Exponential Sums and Their Applications

,

=

-

Hence by Theorem 5 it follows that

v'd=l JP �

P " ' 4X n L e2 I p

x=l

=

(d

-

1 )p

.

� (d - I )JP .

( 69)

The inequalities (69) show that under fixed n and increasing p the estimate obtained in Theorem 5 has the order which can not be improved. Moreover, it can be shown [19] , that in the estimate 63 it is impossible not only to improve the order, but to replace (d - 1)v'P by the quantity (1 c:)(d 1 ) v'P under any c: > 0 either. Now we shall consider sums with a prime-power denominator.

( )

-

-

LEMMA 4 . Let 2 � 0' � n, p be a prime > n, (at , . . . , an , p) al X + . . + a n x n . Then the following estimate holds:

=

1, and

f(x)

=

.

pQ 2"i f( x) L e pQ x=l

� (n _ l )pa - l .

y and z run through complete residue sets modulo pa - l and p, respec­ tively. Then the sum y + pa-l Z runs through a complete residue set modulo pa and, since 0' 2 and p > 2, we have

Proof. Let

;;;::

[ Ch. I, § 4

Complete exponential sums

26

Therefore,

pQ e2 11'i pQ

f(x )

L

x=l

pQ - I

2 11' i f(y+ p Q - I z)

P

L Le y=I z= l f' (y) z pQ""'- I 2 11'1. f ( y) ""' 211'1. -= w e Q we p z=l y= l (y) f . 2 11'1 =p L e y= l

pQ

=

P

p

P

Q-l

pQ 8p [f' ( y)] .

(70 )

But then f(x)

pQ e pQ L 2 11' i

�p

x=l

pLQ - I 8 [J' (y)] = pOl - l 8 [J' (y)] = pOI-I T, L p p P

y= l

y= l

(7 1 )

where T is the number of solutions of the congruence

f '( y) == 0

( mod

p).

Since (a l , ' . . , a n , p) = 1 and p is a prime > n, then at least one of the coefficients of the polynomial f '( y) = a l + 2a2Y + . . . + na ny n - l is prime to p and, therefore, T � n - 1 . Substituting this estimate into ( 71) , we obtain the lemma assertion. It is easy to improve this result for polynomials of a special form. Let p b'� prime, (a, p) 1, and pQ 2 11'i �

=

S(a, pOl) = L e x=l

Let's show that under

a

� 2 and

pQ

n � 3 the following equalities hold:

S(a, pOl ) = { �:=�S ( a , pOl -n )

if 2 � a � n and if a � n + 1 .

Indeed, from (70) it follows that

l 2 11'i p p QL S( a, pOI ) = P e Q 8p ( nay n -l ) . a y"

y= l

Hence under

(n, p) = 1 we get

the first equality of ( 72 ) :

(n, p) = 1 ,

(72)

Ch. I, § 4)

Simplest complete sums

27

be the greatest power of which divides Then, Now let a � + 1 and +1 � - 1)(3 + 2 and considering separately the cases using the estimate a � a � 2 (3 + 2 and + 1 � a � 2(3 + 1 we obtain

p, n pfJ pfJ (p n (y pcx - fJ -lzt y n npa - fJ-l y n -lz (mod pa ). +

==

n.

+

Therefore ,

y=1 -fl l 11";%=1ayB a2:- 2 p p fJ 1 = p + y= l e Q Dpp + l (anyn - l ). Hence, since (an,pfJ+ 1 ) = pfJ, we have pQ2:-fl-1 2 11" ; ayBp 1 S( a, pCX) pfJ+ y=l e Q Dp( y) a-fl-2 p 2 11"; paya -B 2: +1 pfJ y=1 e B = pn -1 S(a,pcx - n ). =

=

Thus the assertion (72) is proved in full. THEOREM 6 . Let sum

the estimate

n and q be arbitrary positive integers and ( a, q) = 1 . Then for the S(a,q) = :c=2:q 1 e2 1I"'' -qa:c-B (73

holds.

Proof. Since 1 , then under 1 and Lemma 2 and Theorem 3 , respectively:

(a, q) =

n=

2:q e211"; �q = 0, :c=1 Therefore it suffices to consider the case

)

n 2 the estimate (73) follows from 2:q e2 1I" ' -axq-2 � y'2q . :c=1

n � 3.

=

[ Ch. I, §

Complete exponential sums

28

At first we shall show that for any prime p under

a

� 1,

4

n � 3, and (a, p) = 1 ( 74 )

where

Cp (n)

Indeed, under

Let 2 �

a

if p < n6 , if p > n6 •

1 by (60)

a =

� n and

= {�

=

(n, p) p .

if if Then p �

p < n6 , p > n6 .

n and, using the trivial estimate, we get

IS( a , pa ) 1 � pa � pa ( 1 - � ) p � npa ( 1 -� ) . Let finally 2 � a � n and (n, p ) 1 . Then by (72)

=

=

IS(a, pa ) 1 = pa - 1 pa ( 1 - !; ) � pa ( 1 - � ) . ( 74) is satisfied under 1 � a � n . Apply

Thus the estimate the induction. Let under 1 + (k - 1 )n � a � kn this estimate be valid for a certain k � 1. We shall show that the estimate is valid under 1 + kn � a � (k + 1)n as well'. Since it is plain that 1 + ( k 1 )n � a n � 1 + kn then using the equality (72), ' by the induction hypothesis we get -

-

Thus the estimate (74) is proved for any a � 1 . Let now q = pr1 p�. be the prime factorization of q. Using the multiplication formula (34), we obtain •

•

•

n q ""' 211'1 axq L.J e .

-

Then, obviously,

p� .

""'

L.J e

2 11'i

ab.x n p". •

(75)

,,=1 x.=1 are coprime with p" . We determine a" with the help of the x=1

where the quantities b " equalities

=

II8 (v

( a " , p,,) = 1 and by (75)

=

=

1 , 2 , . . . , s).

S(a, q) S( a 1 , prl ) . . . S( a8 ' p�· ) .

Ch. /, § 5)

Mordell's method

29

Hence, using the estimate (74) and observing that the number of primes less than n6 does no t exceed n6 , we get the theorem assertion: I S(a, q) 1 �

Note that under n

>

al ( l - � ) a . ( l - � ) n 6 1 _ 1n . . . . Cp . (n)p s �n q Cp 1 (n)P 1 2 , q = pn and (a, p) = 1 by (72) for any prime p pn 2 .,.. i axn '" e p n = pn - 1 = pn ( 1--n1 ) . L...J x=l

Therefore, in this case

1 = q _ 1n .

S(a, q)

Thus under fixed n and increasing q the order of the estimate (73) can not be im­ proved. Let f(x) = a1 X + . . . + a n x n , (a1 , ' " , a n , q) = 1 and Seq) be a complete rational exponential sum of the general form Se q )

q

= Le

2

. In

f(qx)

-

.

(76)

C(n)l - n ,

(77)

x=l

In Theorem 6 the estimate I S(q) 1 �

1

where C(n) = nn 6 , was proved for polynomials of the special form f (x) = a n x n . With the help of the significant complication of the proof technique, Hua Loo-Keng showed that under certain C(n) the estimate (77) is valid for arbitrary complete rational sums (77) as well. A proof of an estimate close to (77) can be found in [16] and [44] . § 5. Mordell's method Let us consider a complete exponential sum with a prime denominator S(p)

P

= Le

2 1r i

al x+...+an x

n

p

x=l

Mordell [36] proposed a method of such sums estimation based on the use of prop­ erties of the system of congruences . n . .. .. . x f + . . + x : == y f + . . . + y :

�� �. : .. �.�.� � ��..�.... : .. � � . .

.

.

.

.

}

(mod p),

( 78 )

Complete exponential sums

30

[ Ch. I, § 5

where p is a prime greater than n and the variables X l , , Yn run through complete residue sets modulo p independently. First of all we shall prove a lemma about the number of solutions of a congruence system of a more general form. •

•

LEMMA 5 . Let q1 , . . . ; qn be arbitrary positive integers, q be the number of solu tions of the system of congruences

��. .�. : . '. ".�.�.�..�.�� .�.:.'. ".�.�.� ....�:��. .��.� . x f + . . . + x� == y f + . + Y k (mod qn ) 71"1 ( Le .

.

Then

.

9

2

.

•

=

LCM ( q1 , " . , q n ) an d Tk

(79)

}, a1 x + + an xn ... 91 9n

) 2k

x=l

Proof. Since the product

equals unity, if numbers X l , otherwise, then, obviously,

•

L q

Tk =

•

•

, Yk satisfy the congruence system (79) , and vanishes

.

8q 1 ( X l + . - Y k ) .

X l , . . · , lIk = l

. . . Oq" ( x f + . . . - Yk ) '

Hence, using Lemma 2, we get the assertion of Lemma 5:

Tk = q1

1 . qn

--­ .

.

q

Le

2 71"1.

(

a1 x

91

+...+

an x qn

n

) 2k

x= l

In particular, under

Ie

=

n and q 1 = . . . = qn = P ·it follows from Lemma' 5, that

1

Tn == -;

P

where Tn

is

P

P

L L

4 l , • • • ,a n =1

%=1

e

2 ' a 1 x+ ...+a n x

71" 1

the number of solutions of the system

P

(78).

n

2n (80)

LEMMA 6. Under any n � 1 and a prime p > n , the number of solutions of the system (78) satisfies inequality Tn � n ! pn .

Ch. /, § 5]

Mordeli's method

31

Proof. Let .A 1 , . . . , .A n be fixed integers, 0 � .AI' � P - 1 , and let T ( .A1 , " " .A n ) be the number of solutions of the system of congruences

��. �.:.'. :.�.�.� .�. .�.

x

We shall show that

1

.

.

f + . . . + x : == .A n

1 � Xv � p.

(mod p)

}

(81)

(82) Indeed, we introduce the following notation for the elementary symmetric functions nd a the sums of powers of quantities X l , . . . , X n : 0' 1 X l + . . . + X n , . . . , O' n = X l . . . X n , 81 = X1 + " , + Xn , . . . , 8n = xf + . . . + x: . =

Let X l , . . . , X n be an arbitrary solution of the system (81). Then, obviously, 81

==

.AI , " " 8 n == .A n

and using the Newton recurrence formula vO'v = 8 1 0'1' -1 - 8 2 0'1' -2 +

under v

=

1 , 2, . . . , n we have vO'v

==

.

(mod p ),

.

. T 8 1' -1 0'1 ± 81"

.A 1 0'v - 1 - .A20'v -2 + . . . T .Av- 1 0'1 ± .Av

(mod p).

(83)

Since p is a prime greater than n , then (v, p) = 1 and the congruence (83) is soluble for 0'1" From (83) we get successively (mod p)

0' 1 == J.t1 " " , O' n == J.tn

(0 � J.t v � p - l ) ,

where the values J.t1 , . . . , J.tn are determined uniquely by setting quantities .A I , . . . , .A n . But then every solution of the system (81) coincides with one of the permutations of the roots of the congruence n X n - J.l.1X -1 +

•

.

.

± J.l. n == 0 (mod p)

with fixed coefficients and, therefore,

T(.Al ' " . , .An ) � n ! .

Now, since Tn =

p L

T( Y1 + . . . + Yn , . . · , y f + . . · + y� ) ,

yl . . . . . y n = l

we get the lemma assertion:

Tn �

p

L

yl . . . . . yn = l

n! = n! p n .

[ Ch. I, § 5

Complete exponential sums

32

Note. From this lemma and the equality (80) it follows immediately, that under any � 1 and a prime p > n the following estimate holds:

n

p I: al t···,an =1

THEOREM 7 . Let al X +

.

.

.

+ an x n .

� 2, p be a prime greater than Then

n

=

1 and f( x )

=

'

f{ x ) 2 I p- ::;; p1 n e "" I: P

(a l , " " an , p)

n,

n

-

.

x=1

Proof . At first we shall consider the case (an ' p) = 1 . Let integers >. and ft vary in the bounds 1 ::;; >. ::;; p 1 , 1 ::;; ft ::;; p. Arrange the polynomial f (>.x + ft ) in the -

ascending order of powers of x

(84) and observe that and Denote the number of solutions of the system (85) by H ( b 1 , , bn ) . It is plain, that H(b1 , , b n ) does not exceed the number of solutions of the system made up of the last two congruences of the system (85): •

•

•

•

and, therefore, since (nan , p)

=

.

