, R. C. 115, 143, 152, 256, 454, 455, 464, 471
Index
494
Vinogradov, I. M. 4, 41, 93, 143, 152, 256, 454,
Wright, E. M. 453
471
Voronoi, G. 4, 41, 128 polygon 2, 40-1, 167, 372 summation formula 128, 203, 451
x-vectors 154, 178, 193, 231 y-vectors 154, 178, 194, 231
Waring's problem 3, 152, 471 Watt, N. 2, 4, 20, 241, 246, 272, 297 Weil, A. 199, 481 Weyl's criterion 13, 96-7 Wilton summation 199, 200-10, 451 translated 440 used 210-13
Zaharescu, A. 8 zeta functions 440-1 Minakshisundaram-Pleijel 440 Riemann 2, 3, 64, 111-13, 160, 237, 432, 435, 438-451 Selberg 439-40