•

1 and (>' , p) = 1 ,

(86) B y (25) for complete sums the equality

p 2 ,.., f{ x) ""' L.J e p •

x=1

2n =

P 2 ""' L.J e x=1

'

1ft

n f{>'X+/l ) 2 P

Ch . I, § 5]

Mordell's method

33

>. and j.L, we have p - 1 p '"' p 1. /( '\ x+ p ) 2n p 2 1rl. /( x) 2 n L: L: 2 1 p > L...J e 1r p I p p 1 ) e - '\=1 p=1 x=1 x =1 p . b 1( '\,p)x+ ... +b n ().,I')x n 2n p-1 P 1 p '"' L...J e21r 1 pep - I ) {; � x=1 G rouping the summands with fixed values b 1 ( >., j.L), , bn(>.. , j.L ) and using the esti­

holds. H ence by (84) after the summation with respect to =

=

.

•

.

mate (86), we get

=

Hence by the note of Lemma 6 we obtain the theorem assertion for the case 1:

(an, p)

p ./( x ) 2 n n n ! 2n n 2 n 2 n -2 '"' 21r' L...J e P � p ep - 1 ) p < p , x=1 P 1 ' / ( x) L: e 2 1r ' -p- < np1 - n . x=1 Now we show that the general case ( a 1 , . , a n , p ) = 1 can be reduced to the case when leading coefficient of the polynomial is prime to p. Indeed, let (a8 , p) 1 and a 8 +1 == . . . == an == 0 ( mod p) , 1 � � Then we obtain P 1r ' !( x l '"' � alx + ...P+ a.x· L...J e 2 1 P L...J e21ri x=1 x=1

=

.

.

s

n.

=

The theorem is proved in full.

=

Note. A substantial improvement of Mordell's estimate was obtained by A . Weil [48] , who showed, that under prime p > n and (an , . . . , an , p) 1 the estimate

� 21ri atx + ... +a n x n L...J e

x=1 is valid.

P

� (n - I ) vp

§

[ Ch. I, § 6

Complete exponential sums

34

6 . Syste ms of congruences

One of the main points of Mordell ' s method (§ 5) is the use of the estimation for the number of solutions of the congruence system

�l . � : � '.� � � .� �� � . .. : �.�n. x f + . + x� yf + . + y� .

.

==

.

.

}

(mod p ) ,

.

where p is a prime greater than n. Hereafter, congruences of the same form but with respect to distinct moduli being equal to growing powers of a prime p will be of great importance. For the first time such systems of congruences were applied by Yu. V. Linnik [34] for the estimation of Weyl's sums by Vinogradov 's method. LEMMA 7. Let

n

� 1 , k � n( n4+1 ) , p be a prime greater than n, and let Tk (pn ) be

the n um ber of solutions of the system of congruences

l

. Yk � .� : '. � � � .� �� �.' ' : � . . . �����).n x f + . . + xi: yf + . + Yk (mod p ) ' .

.

Then

.

==

.

.

(87)

},

l=

Proof. Under n 1 the lemma assertion is evident, so it suffices to consider the case n in Lemma 5. n � 2. Take q = p, . . . , q n = p

Then we obtain

We split up the domain of the summation over

where the summation in

al, . . . , an into two parts:

L: l is extended over n-tuples al . .

.

a n which sa�isfy

and for L: 2 only those n-tuples are taken into account , for which at least for one of l v in the interval 2 � v � n p ,, - is not a divisor of a" .

Systems of congruences

Ch. /, § 6 ]

In the first case, determining b1 ,

•

•

•

35

, bn with the help of the equalities

p n 2 ' ( atz + . .. + a n z n ) p n 2 1' b t z + . . . +b n z n l: e p pn = � p L... e 11'

we get

11'1

z=1

z=1 Therefore

p n e2 11" ' ( aptz + .. . + a n ", n ) pn � L... ",=1

2k

= p2nk-2k

and using the note of Lemma 6, we obtain

an zn ) 2k p n e 2 11' 1 ( � p + + pn L ",=1 n p 2 11' i bt ",+ . ..p+ b n ", 2nk-2n l: � e �p L... bt ... . . b n =l z=l •

...

In the second case, there exists an integer

pn - l � a v p n - v . Therefore,

( a l P n -1 , a 2 Pn-2 , where 2 �

a

•

•

.

v

2n

� n! p2nk .

in the interval 2 �

, a n , pn ) = p n-I> ,

� n. But then

and using Lemma 4, we get

� e 2 11' i b p: + ...p O+ b n ", n

= L...

z=l

= pn - I>

� e 2 11'i btz + ..p.O+b n z L...

",

=1

n

v

(88)

� n such that

36

Complete exponential sums

Hence, since k � n ( n/l ) , it follows that p " 2 11"'. � e L..J

( �p +

.•.

+ a " x" p"

( Ch. I, § 6

) 2k

x=l

(89 ) Now, observing that under n � 2 n 2 k _ ( n _ 1 ) 2 k � 2k ( n - 1 ) 2 k - l

> n ( n - l ) n-l

� n!

from ( 88 ) and ( 8 9 ) we obtain the lemma assertion:

l) l) 2 n k - n ( n+ 2 n k - n ( n+ n 2 2 � !p + (n _ l ) 2 k p n(n +l) 2 n k - -2 � n 2 kp

Note. Let Tk ( P ) denote the number of solutions of the system ( 87 ) , when the domain

of variables variation has the form 1 �

Xj

� P,

1 �

�P

Yj

(j = 1 , 2, . . . , k ) .

If m is a positive integer, then under P = mpn

Indeed, using the complete sums property ( 26 ) , we get

(

mp �" 2 11"'. � + + a " x" p p" e L..J x=l •••

)

p 2 11" ' = m l: e

Tk ( mp n ) = p

n(n+ l ) p -l: 2

= m2 kp-

p mp . . . l: l: e 2 11" ' "

2

"

x=l p"

l: . . . l:

l) p n (n+-

)

(

) 2k

,

x=l

and, therefore, -

(

. a1 x + + a n x " p ... p n

"

(90)

. �+ + a n "x " p . .. p

Systems of congruences

Ch. I. § 6 )

37

Hence, since by Lemma 7

we obtain the estimate (90). run Let E�:, ... , x n denote the sum, in which the summation variables through complete residue sets modulo and belong to different classes modulo

Xl , , Xn . p L EMM A 8 . Let p be a prime greater than n , � 2, and I ( x ) = al x + . . . + a nx n . Let SOl ( al, . . . , a n ) be defined with the help of the equality . !( X l) + ... +J( x n ) 71' 1 '" 2 SOI( al , . . . , an) = pOi

•

•

•

Q

p

O

L...J

po

e

Then S01

( a1 , . . , an ) -- { p(OI - l0) n Sl ( bl , . . . , bn ) ·

a" = pOl- l b,,

if otherwise.

(v = 1 , 2 , .

. . , n),

Proof . Let us change the variables (v = 1 , 2, . . . , n ) .

Xl , . . . , Xn

Since by the assumption the quantities belong to different Classes, then the belong to different classes modulo p as well. Therefore, using quantities that

Yl , . . . , Yn

we obtain

Zl' . " , %n= l

Yl, . ·, Yn

(9 1 ) Since f'(y ) = a l +2 a 2 Y +" .+n a n yn - l , then under prime p > n and ( a l, . . . , a n ,p) = 1 the congruence f' ( y ) 0 (mod p ) can be satisfied by at most - 1 values of y from different classes modulo p. In the sum (91) the quantities Yl, . . . , Y n belong to different classes and, therefore, if ( a l, . . . ,a n ,p) = p, if ( a l , . . . , a n , p) = 1 . Y l ' ' ' ' , Yn

==

n

Complete exponential sums

38

But then by (91 ) the sum then

[ Ch. I, § 6

So-(at. . . . ,an ) vanishes under (at. . . . ,an ,p) =

(at. . . . ,an , p) = P,

1-.

If

yt .. · · ' yn

Thus

l ) if (al , . . . ,an ,p) = P t n p{ p , . ,a (a S . . lP _l n o(92) S ( al, · · . , an ) - 0 ·f ( a I , . . . , a n , P ) 1 Applying the equality (92) to So- - 1 (a lP - I , . . . , a nP - l ) we get (al , . . . ,an ,p2) = p2, S (at. . . . , an ) - { P2ns00- -2(a lP-2, . . . , anP-2) ifotherwise. _

0-

=

1

.

0-

Continue this process. Then after

a-I

step we obtain the lemma assertion:

So- (al, . . . , a n ) · f ( at , . . . ,an ,p0--1 ) = p0--1 , = { p( o- - � n 51 ( alP- ( o- - I ) , . . . , anP- ( o- - I») otherwise . . LEMMA 9 . ( Linnik's lemma ) . Let AI, . . . , A n be fixed integers, p a prime greater than and let T*(Al' . . . ' A n ) be the number of solutions of the system of congruences � 1 . � � •. • . � �� .� �� . ( �o d �) . .. n xf x: A n (mod p ) where the variables run through complete residue sets modulo p n and belong to different classes modulo p. Then n n- l ) -T*(A l , . . . , A n ) :::; n!p ( 2 n and according to the notation x a Proof. Let f (x) = alpn - l x a2p n -2x2 n of Lemma 8 1

n

},

+...+

==

+...+

+

%1 ,

Using Lemma 2 we obtain

p n

.. · , X n

n

T* ( A l , . . . , A n ) =

L II %1,

... , X n

v= l

Spv(x� . . . +

+ x� -

All),

Ch. I, § 6 )

Systems of congruences

39

where the summation with respect to al , " " a n is extended over the domain 1 � al � p, . . , 1 � an � pn . Hence observing that x�+ " ,+x� p I' 211"1. a t X l +"'+X" + .. . + a " P '"'" e pI' � X l ,···,Xn 2 11" i f(X l )+ . . . + f(xn ) pn n = 2:: e p = Sn (a l pn-l , . . . , a n ) , X l " ",Xn we have

)

(

.

T* ( A l " . . . , A n ) , n(n+ l ) p pn i a l Al + ... + a n An -211" - --'"'" S ( a n- l , . , a ) e p pn 2 =p .. n L � n lP a l=l a n=l Determine the quantities b l , , b n with the help of the equalities a l = b 1 , a 2 = pb2 , · · · , a n = p n- l b n '

(

. .

•

According to Lemma 8

Sn (a l pn - l , . . . , a n ) n(n 1 ) Sl ( b l , . . . , bn ) = p

{

�

if a" = p,,-l b ,, ( v = 1 , 2, . . . , n ) , otherwise,

where X l " " , Xn

Therefore, n(n-3) p - -2- T* ( Al , " " A n ) p L S1 (b1 , . . , ... bt ,bn = 1 P

L

P

L

e

.

2 . , bn ) e- 11" 1

b I At + ... +bn An p

. . . ·+bn (x�+ . . . + x� -An ) 2 11" i b l (XI + . . +xn -Ad+ P

Now, using Lemma 2, we obtain

n(n- l ) 2T*(A l , . . . , A n ) = p-

L II 8p ( xr + . . . + x� - A ,, ) P

Xl " " , Xn n(n-l ) p �p 2 L

=p

n

v= l

n

II 8p(xr +

n(n-l) 2 T(A l , . . . , A n ) ,

.

. . + x � - A ,, ) ,

)

.

where

[Ch. I, § 6

Complete exponential sums

40

T(Al , ' . . , A n) is the number of solutions of the system of congruences 1 ::;; X II ::;; p.

Hence, because

T(Al " . , A n ) ::;; n!

by (82), the lemma assertion follows:

.

n ( n- l )

T* (A l " . . , A n ) ::;; p-2- T(A l , ' . . , A n ) ::;; n! pn ( n- l) / 2 . COROLLARY . Let T� (mp n ) be the number of solutions of the system of congruences xj , Yj ::;; mpn , i i- j => X i ¢. Xj , Yi ¢. Yj (mod p) . 1 ::;;

Then (93)

Proof. Since each variable among X l , , Y n runs through a complete residue sys­ tem modulo p n m times (under the additional conditions i i- j => Xi ¢. x j , Y i ¢. Yj ( mod p )) , then using the lemma we obtain •

= m 2n

.

•

pn

L

Yt ,· . .

,Y n

2

::;; m 2np n nI p

T* ( Yl + · · · + Y n , · · · , Yr + · · · + Y� )

n(n - l ) 2

--

= n! m 2n p

2 n +l) 2n - (n 2

--

§ 7. S ums wit h exp onential function

Let form

a be an integer, m

� 2 and

q

� 2 be coprime positive integers. Sums of the P

S( P)

=

a q'"

L e 2 7U m

%=1

.

are called rational exponential sums containing an exponential function. In the inves­ tigation of such sums we shall need some properties of the order of q for modulus, m.

Sums with exponential function

Ch. I, § 7)

41

Let P be a prime, m = pm1 ' T and T1 be the orders of q for moduli m and m1 , res pectively. We shall show, that if T of. T1 and p\m1 , then the equality (94) hol ds. In deed, since m1 \m, then from the congruence qT == 1 ( mod m) we get qT == 1 ( mod m1 ) and, therefore, T1 \T. On the other hand, from the congruence qTl == 1 ( mod m1 ) we obtain qTl = 1 + U 1 mt , where U1 is an integer and m1 is a multiple of P by the assumption. But then qP T!

( 1 + u 1 m dP == 1 ( mod m)

=

and T\ PT1 ' Since T1 \T, T1 of. T and P is a prime, the equal h y (94) follows:

=

1 TT1- 1 \ P, TT1- = p, T PT1 . Let now m be odd, m pr 1 p� . be the prime factorization of m, T and T1 be the orders of q for moduli m and PI . . . Ps , respectively. We determine the quantities f31 , . . . , f3s with the help of the conditions q Tl 1 - U 0 pf31 1 . . pf3• (95) ( u o , PI . . . P s ) 1 . s

=

_

•

•

•

-

=

.

For definiteness we suppose that in the prime factorization of m those primes, which satisfy the inequality av > f3v , are put at the first r places (0 � r � s ) , so av > f3v under v � r and av � f3v ur�der v > r. Further let m 1 = p� l

. . . p�r p�.;:.r . . . p� • .

From the definition of T1 and the equality (95) it follows that the order of q for Ip.odulus mi is equal to 7"1 and Let us show the validity of the equalities qT

=

1 + u m,

(u ,

PI . . . Pr)

=

1 and 7"

m m1

1

= -T

(96)

.

Indeed, let m2 = pm1 ' where P is any number among the primes PI , . . . , Pr ' Let '7"2 denote the order of q for modulus m2 . Obviously 7"2 of. 7"1 ( for otherwise we would have m2 \qTl - 1 and pm1 \U1ml , which contradicts the condition (U1 , PI . . . Pr) = 1). Since, besides that, p\ml , then by (94) 7"2 = P7"l . But then qPTl q T2

where U2

==

=

=

U 1 ( mod Pl

( 1 + U 1 m1 )P 1 + U2m2 ,

==

1 + U1m2

( mod p1 " P m2 ) ,

. . . Pr), and, therefore, ( U2 ' Pl " . Pr)

'

r

= 1.

Thus

Repeating this process av - f3v times with P being equal to each Pv ( v we obtain the equality (96) .

=

1 , 2, . . , r ) , .

Complete exponential sums

42

[ Ch. I, § 7

THEOREM 8 . Let m � 2, p be a prime, m = pm 1 , T and 71 be the orders of q for moduli m and m 1 , respectively. If 7 '" 71 and p2 \m, then under any a not divisible by p Z

aq m - 0. 2: e211'i x=l T

(9 7 )

Proof . Let T denote the number of solutions of the congruence q

X

== q

( mod m ) ,

Y

1 �

X, Y

�

T.

U sing Lemma 2, we obtain

a q_'"_-=-qU-,-) e 211' i _(.:..:: m q'" 2

m a 1 2: � e 211' i ­ m L... .J m a=l x=l T

=

On the other hand, obviously, T = 7 . But then

m aq'" 2 211' i 2: e m 2: a=l x=l T

=

mT =

(9 8 )

mT.

Therefore, b y (94)

a q '" 2 211'i m e L....J ( aa=l , p)=l x=l m = 2: a=l m

Hence for any

a

T

�

m � 211'i a q"' 2: L... .J e m a=l x=l ( a ,p) = p T

2

not divisible by p we obtain the theorem assertion T a q'" 2: e 211't. m -- 0 . x=l

Let us show that if at least one of the theorem conditions

Sums with exponential (unction

Ch. I. § 7]

43

is not satisfied, then the sum (97) might be not equal to zero. Indeed, let p > 2 be an arbitrary prime, m = 2p, ( a, m ) = 1 , and root of 2p. Then we have

11' . a q" '"'" m .L..t e 2 1 x=1

q

be a primitive

a ( 2x -1) p 211'1. ---2p = -e 211' i �2 1 . L e = x=1 x� P+2 1 In this example the condition T i- T1 is obviously satisfied, but p2 � 2p , and the second r

=

condition is violated. Let now m = p2 , 9 be a primitive root of p2 , and using the equality (98), we obtain

q

= fl . Then

T

=

P

-

1

and,

p 2 p - 1 211'i agP"2 2 a q" 2 11'i 2 m 1 L Le p >.: max L e p(p - 1 ) a=1 ( a, p )=1 x=1 ( a,p)=1 x=1 a 1g " 2 p 2 p - 1 2 11'i agP"2 2 P P -1 211'1. --1 1 '"'" L Le p -:---. � L e p = p - 1 , p( p - 1 ) a=1 x=1 p( p - 1 ) a 1 x=1 211' i a q" max L e m � JP::J. . ( a, p )=1 x=1 r

?"

=

r

In this case the condition p2 \m was fulfilled, but T = T1 . ; Another form of conditions, under which the complete sums S( T ) vanish, is shown in the following theorem. THEOREM 9 . Let m = pf1 . . . p�. be prime factorization of odd m ,

T be the order of for modulus m and the quantities P1 , . . . , Ps be determined by the equality (95). If 0 there exists v such that av > PI' and a t= ( mod p�. -p. ) , then

q

a q" '"'" e211'i m .L..t x=1 r

Proof. Chose that value

0

= .

v , which satisfies the conditions

and write a in the form a = pOl. -p. - 'Y a 1 , where 'Y � 1 and ( a ' , PI' ) = 1 . Let m = p�. - P. -'Y m', m' Pv m " , T ' and T " be the orders of q for moduli m' and m il , respectively. Since p�' \m, then pe · +'Y\m' . But then pe ' \m " and by (96 )

=

T

,

=

m' m' " - T1 = -il T " = P v T . m1 m

Complete exponential sums

44

From the divisibility of m' by pe p + 'Y it follows also, that m' = P v m " ,

r

'

Therefore, by Theorem 8

Since q r

f.

r

"

[ Ch. I, § 7

p� \m' . Thus

p! \m' and (a' , pv )

,

=

1.

a ' q�

r'

m' = 0 . l: e211'i -

x=1

==

1 (mod m) and m'\m, then q r

1 (mod m') and

==

Now, using the property (26), we obtain the theorem assertion: r

a q�

� 211'i m

L..J e x=1

T

=

� L..J

x=1

a' q �

e 11'1 -, = 2

•

T

a ' q :D

211'1 L..J e m , x=1

r'

r

m

�

'

.

=

r

'

is a divisor of

r.

O.

Note that the Theorem 9 requirements can be relaxed, namely, the condition of m being odd may be omitted. In order to prove that it suffices (see [32] ) in determining the quantities PI , . . . , Ps to use the equality q (I'H) r1

_

1

l3t fJ. - u0 p 1 . . . p s ,

where J-I. = 1 , if m == 0 (mod 2), instead of the equality (95) .

T} ==

( uo , P I " . P s) = 1 ,

1 (mod 2),

q ==

3 (mod 4), and J-I. = 0 otherwise,

THEOREM 10 . Let m � 2 be an arbitrary integer, (a, m ) the order of q for modulus m . Then the estimate

( a q., )

bX 2 11'i � m +r L..J e

x=1

=

1 , (q, m)

= i,' and

� rm

r

be

(99)

holds under any integer b. Proof. Since the fractional parts

{ �X }

and

c:}

,

have the same period T , then by (28) the sum (99) is a complete exponential sum. But then under any integer z r

2 L..J e 11" x=1

�

'

( amq., + b X ) - r

=

=

r

",,

L..J e x=1 r

2

11"

'

q"m+' + b X+bZ (a) r , qm., q. + -brX ) . (a--

"" e 2 11'1 L..J x=1

-

Ch. I, § 8]

Distribution of digits in complete period of periodic fractions

Therefore,

45

q' b�): 2 T e2 11"1. (aq'"--+....: ""T 2 11"i (aq'"-+-.!b ) 2 = LT "" L...J z=1 z=1 z=1 T e211"1. (aq"'Z bz ) 2 "" --+"" � L...J L...J L...J e

r

m

T

T

m

m

m

T

%=1 z=1

Hen ce the theorem assertion follows, because the congruence x, y T, q Z == q Y ( mod m ,

)

1� � is sat isfied for x = y only: T 2 1r1 (aq'"m + rb"') 2 T 2 1 (aq'" Z + bZ) 2 L e 11" -;n- r e � L L ",=1 :=1 z= 1 T . T "" e2 11"1. -qY)z = -rI """" L...Jy=1 e2 11"1 -L...J :=1 b ( z - y) T Om (q'" - ) m, = -:;: """" L...Jy= 1 e2 11"1 -(m ag'" + ) 2 "" 1 211" r � ..;m. e L...J z=1 1

•

m

•

T

b(z-y)

m

T

§

T

( q'"

m

m

•

qy

=

b"'

•

8. Distribution of digits in complete period of periodic fractions

Let ! be an irreducible fraction and q � 2 an arbitrary integer prime to m . In writing the q-adic expansion of the number ! , the following infinite pure recurring "decimal" to the base q arises:

: [ : ] 0 ·")'1 1'2 . . ")' +

=

.

z · . .

")'''' + T = ")'z

,

(

x

�

1 ),

( 1 00)

with a period T being equal t o the order, t o which q belongs for modulus m . Let N!:) ( 01 . . denote the number of the times that the following equation is satisfied: .

On )

")'z +n = 01 . . . On ( x 0, 1 , . , P - 1 ), . . . On i s an arbitrary fixed n-digited number i n the scale of In On) is the number of occurrences of the given block 01 . . . On among the first P blocks , ")'P+n-1 , ")'1 ")'n , ")'2 ")'n+1

"),,,, + 1 where P � and 0 1 other words, N!:) (01 . r

of digits of length

n

•

.

.

=

•

.

q.

. .

•

•

•

•

•

•

,

•

.

•

,,),p

•

•

•

[ Ch. I, § 8

Complete exponential sums

46

formed by successive digits of the expansion ( 100). The question about the nature of the distribution of digits in the period of the fraction 1il is closely connected with properties of rational exponential sums contain­ ing exponential function. This connection is based on the possibility to represent the quantity N!'; ) ( 81 8n ) in terms of the number of solutions of the congruence •

•

•

a q X ==

y

+ b ( mod m ) ,

o

�

< P,

x

1�

where b and h depend on a choice of the block of digits 8 1 number of solutions of the congruence ( 101) by T!'; ) ( b, h) .

y •

.

� h, •

( 10 1 )

8 n . We denote the

LEMMA 1 0 . Let quantities t , b , and h be defined by the equalities b=

[:�],

Then Proof. Let x be any solution of the equation

(0 � x < P ) .

( 102)

Then we obtain from (100)

{ } aq X m

Ox = O.1'x+l . . · 1'x + n . . . = O .1'x + l . . · 1'x + n + -n q t + Ox Ox = -= O. 81 . . . 8n + n , qn q

where 0 < Ox < 1. Hence it is plain that the equality ( 102) is satisfied for those and only those x , for which O � x < P.

( 1 03)

Since from the definition of b and h it follows that

� m

�

� qn

<

b+1 111

and

b+h m

�

t+1 b+h+l n < , m q

then the inequalities ( 103) are equivalent to the inequalities

� m

<

{ } aq x 111

�

b+ h m

,

O � x < p.

( 1 04)

Ch. /, § 8]

Distribution of digits in complete period of periodic fractions

47

We use y to denote the least non-negative residue of at: to modulus m . Then gg:. == JL and the inequalities (104) are satisfied for those and only those x, which m m s atisfy the congruence

aqX == y

o�

(mod m),

x

P,

<

b

< y

� b + h,

or , th at is just the same, the congruence

aqX == y + b

o�

(mod m),

x

<

P,

1 � y � h.

( 1 05)

But then the number of solutions of the congruence (105) coincides with the number of solutions of the equation (102). The lemma is proved. ' Let m be odd, m = pr 1 p�. the prime factorization of m, T1 the order of q for mo dulus P 1 . . . P a and quantities P1 , , Pa be determined as in § 7 with the help of the conditions q T1 1 - uOP1lit . . . PsP. , ( u o , P1 . . ' Ps ) = 1 . •

•

•

. .

•

_

-

We assume that

a"

> p"

under

v�

r

and

a"

� p" under

v>

r

(0 �

r

� s ) . Choose

O' r + 1 A. m 1 - P1Ih " ' PrPr Pr+l · · · Ps . _

Then the order of q for modulus m 1 should be equal to T1 and T = :::1 T1 by (96) . LEMMA 1 1 . Let b == bI (mod mt ), h == h 1 ( m od m 1 ) ' and h � h 1 . Then

Proof. Using Lemma 2 we get T!,;) (b, h ) =

T- l

h

L L 6m (aqX X= o y=1 m ""'h 1 ""'

= - L...J

m z=1

(

L...J e

y=1

-

.

y - b)

(YH)Z ) (

- 2 71'1 -

m

T

""'

L...J e

a zq

",

2 71' -

m

)

•

x=1

By Theorem 9 the inner sum of the right-hand side of this equality m ay not vanish only for values z, which satisfy the congruences

v = 1 , 2, i.e. , for

.

.

.

, r,

[ Ch. I, § 8

Complete exponential sums

48

Therefore, using r

=

..!!L ml r1 and

b b1 ==

( mod m 1 ) , we obtain

x=1 y=1

x=1 y =1

Since the difference h - h l is a multiple of m 1 and, therefore,

h h-h l h - h1 X b ( b = q , ) a - Y 1 L bm1 ( Y ) = -L m1 m1 y=1 y =ht + 1 -

we obtain the Lemma assertion: Tt hl

T� ) ( h ) =

L L bm 1 ( a q X - Y

-

Tl bd + L x=1 y=h 1 +l

Let us consider the question concerning the distribution of blocks of digits in the complete period of the fraction ;; . Since there exist q n distinct blocks b 1 . . . bn, then the mean value of the number of occurrences of a given block of n digits equals r.

in

THEOREM 1 1 . Let value:

Rn be the deviation of the quantity N�) ( b1 . . . bn ) from its mean R N� ) ( b1 . . . bn ) = �T q n + n.

Then under odd m , any n � 1 , and any choice of the digits estimate is valid:

b1 . . . b n

the following

(106) where rl is the order of q for modulus being equal to the product of primes entering into the prime factorization of m . Proof. By Lemma 10

where

N!:) ( b1 . . . bn ) = T�)(b, h ) ,

T�) (b, h ) is the number of solutions of the congruence o � < T, 1 � Y � h , a q X == Y + b ( mod m ) , x

Ch. I, § 8)

Distribution of digits in complete perio d of periodic fractions

b and h are determined with the help of the equalities

an d the quantities

[:7],

b= Let

49

h1 denote the least non-negative residue of h to modulus m1 : h

Observing that

==

h1

0 ::;; h < m1 .

( mod mI ) ,

h��' is an integer and

h we obtain

=

m

qn

_

{(t +qn1 )m} + { tmqn },

[ ] [

h - h1 m m h - h1 = q q n n � m1 - m1 �

I

]

But then it follows from Lemma 1 1 by virtue of the equality

Since 0

::;;

h1

::;;

ml

m1 - 1 , then

if h1 = 0 and otherwise and therefore

7 = .!!!:.. 71 , that

tm } { (t +qri1 )m } { (j" ,

T�) (b , h) - ;n 7 = T�:) (b1 ' h I ) - { rn%n } 71 +

{

71 0

if h1 0 and otherwise.

=

<

{ (t �J )m } { , } , <

Complete exponential sums

50

Since

o

�

Ttd(b, hd � 71

and

[ Ch. I, §

8

Tml( 1'I)(b ' 0) = 0 ,

we obtain the theorem assertion:

It is easy to ascertain that the estimate ( 106) can not be substantially improved. Indeed, let q = 2, 7 > 1 , a = 1 , and m = 21' - 1. The 2-adic expansion of 1:i has period 7 a 1 - = -- = 0.(0 . . . 0 1 )0 . . . 01 . . . . 21' - 1 m Choose

01 .

. .

On = 0 . . . O. Then we get

N!:)(01 . . . on) = 7

-

n and

Let us note also, that under 71 = 1 it follows from Theorem 1 1 , that

where I Bn I < 1 . It is so, for example, under q = 4 and m 301 . In general, for m = PI 01 1 . . . ps''' ' under fixed primes PII and growing all , the magnitude 71 IS bounded and the following asymptotic formula is valid by (106): =

Now, let us establish the correlation between the occurrence of a given block of digits in period of the fractions

and a

-= ml

where the quantity

ml

[a] " -

ml

+ 0 " 1 /2 "

,

' Ix ' "

is determined as in Lemma 1 1 .

Ch.

I, § 8 1

Distribution of digits in complete period of periodic fractions

TH EOREM 12. If qno \m - m l under (r) ( U"l Nm

a

51

certain n o � 1 , then

- Tl

T " " + Nm(rd · · · Un ) = -l ( Ul · · · On ) qn "

under any n � n o and any choice of a block of digits 01 . . . On . with the help of the equalities Proof. Determine integers t, b, h, b . . and

hI

Then obviously qn (b

_

bt } = t(m

m. )

_

_

qn ( { (t :

q n ( h - h. ) = m - ml

�)m

_

Using the congruence m

( { :� } { t;1 } ), } {t } { t;1 } { :� } ) .

qn

_

_

( + �)m l q

_

m l (mod q n ), which is satisfied under

{ :�} f;I } , - b. ) t - ml , =

+

n

� no , we get

=

Therefore,

q n (b

= (m

)

and, since ml \m and (q, ml ) = 1 ,

But then, according to Lemma 1 1 ,

( 1 07)

h

=

hI

( mod m . ) .

Multiply the second equality of (107) by ---.n. ..,. . Then, observing that T = mlq . ob tam

h m- hI l

-- Tl =

and, therefore,

T(r) (b m

'

h)

=

mI qn

m - ml T - Tl Tl = -qn

T - Tl n + Tm(rdl (bl "

q

,

hI )

l , we

..!!L T ml

[ Ch. I, §

Complete exponential sums

52

8

Hence, applying Lemma 10, we get the theorem assertion I: N(T) m (ul

' "

1:

un )

7 - 71 N(Td 1: + m l ( ul

= --

q

n

' "

I: )

un

•

7)

Let us notice particularly the case q = 2, m = pOi , and m l = pP , where p is a prime greater than 2. Suppose further f3 = 1 (it is so, for instance, under p = 3 , 5, and under n ::::;; no compare the numbers of occurrences of any n-digited block in the period of 2-adic expansion of the fractions pI", and �. The former exceeds the latter by one and the same quantity (being equal to T;:I ). SO, for example, under m = 2 7 we get m l = 3, 7 = 1 8 , 71 = 2 and n o = 3. Therefore, the number of occurrences of any block of digits of length 1, 2, or 3 in the period of the fraction

;7

=

0.(0000100101 1 1 101 101)00 . . .

exceeds by 8, 4, and 2, respectively, the number of occurrences of the same block of digi ts in the period of the fraction 1

3"

=

0.(01)01 . . . . .

Analogously under m = 25 we obtain n o = 2, and the number of o�currences of any block of digits of length 1 or 2 in the period of the fraction 1 25

=

0.(00001010001 1 1 10101 1 1 )00 . . .

is 8 or 4, respectively, more than the number of occurrences of the same block of digits in the period of the fraction 1

"5

=

0 . (0011 )00 . . . .

These relations can be observed under p table given below.

=

3, 5, 7,

n =

1 , 2, 3 , and

p Oi . ::::;; \

125 in the

Ch.

Exponential sums with recurrent function

I. § 9 )

� 0 1 00 01 10 11 000 001 010 011 100 101 110 111

§

1

I

9

1

1 27

81

1

! 5

1

3

25

_I_ 125

.,

49

1 1 0 1 1 0 0 0 1 0 0 1 0 0

3 3 2 1 1 2 1 1 0 1 1 0 1 1

9 9 4 5 5 4 2 2 3 2 2 3 2 2

27 27 14 13 13 14 7 7 6 7 7 6 7 7

2 2 1 1 1 1 0 1 0 1 1 0 1 0

10 10 5 5 5 5 3 2 3 2 2 3 2 3

50 50 25 25 25 25 12 13 12 13 13 12 13 12

2 1 1 1 1 0 0 1 1 0 0 1 0 0

11 10 6 6 5 5 3 3 3 2 3 2 2 3

!

pO<

61 · · · 6 n

53

9. Exponential sums wit h recurrent function

Let us consider functions "p ( a; ) satisfying the linear difference equation with con­ stant coefficients "p ( x ) = a 1 "p ( x - 1) + . . . + an"p(x

-

n)

(x > n ) .

( 108)

It is known ( see, for example, [11]) that any function "p(x) determined by the recur­ rence equality ( 108) can be represented in the form

where

r

�

>' 1 ,

n,

.

.

.

, >'r are distinct roots of the characteristic equation ( 1 09)

and PI (a;), . , P r ( a; ) are polynomials whose degrees are unity less than the multiplic­ ity of the corresponding roots of the equation (109). In particular, if the characteristic equation has no multiple roots, then .

.

( 1 10) where C1 , , en are constants depending upon the choice of initial values of the function "p ( a; ) . If coefficients of the equation (108) and initial values "p ( 1 ) , . , ,,pen) •

•

•

.

.

Complete exponential sums

54

[ Ch. I, § 9

are integers, then, obviously, under any positive integer x the function 1jJ( x ) takes on integer values. Let m > 1 , ( a n , m) 1 , and at least one of the initial values 1jJ(1), . . . , 1jJ(n) be not a multiple of m. In the equation (108) we replace x by x + n and transit to the congruence to the modulus m :

=

1jJ (x + n) == a 1 1jJ (x + n - 1) + . . . + a n 1jJ (x) (mod m).

( 1 11)

Since ( a n , m ) 1 , so in this congruence 1jJ(x) can be expressed in terms of 1jJ(x + 1), . . . , 1jJ( x + n ) and, setting x = 0, - 1 , -2, . . . , we may extend the function 1jJ( x) for integers x � O. A function 1jJ(x) determined for integers x by the congruence ( 1 1 1 ) and initial values 1jJ ( 1 ) , . . . , 1jJ(n) (see [21] ) is called a recurrent function of the n-th order to the modulus m, and the sum p , ,,, ( x) S (P ) = :E e 2 1f1 rn x=l

=

a recurrent function. It is easily seen that under n = 1 these sums coincide with considered in § 7 sums with an exponential function. Let us show that a sequence of least non-negative residues of the function 1jJ(x) to modulus m is periodic and that its least period does not exceed the quantity· m n - 1 . In fact, let us denote the least non-negative residue of 1jJ(x) to modulus m by 'Yx :

. an exp onential sum with

.,p(x) == 'Yx

(mod m ),

o � 'Yx � m - 1 .

Then by virtue of ( 1 1 1 ) 'Yx + n == a l'Yx + n

-

+ . . . + a n'Yx

l

(mod m).

( 1 12)

Consider blocks of n digits with respect to the base m (x = 0, 1 , . . , m n ) .

'Yx + 1 . . . 'Yx + n

.

( 1 13)

Since the number of distinct blocks of n digits is equal to m n , then among the blocks ( 1 1 3) there exist two identical blocks ( 1 14) We determine

r

by the equality

r

=

X2

'Yx + 1 . . . 'Yx + n

- X l and will show that under any x �

= 'Yx + r+ l . . . 'Yx + r + n·

Xl

( 1 15)

Exponential sums with recurrent (unction

Ch. I, § 9 ]

55

In fact, under x = Xl this equality is fulfilled by virtue of ( 1 14) . Apply the induction. Let us suppose that the equality (1 15) holds for a certain x � X l . In the congruence ( 1 12 ) we replace x by x + 7 + 1 . Then using the induction hypothesis we obtain

')'z+r+n+l and, therefore,

a l')'z+r+n + . . . + a n')'z+r+l a1"Yz+n + . . . + a n')'z+l == ')'z+n+l

== =

')'z+r+ n + l

=

')'z+n+ l .

(mod m ) ,

But then

')'z+2 . . . ')'z+n+ l

= ')'z+r+2

. . . ')'z+r+n+ l ,

hence the equality (1 15) is proved for any x � X l . By me ans of such considerations we get this equality for x < X l as well (but in this case, ')'x should be expressed from t he congruence (1 12) in terms of ')'x +l , ' . . , ')'z+n beforehand, and that could be done because of ( a n , m ) = 1 ) . Hence it follows that the least non-negative residues of the function 1/J( x ) have a period 7 , where 1 � 7 � m n . Let us assume that the least period is equal to m n . Then any block of n digits should occur among the blocks (1 13), and, in particular, the block 0 . . . 0 being formed by zeros only is present among them. But then by ( 1 12) all terms of the sequence of the residues will equal zero and its least period equal that contradicts to the assumption. Therefore, the least period of the function 1/J( x ) does not exceed m n - 1 . Henceforward we let 7 denote the least period of the sequence of the least non­ negative residues of the function 1/J( x ) to modulus m. It is easily seen that 7 is the period of fractional parts of the .function "'�) :

1,

Therefore, the sum

S(7)

=

r 2 . ---;n x) :�::> 71"1 ",( x =l

is a complete exponential sum. Since under integer

then by (28) under any integer

a the sum r ' ("'(X ) + B X ) ""' 2 L.J e

z=l is a complete sum as well.

71"1

m

r

a

[ Ch. I, §

Complete exponential sums

56

9

Let .,pI ( X ) , . . . , .,p n (x) be recurrent functions satisfying the equation ( 108) and de­ termined by initial values if x = j, if 1 � x � n,

x i= j

(j = 1, 2, . . . , n ) .

It is easy to show that .,p(x + z) = .,p(z + l ).,p l ( X ) + . . . + .,p(z + n).,p n (x).

( 1 16)

In fact by virtue of the linearity of the equation ( 108) , any linear combination of its solutions is a solution too . In particular, the sum in the right-hand side of the equality ( 1 16) is a solution of the equation (108). From the definition of the functions .,pj (x) it is seen that under x = 1 , 2, . . . , n the initial values of this sum are equal to .,p(z + 1) , .,p ( z + 2), . . . , .,p (z + n ) , respectively. The solution .,p(x + z) has the same initial values. But solutions, which have the same initial values, coincide. Hence the equality ( 1 16) is proved.

THEOREM 1 3 . Let .,p(x) be a recurrent function of the n-th order to the modulus m,

'T

be its lea.s t period, and P � T

•

� 2 11"1

L.-J e

x=1

x

",( ) m

T.

Then we have the estimates P

n

Le x=1

� m2 ,

Proof. Since under an integer

a

",(x) 2 1I"1. --;n

!!.

� m 2 ( 1 + n log m).

the sum

is a complete sum, then under any integer z

( ",( x +z) + a x +z ) T m , x=1 ( "'(x + z) + "T ax ) mISa(T) 1 L /1r1 x=1 Sa(T )

T

=

L

e

11" 2 i

T

•

=

Squaring and summing over z yields T-l

T ISa( TW

=

L

T

Le %=0 x=1

. ( ",( x +%) + a x ) 2 11"1 m

-- "T

2

I

( 1 1 7)

Ch. /, § 9]

Exponential sums with recurrent function

57

We let 'Yz denote the least non-negative residue of the function ,p(z) to modulus Then by ( 1 16 )

m.

,p(z + 1 ),p1 (X) + . . . + ,p(z + n),pn(x) == 'Yz+ 1 ,p1 (X) + . . . + 'Yz+n,pn(x) (mod m),

,p(x + z)

=

and, therefore,

T -n '" 2 11'1. 1/1(X) L...J e -m = I Sa( T ) 1 � m 2 • ( 1 18 ) x=l Since T is the least period of 'Yz to modulus m, then under z = 0, 1 , . . . , T - 1 all block s 'Yz+ 1 . . . 'Yz+ n of n digits are distinct . Therefore, exlending the summation to all p ossible blocks Zl . . . Z n of n digits, we obtain

p-1 T I Sa ( T ) 1 2 � %1 ,

L •••

, %n = O

T

=

a (x-y)

2 71'1 -'" T L...J e •

x ,y=l

� mn

T

m -l L

%l , . . . , Zn =O

L 15m [,pI (x) - ,p1 (Y)]

x , y= l

.

. .

15m [,pn (x) - ,pn (Y)]

=

m n T,

( 119 )

where T is the number of solutions of the system of congruences

: ¢1 (�� }

�

�1 ( ) ,pn(x)

=

,pn {Y)

(mod m ) ,

1 � X , Y � T.

( 1 20 )

Let us assume that this system has a solution with Y -=I x . Without loss of generality, we may assume that Y > x . Using the equality ( 116 ) , we get

,p{Z) Y ,p{z + - x)

,p{z - x + 1 ),p1 {X) + . . . + ,p (z - x + n),pn{x) , = ,p{z - x + 1 ),p1 {Y) + . . . + ,p(z - x + n ) ,pn (Y) .

=

Hence by (120 ) it follows that under .any integer z

,p(z + Y - x)

==

,p(z) (mod m).

But then Y - x is a period of 'Yz , and since T is the least period, then Y - x � T , which leads to a contradiction. Thus, the congruence system ( 120 ) has no other solutions except for solutions with Y = x and, therefore, T = T . Now from ( 119 ) , we get

( 121 )

Complete exponential sums

58

Hence under

a =

[ Ch. I, §

9

0 the first assertion of the theorem follows:

T 1/I(x) L: e 2 '1r1 --;n x=1 •

=

!!

ISo( r)1 � m 2

.

The second assertion of the theorem follows immediately from Theorem 2 and the estimate ( 12 1 ) :

T ' ( 1/I( X ) ax) P . 1/I(x) L: e2 '1r 1 --;n � max L: e 2 'lr l --;n + T (1 + log r) l �a�T x=1 x=1 n = max I Sa (r) l ( l + log r) < m 2 ( 1 + n log m) . l �a�T Note that in the general case the order of the estimation

can not be improved further. Indeed, using considerations from the theory of finite 'fields (see, for instance, [33]) , it can be shown that under any prime p > 2 and positive integer n < p there exist recurrent functions t/J(x) of the n-th order to the modulus p with the period r = p n - 1 . Besides, roots of the corresponding characteristic equation ( 1 09) are distinct and by ( 1 10) •

( 1 22) By virtue of properties of symmetric functions there exists an equation with integral coefficients J.t n b 1 J.t n- 1 + . . . + b n ,

=

whose roots J.t l , . . . , J.t n equal A � , . . . , A � , respectively, and the free term is relatively prime to p. Consider the functions t/J (2x) and t/J(2x + 1 ) . It follows from ( 122) that t/J(2x) t/J(2x + 1 )

= C1 J.t f + . . . + CnJ.t!, =

C1 A I J.t f + . . . + Cn A n J.t !.

Thus, t/J(2x) and t/J(2x + 1 ) are recurrent functions of the n-th order t o the modulus satisfying the equation

p

t/J*(x)

=

b1 t/J*(x - 1 )

+ . . . + bnt/J*(x - n).

Denote by "Ix the least non-negative residue of t/J(x) to modulus p . Under rl = ! r we get * "I2( X+ Tt } "I2 X+ T 12 x and, therefore, 12x has a period being equal to rl .

=

· Since p is o d d , then

=

T:! = pR2- t

is an integer .

Exponential sums with recurrent function

Ch. I, § 9 ]

1'2

59

Let us assume that 1'1 is not the least period. Then we can find a positive integer < 1'1 , such that under any integer x ( 1 23)

A pplying the equality ( 1 16) we obtain

+ 1) + . . . + .,p (2x + 1 ).,pn (2 2 + 1 ) , (1) + . . . + .,p (2x + n - 1 ) .,pn ( I ) .

.,p (2 x + 21'2 ) = .,p (2 X) .,pl (21'2 .,p (2x) = .,p (2x ) .,pl

n -

1'

But then by ( 123) the congruence

should be fulfilled under any integer x . From properties of solutions of the system (120) , it follows that at least one of square brackets in ( 1 24) is not congruent to zero to modulus p and, therefore, the number of solutions of the congruence ( 124) does not exceed p n - l - 1. On the other hand, according to the definition of the function ?jJ (x) under x = 1 , 2, . . . , 1'1 n-tuples 12x , 12x + l , . . . , ')'2x + n -1 yield distinct solutions of this congruence. Since obviously

1'1

=

n-1 p2

> pn-1 - 1 ,

tpen w e arrive at a contradiction and, therefore, 1'1 i s the last period of ')'2 x Analo­ gously we get that the least period of ')'2x + 1 is equal to 1'1 as well. Now, in the same fashion as in the deduction of ( 1 18) , we arrive at the equalities •

Tl

•

2 '11' 1 L.J e 1'1 "" x=1

!JI(2x+1) 2 P

_

Tl -1 "" L.J z=O

Hence by virtue of the choice of the function ?jJ( x ) , it follows that

1'1

(

)

2 . 1/>(2x) 2 2 71' 1 -271'1. 1/>( 2x+1) "" "" P e e P L.J + L.J x=1 x=1 }I",, zl 1/>1 (2x)+". + zn 1/>n (2x) 2 - I' 2 '11' 1 "" = P , L.J e L.J Zl , , , . , Zn =O x=1 Tl

TI

TI

•

( 125)

Complete exponential sums

60

[ Ch. I, §

I

Z1 ,

9

, Zn, where the sign I in the sum L: z 1 " " , Z n indicates the deletion of n-tuple formed by zeros entirely, from the range of summation. Let denote the number of solutions of the system of congruences

T1

1

1

� ?�). �.� ?�)

1P n(2 x ) == 1P n(2y )

(mod p)

}

In the same way as in the system ( 120) , we have

p-1

1 � x, y �

T1

71

=

•

•

•

7.

and, therefore,

L

%1 " " , Zn =0

But then (125) can be rewritten in the form

71 TI

(

:1: = 1 '"""

L....J

We determine

TI

L....J e

1

'"""

x=

.

•

2 11'1

t/t(2x)

2 7f 1 -P e

I S* ( 7d l

--

t/t ( 2 :1:) 2 P

TI

+ TI

2

+

:1: = 1

L....J e '"""

.

7f 1 '""" e 2

:1: = 1

L....J

•

2 7f1

t/t ( 2 :1:+1 ) 2 P

t/t(2 :1: + 1 ) P

2

=

P

)

n - 71

=

-1 ' -+ 2

pn

(126)

with the help of the equality

Then from (126) we get

1 .!!:. I S*(7dl > "2 p 2 . Hence by ( 1 17) it follows that under any prime p > 2 and any n > 1 there exists a recurrent function of the n-th order to the modulus p such that for the exponential sum S *( 7t ) the following estimates

1

.!!:.

"2 p 2 < hold.

I S* ( 7d l

.!!:.

� p2

Ch . I, § 1 0 ]

Sums of Legendre's symbols

61

§ 10. Sums of Legendre's symbols Let p > 2 be a prime, I ( x ) ao + al x + . . . + a n x n be a polynomial with integral coeffi cients, n < p , and ( a n , p) = 1 . Let U n denote the sum of Legendre ' s symbols =

Un = t x=1

( I(pX) ) ,

( 127)

and Tn denote the number of solutions of the congruence

y2 The quantities

==

I ( x)

( mod p) .

( 128)

Tn and U n are connected by a simple relationship p Tn = � 1 + I ( x ) p + Un . P

[ ( )]

=

( 1 29)

This relationship reduces the question on the number of solutions of the congruence (128) to studies of sums of the Legendre symbols. The sums ( 127) are easily evaluated for polynomials of the first and the second degree. Indeed, since Legendre's symbol ( : ) is a periodic function with a period p and under (a I , p ) = 1 the linear function ao + al x runs through a complete residue system modulo p , when x runs through a complete residue system modulo p, then

t ( ao + a1 x ) t ( :. ) p P =

x=1 In order to evaluate the sum 17

we consider the congruence

2

=

o.

x=1

=

t ( ao + a X + a2 x2 ) , l

p

x=1

y2

==

x 2 + a (mod p)

and denote the number of its solutions by

p

T( a ) .

Obviously,

T ( a ) = L Dp ( X 2 - y 2 + a) x , y=1

X2_ 2 1 p - l e 2 11'1. a % P e 2 11' 1' Z( p y ) =-L P L P z= o "" y= 1 l 2 2 1 p- e 2'11'i � P e 2'11'i !!P L L p = P + pz= 1 x=1

[ Ch. I, § 1 0

Complete exponential sums

62

Using the fact that the modulus of the Gaussian sum equals -/p , we obtain

a + p-lL.,.. €2... p

T( ) = p and, therefore, by ( 1 29)

P

Let

a2

L :c = 1

. az

'"'

z= 1

(X2 +a ) -p

a

= p + pop ( ) - l ,

a

= pop ( ) - 1

( 130)

.

¢. 0 (mod p) . Then observing that

( � ) ( 4 ;2 )

= 1,

we get

( 131 ) Note that , in particular,

� (x - a ) (X - b) L.,..

:c=1

-p

p

--

= pop(a - b) - 1 .

( 132)

Indeed, this equality follows at once from ( 1 31 ):

ab - (a+ b)x + x2 ) (a [4 ab - (a +

= t( ) � ( ) � ( t p p :c= 1 :c = 1

= pOp

Under n � 3 the investigation of the sums for some special cases.

p

(T n

b) 2 ] - 1 = pOp

- b) - 1 .

is much more complicated, except

Sums of Legendre's symbols

Ch. I, § 1 0]

Consider one of such special cases. Let n � and

63

3 be odd, p

>

n be a prime,

Let us show that

( aI, p) = 1, ( 1 33 )

zx

runs through a complete In fact , since under z ¢. 0 ( mod p) the linear function residue system modulo p when runs through a complete residue system modulo p, then

x

Therefore,

l un (al Z n - 1 )1 = t ( z n x n +p a1 z n x ) t ( x n +p a1 x ) I un ( ad l . x= 1 x= 1 Squaring this equality and summing over z, we obtain p- l n 2 p - l (134) (p - 1) l u n ( a dl 2 = L l u n (a lZ - 1 ) 1 = L teA) I U n ( A ) 1 2 , %=1 �= 1 where t( A) is the number of solutions of the congruence alzn - 1 A (mod p) . Since teA) � 1 , then from (134 ) it follows that pL- l (p - 1) l u n ( a l ) 1 2 � ( n - 1 ) I U n ( AW �= 1 n + Ax y n + AY = (n - 1 ) � t ( x �=1 X, I/= 1 P ) ( P ) t ( A + xn-1 ) ( A + yn- 1 ) 1 ) x,�l/= 1 ( XpY ) �=1 =( P . P Hence, using the equality (132), we get the estimate (133 ) : p-l Y n d ( l un a l 2 � p = � X,I/L= 1 ( Xp ) [p.sp (x n - l yn - 1 ) - 1] ( n __ �P � xY .sp ( n - 1 y n - 1 ) � ( _ 1)2p, = p X, 1/= 1 ( P ) x l un ( adl � ( l )JP =

=

==

n

-

n

_

_

_

n

-

.

n

Complete exponential sums

64

Under odd

n

� 3 and

( an , p) = 1

[Ch. I, § 1 0

the same estimate holds for the general case also: ( 1 35 )

For n = 3 this estimate was obtained by Hasse [ 1 3) , under an arbitrary n it follows from more general results of A. Weil [48) . One can acquaint with elementary methods for obtaining estimates of the sums ( 1 35) by papers [35) , [42) , and [31 ) . A s i t was shown above, sums of Legendre's symbols for polynomials of the second degree can be evaluated with the help of Gaussian sums. Let us show that G aussian sums can be used in estimating the simplest incomplete sums of Legendre ' s symbols:

O'(P ) = x=L1 ( :.P ) (P p

<

p) .

Indeed, in the same way as in the estimation of incomplete exponential sums (see Theorem 2), we obtain

p-1 ( P = ) (P O' � � ) �6p ( x

_

y) =

p-1 P p - 1 . z(x-y) � � ( � ) � � e2 "' -p- .

Hence, after interchanging the order of summation and singling out the summand with z = 0 , by (41 ) and (42) it follows that P . zy I O'(P ) I � p- p-1 z=Lp1 y=L1 /"' -1 � � � min (P, 2 11� 11 ) � ..;p log p. 1

Thus we obtain the estimate

P

x=L1 ( :.P ) � ..;p log p. P

P

Plainly, under > vP log p this estimate is better than the trivial one. The availability of a nontrivial estimate

� (�) P P

<

( 1 36)

Ch. I, § 1 0]

Sums of L egendre 's symbols

65

si gnali zes that on the interval [1 , P] there is at least one quadratic nonresidue modulo Let Po denote the least quadratic nonresidue . From ( 136 ) it follows that

p.

Po � 1 + [JP log p]

=

1

0 (p 2 log p) .

Accordi ng to the conjecture enunciated by I. M. Vinogradov, under any e estim ate Po = O (pE )

>

0 the

is valid, where the constant implied by the symbol "0" depends on e only. In this direction, the strongest result has the form

47e'

where 'Y is any number greater than A proof of this result [4] is based upon the use of the Hasse-Weil estimate ( 135). Let Nt and N2 be, respectively, the number of quadratic residues and quadratic nonresidues to modulus p among the first P positive integers. It is easy to obtain asymptotic formulas for Nt and N2 with the help of the estimate (136) . Indeed, observing that the number of solutions of the congruence y 2 == x under P < p equals

where 1 61

( mod p ) ,

2Nt , we get

� 1 by ( 136 ) .

Hence, since

1 � y � p , 1 � x � P,

Nl + N2 = P, we have

A question concerning the distribution of quadratic nonresidues in a sequence of values of recurrent functions is worked out just as easily. Let p > 2 be a prime and .,p( x ) be a recurrent function of the n-th order ( n � 2) with a period l' to modulus p. Let No, Nt , and N2 denote, respectively, the number of zeros to modulus p, quadratic residues and quadratic nonresidues in a period of the function .,p ( x ) . Obviously,

Hence, using Theorem 13, we get

I NO

where 1 80 I

[ Ch. I, § 1 0

Complete exponential sums

66

-

( )

.

p-1 T 2 z .p(1:) � � L: L: e 71" I -p- � 1 - p� p� , p z=l x=l 1 �2 r p , No = p + 80 1 �I

P

( - P)

(13 7)

� 1 . In the same way, using Theorems 3 and 13, we obtain T P I T P p-1 / z [ .p( X ) _ y 2 j 2 P No + 2N1 = L: L: 8p [1/J(x) _ y ] = - L: L: L: "" x=l y=l z=O x=l y =l � 2 ,..i z .p ( X » -2,..i 1�

Therefore,

INo + 2 N1

P

(� :;;Y2) ( x=l

= r + - L....J L....J c :;;= 1 y=l

P

.

P

-1

.

L....J e

zy P '1 p""'" ""'" r l � L....J L....J e 2 71" P

- - z=l y=l p

1 ) p n +2 1 , 1 p ( �

2

)

•

T 2 ,..1 -:;; .p( x ) ""'" P e L....J •

x=l

( - P1 ) p n+2 1 ,

No + 2 N1 = r + 0 1 1 Now, observing that formulas for Nl and

P

( 1 38 )

No + N1 + N2 = r, from (137) and (138) we get the asymptotic N2 :

If the period of the recurrent function is sufficiently large,

r > p 2 (y'P + 1), n

( 139)

then the magnitudes N1 and N2 will be positive and, therefore, both quadratic residues and quadratic nonresidues to modulus p will occur among terms of the recurrent sequence. Note that for sequences of the third order the condition (139) is nearly of the best possible kind. By (139) recurrent sequences of the third order, whose period is greater than p2 py'P, contain quadratic nonresidues to modulus p. _

+

Ch. I, § 1 0]

Sums of L egendre's symbols

67

shall show that there exist sequences of the third order with period ! ( p2 - p ) , whi ch �o not contain quadratic nonresidues. Indeed, let g be a primitive root to modulus p. Consider the function t/J( x ) satisfying the equation of the third order

We

t/J(x) = 3g 2 t/J(x - 1 ) - 3ltfJ(x - 2) + It/J(x - 3)

and determined by initial conditions tfJ(2)

= 4g4 ,

t/J(3) = 9 g 6 •

It is easy to verify that t/J( x ) = x2 g2x . Obviously, the se'quence of values of this functi on does not contain quadratic nonresidues. Let T denote the least period of the fun ction t/J(x) to modulus p. Then the congruence

holds for any integer x . Hence we get without difficulty that T = ! (p2 - p). Thus, under n = 3 the bound (1 39) for the magnitude of periods of recurrent functions, in whose values quadratic nonresidues occur, has the precise order and the constant in it cannot be improved more than twice.

CHAPTER

II

WEYL 'S SUMS

§

1 1.

Weyl's method

In Chapter I, the Weyl sums of the first degree were considered and it was shown that the estimate 2 1 11' i ax ./ "=:: mm (140)

�e

•

(p' 2 11 a ll )

. holds for them . The basic idea of Weyl's method consists in reducing the estimation of a sum of an arbitrary degree n �

2

p

S( P) = L e2 11'i ( a l x + . . . + a " x " ) x=l

-

to the estimation of a sum of degree n 1 and, ultimately, to the use of the estimate ( 140). We have already met the reduction of the degree of an exponential sum in proving the theorem on the modulus of the Gauss sum. In the Gauss theorem, the square of the modulus of the exponential sum of the second degree was transformed with the help of linear change of variable in summation into a double sum, in which one of summations was reduced to the evaluation of a sum of the first degree. Similar but technically more complicated considerations are used for the reduction of the degree of sums in the Weyl method as well. In deducing estimates of Weyl's sums we shall need the following inequalities:

( t, u. v.), � (t, ··t' t, v;, ( t, v.) ' U·-1 t, v;, ( t, ..v.) ' � t, .: t, v: . •.

(141) ( 1 42) ( 143)

Weyl's method

Ch . II. § 1 1 ]

69

These inequalities hold under U x ;;:: 0, V x ;;:: 0, and an arbitrary positive integer k. Let us prove the inequality (141 ) . Denote by ak the sums ao

=

p

( k = 1 , 2, . . . ) .

L Ux ,

x =1

If a o = 0 or ao # 0 and k = 1, then the inequality ( 141 ) is trivial. We shall assume that ao # 0 and k ;;:: 2. Since

then, obviously,

o

p

� L

x , y= l

U X u y ( v ! - vyv ! -1 + v:

- vx v: - 1 )

= 2 ( aO a k

- a1 ak-

d·

B ut then

and, therefore, The last inequality coincides with ( 141 ) . . The inequality ( 142) is obtained from ( 141) under U 1 = . . . = U p = 1 . The inequal­ ity (143) follows from (141) also. Indeed, denote by E* the sum extended over those values x, for which Ux # O. Then, setting k = 2 in ( 141) , we obtain the inequality (143):

Let

Y1

;;:: 0 be an integer. We shall use D. f (x) to denote a finite difference of a

function f ( x ):

Yl

D. f (x) = f (x + yt } - f( x). Yl

Under k ;;:: 1 we determine a finite difference of the k-th order help of the equality

D. f (x) with the

Y l l " " Yk

Weyl's sums

70

It is easily seen that

[ Ch. II, § 1 1

b. f( x ) does not depend on the order of the arrangement of

Y t , · " , Yk

YI , . . . , Yk . So, for instance, � f(x ) = b. [� f(x )] = b. [f(x + yt ) - f(x)]

quantities

Yt , Y2

Let

Y2

Yl

Y2

f(x ) be a polynomial of degree � 2: f( x) = ao + alX + . . . + an xll• n

We shall show that for its finite difference of the order

n

-

1 the equality

b. f(x) = n l an YI . . . Yn - IX + f3 , (144) where f3 depends only on coefficients of the polynomial f(x) and on quantities YI , . . . , Yn - l , is valid . Indeed, for polynomials of the second degree h ex) = a 2 x 2 + a l x + ao this equality follows immediately from the definition of a finite difference: �Yl f2 (X ) = a2 ( x + YI? + a leX + YI ) + ao - (a2 x 2 + alx + ao) = 2 a 2 YIX + a 2 Y� + a lYI = 2 a 2 YIX + f32 ' Apply induction. Let under a certain k � 2 the equality ( 145) b. fk (X) = k l ak Yl . . · Yk-I X + f3k be valid for every polynomial fk (X) = a k x k + . . . +ao. Then for fk+l ( x ) = a k+lx k+ l + . . . + ao we get Yl , , , · , Yn - l

Y l , , , · , Yk - 1

= (k + 1)1 ak+lYI . . . YkX + f3H I , by that the equality ( 145) is proved for any k � 2 . In particular, under k = obtain the equality ( 144). The following lemma is central in Weyl's method. LEMMA 12. Under any k � 1 we have PI = P and under = P,,+ l = P" - y" .

where equality

v

.

1 , 2 , . , k quantities .

P,,+I

n

we

are determined by the

Weyl's method

Ch. If, § 1 1 ] Proof .

In deed,

71

At first we shall show that the assertion of the lemma holds under k

PI

L x=1

e 211" ; f(x)

2 x, y=1 =

PI +

=

l.

e 211"; (f( y)- f( x »)

L e211"; [ f( y) - f( x ») + L e 211"i (f( y) - f( x »)

x y Pl - l PI - X � PI + 2 L L e211"i (f( x + y)-f( x ») x=1 y=1

•

Hence, after interch anging the order of summation, it follows that

Raise this inequality to the power 2 k - 1 • Then according to (142) we obtain

PI

L e211"i !( x )

(146)

x=1

Applying the inequality ( 146) to its right-hand side in succession and observing that PI = P and Pv � P, we arrive at the assertion of the lemma:

LPI e211"i f( x )

x=1

( 11k 22k .

- 1 ) PLI - 1 ' " PkL- 1

Pk + 1 2 11"; a f( x ) L e �l " "'�k � v=1 YI =O Yk =O x=1 PI - 1 Pk - 1 Pk + 1 2 ".; a f( x ) k k � 22 - 1 p2 -( k+1 ) L . . . L L e �l'.···.�k YI = O Yk =O x=l k

. - P; -

•

LEMMA 13. Let >. and X l , of solutions of the equation

•

•

.

Xl

, x n be positive integers. Denote by 7'n ( >' ) the number X n = >.. Then under any e (0 < e � 1 ) we have •

.

•

where the constant Cn (e) depends on n and e only. Proof. Let

a

� 1, p �

2,

and 0

< e

� 1 . Since

1 + ae log p

<

e ao: log p pae, =

Weyl's sums

72

[ Ch. II, § 1 1

then for any P � 2 1+a If P �

1

e€ ,

<

1 1 - ( 1 + ac log p) c og 2

then the coefficient e

l�g 2

<

1 al!: 1 -p . C og 2

( 147 )

in this estimate may be omitted: ( 148 )

Under >' = 1 the assertion of the lemma is obvious. Let >. � 2 be given by its prime factorization: (P I < P2 < . . . < P . ) · 1 ::;; Pr+ I ' Then, applying the Estimate the number of divisors of >.. Let Pr < estimate ( 147) for P I , . . . , Pr and using the estimate ( 148) for Pr+ 1 , . . . , P s , we obtain

ee

1

Hence, because the number of primes, which are less than e e , does not exceed follows that r( >') ::;;

C� ) l g2

1 e€

, it

1.

e

e

>. e = C (c)>.e .

Replace c by � in this estimate. Then, observing that rn ( >') ::;; [r( >. )] n , we get the assertion of the lemma:

LEMMA 14 . Let P � 2 and a () a--+­ q q2 '

( a, q) = 1 ,

I()I ::;; 1 .

Then under any positive integer Q and an arbitrruy real f3 we have

1°.

t ( t, (

min p,

x-I

2° .

min p 2 ,

lI a X

� ) ( �) : ) ( �)

lIa x

f3I1.

(3 11 2

::;; 4 1 +

::;; 4 P 1 +

(P + q log P) , ( p + q).

Weyl's method

Ch. tI, § 1 1 ]

Proof.

Let us represent f3 in the form

b + (h ,

f3 = q

b

73

-q

1 (1t I

< 1 and the sign of (} 1 is opposite to the sign of (}. Then where is an integer, un der 1 :;:;; :;:;; q we obtain

x

I

ax + b + 81 8 x2 + -, a x + f3 = -q q ax + b = ax + f3 + (} q

q

I lI

-

(:� : ) I :;:;; lI ax

4-

f3 1 + � .

(149)

At first we shall show that

Indeed, if q or

�

min

(p, lIax � (3 1 )

:;:;;

3P + 4q log P.

( 1 50)

P is less than four, then this estimate is trivial. x, under which ax + b �

P � 4. Then, according to (149) for those values

I

w�

have

l I a x + f311 �

l I ax b I .

�,

q

+ q

Let q � 4 and

_

� � � I a x q+ b I .

This estimate may be used for all x within the interval 1 :;:;; of those, for which + == 0, ±1 ( mod q ) .

x

:;:;; q with the exception

ax b

Since q ) = 1 , then runs through a complete residue system modulo q, when runs through a complete residue system modulo p. Therefore

ax + b

( a,

x

t

x=1

min

(p, lI ax � ) (3 1 1

:;:;;

3P +

L

a x + b�O , ± 1

= 3P + L

2 �x
:;:;; 3P + 2

min

(p, I � II ) (p, I !2 I )

min

q

q

min ( p, : ) . q 2 �x � 2

L

(151)

Weyt's sums

74

[ Ch. II, § 1 1

Hence, observing that

( )

L min p, 2xq 2�x� 2q

'!.

2

�

1

P + 2q

�

x

: � 2q log P,

2q + 2q / d 2q P

we get the estimate ( 1 5 0 ). replace the quantity x by qX l + X 2 in the estimate 1 0 . Now choose Q l = 1 + [ !q Q] and . Then, using the inequality (15 0 ), we obtain

Q

?;

min

(

P, ax 1+ ,8 11 !!

) x�o � (P, !! ax 2 + aqx l + ) Q +4 P). �

�

Ql -l

q

X

l (3P

l

1

min

,811

q log

�,

Hence, since Q l < 1 + the first assertion of the lemma follows. To prove the assertion 20 , at first we will show that ( 1 52)

In fact , as in the proof of the estimate ( 1 5 1 ) , we get

Hence, because of

L 2�X<

p2 + 4q2

� +1

� 2P q + 4 q 2

00

/ dxx2

2q p

=

4P q ,

Weyt's method

Ch. fl. § 1 1 ] we

75

ob t ain the estimate ( 1 52) . Therefore,

thus completing the proof of the lemma. Not e. Let m be an arbitrary positive integer and a

() + -2 , q q

a =

Then under any f3 and any e (0

2:

(

< e

181 � 1.

� 1 ) we have

1 II a m x +

min p,

1 :r; I < Q

( a , q) = 1 ,

-

f3 1 )

8m

�

e

( ) Q

1+

q

(P + q ) P � .

Really, under P < 3 this estimation is trivial. Let P � 3. Then 1 and min p, min p, I l am + l I ax

2:

1 :r; I < Q

: f3 1 ) �

(

=

( � f3 1 ) 2: (p, ) f3 ( � 1)

1 :r; I < m Q 2 mQ l -

1

min

."...-..--:----------: :7." --:

l I ax +

2m �4 1+

THEOREM 14. Let n � 2, f(x) = a l x + . . . + anx n , and () a + -, q q2

( a , q) = 1 ,

If P � q � p n- 1 , then under any positive e

Ip

<

:r; = 1

1 81 � 1 .

1 we have

1

2: e 2 :J\'i /(:r;) � G (n, e ) p

where the constant G (n, e ) does not depend on P.

l-

l

-

a m Q II

(P + q log P) .

-

Hence the estimate ( 153) follows obviously.

-

log P

2:

:r; = 1

an =

<

( 153)

-�

2n- 1 ,

<

�

pe

Proof . It follows from Lemma 12 under k� � 22

where PI

[ Ch. II. § 1 1

Weyt's sums

76

=

- p2 - -n

n l

n l

P and P,,+ I = P" �

-

(v

y"

P1�-I L...J

=

.

.

nPn - l - I -

=

Pn 2 'I1'I' ... f( � 1 " "'�n- 1 L...J L...J e Yn-l =O x=1 �

•

Yl =O

1 , 2,

1 that

. ,n .

.

-

�

A

X

)

( 1 5 4)

1 ) . Since by ( 1 44)

Yl '''',Y n - l !(x) = Q n n ! YI . . . Yn- I X + f3 ,

then using the estimate ( 1 1 ) , we obtain

Pn '11' ; L e2 '1 '''''' n - l f(x) 6.

=

x=1

an n! Yl "'Yn_ I X . (pn , I Qnn'. I I ) .

Pn L e2 '11'i x=1

1 2 YI . . . Yn Substituting this estimate into ( 1 5 4) and singling out the summands, in which quan­ tities y" being equal to zero occur, we get � mm

n -n

n-1 P2 - 1

P

�

( Yl , ... ,Y n - I =1 YI . ..,XYn - I . Y -l n Tn- l('x) Cn_l(e),Xe e t (p, l i n! Yl .. .IYn_ l an I ) YI "",Yn- l -l p n- 1 (; Tn _I (,X) min (p, I n ! �Qnl l ) p n- I n l Cn _ l(e)p( - )e (; (p, l i n ! �Qn l ) . q pn- I t (p, li n ! IYn _ l Qn l ) YI , 8 n ! Cn�l (e) ( : ) (p q)pen Cn- I (e) pn - 1+en , 2 +2

min P,

L...J

Collect summands with the same values of the product 1 3 for the number of solutions of the equation YI . . � under any positive � 1 , then

=

.

_

1 , li n . YI . . . Yn- I Q n ll

)

.

Sirlce by Lemma we have the estimate

min

�

�

min

Now, using the condition P � _

..

·,Yn - I - I

and the note of Lemma 14, we obtain

�

min

YI

1+

�

� 32 n !

e

.

.

P

.

-

I

+

( 1 55 )

Weyts method

Ch . II, § 1 1 J an d ,

77

therefore, p L: e 211"i / ( x)

2n - 1

x=1 Hen ce

:s:;;

(c 1 1 n 2 2 n - l p 2 n - l _ 1 + 32 n ! 2 2 n - Cn -1 ) p 2 n - -1 +en . C

after replacing c by -; we get the theorem assertion: :s:;;

=

(

)

-

_ n - nC ( £ ) n - 1 1 - l -e 64 n ! 2 2 l n � 1 n 2 p 2 n- 1 _1

,

l -E 1 - -C(n, c ) P 2 n- 1

No te. A nontri vial (with respect to the order) estimate of Weyl ' s sum of degree n � 2 can be obtained not only on the interval P :s:;; q :s:;; pn -l indicated in Theorem 14,

bu t also on the interval

(0

<

Cl

(156)

1)

<

containing the former within it. Indeed, in this case by ( 155) we have the estimate

_ E1 - e

p

I L: e2 11"i / ( X ) :S:;; C(n, c, c t }P 2 n - l ,

x=1

having nontrivial order under any positive c < C l . The further extension of the interval ( 156) to q :s:;; p n is impossible, because under q = pn there exist Weyl ' s sums of degree n, for which

In order to convince ourselves in it, let us choose, for example,

Then, since

l e 2 11"'. /(x) - 1 1

=

21 si n 1l'f( x ) 1

:(

'"

21l' x

n � 12,

�--'(n l) p n

n+l

_ (x

--

+

_

l ) n+l

-

'

q

=

p n , and

Weyt 's sums

78

[ Ch. II, §

11

so, obviously,

and, therefore, P+

=

L (e2 11'i !(x) - 1) p

x=1

§ 12. Syste ms of equations A method proposed by Mordell for the estimation of complete rational sums consists in the reduction of the estimation of an individual sum to the estimation of the mean value

under k = n or, in other words, to the estimation of the number of solutions of the system of congruences

�.1. � .' .' .' .: . �� .�. ��. :.::: � �k

.

}

(mod p ) .

x f + . . . + x k == y f + . . . + y�

Similarly, in Vinogradov's method the estimation of Weyl ' s sum is reduced to the estimation, under a certain k, the mean value of the quantity

e211'i (Q'l:t+ ... +Q' n x n ) L x=1 p

being equal to

1

1

/.../ o

0

e211'i ( Q' l X+ .. . +Q' n :t n ) L x= 1 p

2k

2k

Systems of equa tions

Ch. /I, § 1 2)

79

and , as it will be shown below, coinciding with the number of solutions of the system of equations

�1. � .. .. : .:.��. �. ��.:.::: � .�k

.

},

(157)

X l + . . . + xi: = Y I + . . . + y� Denote by

S(a l , ' " , an) the Weyl sum P S('" ""'" n ) -- '" e2 11'i ( 0I 1 X+ ... +OI n X n ) ,-, 1 , · · · ,

L...J

x= l

Let n � 2 , A I , . . . , A n be fixed integers, and integral solutions of the system of equations

N�� ( A I , . . . , A n)

� l � : �� � � �� � "" ' : �� : � � l . . ::: . . . . . . . . . . . .

xl + . . . + xi: - ( yf + . . . + y� ) = A n

in which the quantities

Xi

.

be the number of

},

(158)

and Yi vary within the limits

1 ::;; xi ::;; P,

1 ::;; Yj ::;; P

(j = 1 , 2, . . , k). .

Obviously, under A l = . . . = A n = 0 this system of equations coincides with the system ( 157). L�t us consider the simplest properties of such systems. First of all we shall show that under any positive integer k we have ) 2 11'i (0I1 A1 + ... +OIn An ) , '" ( P) L...J Nk,n ( A 1 , . . , A n e .

(159)

A1 .. .. . An

where the range of summation is

IAv l

<

kP "

( v = 1 , 2, . . , n ) . .

Indeed, since

LP

e211'i (011 (Xl + . .. +Xk}+ ... +OIn ( X f + ... + X � » ,

then, obviously,

(160)

[ Ch. II, §

Weyl's sums

80

Here we unite addends with fixed

values of the sums x l' + . . . - y l: ( v

�

�

� � . . : : : � .�k. � . � x r + . . . - y� = An

=

12

1 , 2, . . . , n)

}.

Since 1 � Xj � P and 1 � Yj � P (j 1 , 2, . . . , k) , then by ( 158) the number of su ch addends equals N� .� (AI ' . . . , An) and besides =

IA" I

=

Ixr + . . . + x i; - yr - . . . - yA; I

<

kP"

(v

=

1 , 2,

. . . , n

).

Therefore, P

L x l .·

.. ,Yk = l

e 2 11"i (<> l (x d .. · - Yk )+ . . . +<>n ( xj' + · .. - Y k ) )

=

, The equality (159) follows by (160). The relation (159) is the expansion of the function IS( 0' 1 , . . . , an) 1 2 k in the multiple Fourier series. The quantities N�� (Al " ' " An) are its Fourier coefficients. Therefore,

N�� (A1 , . . . , An) 1

1

- f f I S( '-"1� ,

-

•

•

•

o

•

•

0

•

, �'-" n ) 1 2 k e -211"i (<>1>'1+ ... +<> n >' n ) d�<-<1 ·

•

•

�n d'-"

( 161)

and, in particular, 1

..

N�� ( O, . , 0)

=

1

f · · · f IS(O'I , " o

0

" O'n) 1 2 kdO'I . . . dan ·

Hereafter we shall often use the abbreviated notation Nk . n(P) and Nk(P) instead of . . the modulus of Nk(P) • n (AI " ' " A n ). SInce , n (0, . . . , 0), and N (P) (AI , " " A n ) Instead Nk(P) k of an integral does not exceed the integral of the modulus of the integrand, then it follows from (161) that under any A I , . . . , An

N� P) (A1 " ' " An)

1

1

�

f . . f I S( O' I , . . . , 00nWkdO'I . . . dan = Nk (P). ·

o

0

(162)

Systems of equations

Ch. /I, § 12)

81

Let us show the validity of the following equalities:

'" L..J Nk( P) ( AI , . . . , A n ) = Nk( P) (A I, . . . , A n - l ) ,

( 163) ( 1 64)

L

�l , . . . , � n

[N�P>C AI , . . . , A n )r = N2 k ( P ).

( 1 65)

In fact, according to the introduced notation the number of solutions of the system of equations

}

nx l - l ��+ :. ... . Y�kn-l��. �= .An�l - l i s equal to N� P ) (A1! " " An-I). Complete this system by the equation •

•

•

•

•

'

.

•

•

.

.

•

•

\

-

( 1 66)

Nt) { AI, . . . , A n ).

The number of solutions of the completed system equals Every solution of the system ( 1 66) satisfies one and only one of completed systems arising under distinct values and every solution of the completed system satisfies the I extended to all system ( 166). Therefore the sum of the quanti ties is equal to the number of solutions of the system ( 166), i.e. , the possible values equality ( 1 63) is valid. ' The equality ( 164) follows immediately from ( 1'63):

An,

N� P ) ( A , . . . , A n )

An

L

Nt)( AI , . . . , A n ) = L Nt) ( A I , . . . , A n- .) = . . . = L N� P){ A.) p2 k . =

�1

To prove the equality ( 1 65), we consider the system of equations

��.:. : : : � .��� � .�I. � : : : � ��� .� . � } . .

.

( 167 )

. . . - Y�k = 0 The number of solutions of this system is N2 k (P). Collect those solutions , for which under fixed AI , . . . , A n the equations xf +

. . + x�k .

-

yj

-

��.:. : : : � .�k. � � � } , .

x i + . - Yk = An .

.

Weyl's sums

82

[ Ch. 1/, § 1 2

2k . . . - X�k >'n

���.1. � : : : .� .��� � .� ��� .�. '. '. '. � �. . . � � � .

.

k

Yk+l + . . + Y; - Xk+ l

-

=

} [Nt) ( >.! , . . . , >'n)] 2 .

are fulfilled . Obviously the number of such solutions is equal to To each n-tuple of values >' 1 , there corresponds one definite aggregate of so­ lutions of the system ( 1 67) and each solution of the system enters into one and only one of t hese aggregates. Thus, considering all possible n-tuples >'1 , . . . , >' n , we get all solutions of the system (167) and, therefore,

. . . , >'n

L... � [N�P)(>'I , . . . , >'n)] 2 N2 k (P). =

�l ,

n

,

In investigating properties of the system of equations

��.:. : : : � .�k. �.� � } ,

x� +

.

( 1 68)

. . - Yk >' n =

. and in deducing estimates of Weyl's sums, the relationship between the exponential sums

S ( <.<

�

1,

•

•

•

�

, <.< n

p "" e 2 11' i -

)

L...J x= 1

( O t2: + . . . + on x " )

'

a I , . . . , an

and the number of solutions of the considered as functions of n variables equation system ( 1 68) is used essentially. This relationship is seen from the expansion in a multiple Fourier series: of the function I S(

a 1 , . . . , a n)l 2 k 00

L= - 00

( 1 69)

� 1 , . . . , �n

Actually the series ( 169), as it was shown in the equality ( 159), is a finite sum, because if at least one of quantities in absolute value is greater than or equal to then the system (168) has no solutions and the corresponding Fourier coefficient vanishes: � kplI ) .

>'11

kPII ,

( 1)'111

We shall show that the above established properties ( 163)-( 165) of quantities N� P) (>' 1 , . are evident corollaries of this expansion. In fact, setting 1 = . . . = = 0 in ( 169) , we obtain the equality (164):

. . , >'n)

a

an p2 k = L N� P) (>'b " " >'n). � 1 , . . . , ). n

Systems of equations

Ch. /I, § 1 2]

83

, an ) 1 2 k .

The equality (165) follows at once from Parseval's identity for the function IS(al ! ' " : 1

L: [Nt) ( A l ! " " An )f = J

'>'1,·

0

.. , '>' "

=

Finally, setting

an

=

1

.

.

.

1

f [ I S(a l! " " an ) 1 2 k f dal . . . dan 0

1

k dal . . . dan N2 k ( P) . , an)1 S(a , 4 " I l ' f···f =

o

0

0 in (169) , we get

[

� '" )1 2 k - � LJ Nk( P ) ( \ LJ IS( '-"l'" , · · · , '-"n-1 '>'1, ... ,.>.,, - 1 .>. "

]

\ ) e211'i ( O'I'>'I + "'+ O'''_I'>' ,, -d

AI , · · · , An

Hence by virtue of the uniqueness of the expansion of the function in the Fourier series

IS( al , . . . , an - l ) 1 2 k

=

� Nk( P) ( \ '>'), ... ,.>.,, - 1 LJ

.

IS(a l , . . . , an_I)12 k

\ ) e211'i ( 0' 1 '>' 1 + ... + 0',,-1.>.,,- 1 )

AI , · · · , An-1

the equality (163)

follows. The most important question in the theory of the systems of equations

��.� : : : �.�� .� . �

xf +

. . - Yi: = 0 .

},

is a question concerning the character of the growth of the nwnber of system solutions in dependence on the magnitude of an interval of the variation of variables, i.e. , a question concerning the character of the growth of the quantity while increases infinitely. It is easy to establish a lower bound for � (j = Indeed, since 1 � 1 , 2 , . , k ) , the quantities ways. Choosing then YI = , can be chosen in Yk = Xl , we obtain solutions. Therefore we have the estimate .

.

.

•

•

,

Xk

X lpk, X k •

•

•

Nk ( P) . pk

Nk (P) P Xj P ( 1 70)

Weyt's sums

84

[ Ch. II, §

12

Next, by ( 162) and (164)

p2k =

L

Nk P) P' l , " " A n ) � Nk (P )

L

>.t . . . . . >.n

1�

n(n+1) ( 2k ) n P -2- Nk (P ) ,

1>'. I< k p· and, therefore,

Nk (P ) � ( 2k1 n P )

1) 2k- n ( n+ 2

Taking into account this result and the estimate (170), we get the lower estimate for

Nk (P )

( 171) We shall show that under k � n this estimate indicates the precise order of the growth of the quantity Nk (P). Indeed, consider the system of equations

}

�� .:. . . . � �. . � .�� : :::.:. :� , '''

k

x � + . . . + x t = yf + . . . + y:

( 1 72)

In the same way as in the proof of Mordell's lemma (§ 5), it is easy to verify that the quantities satisfying this system coincide with permutations of quantities Since the system , Y k . Hence the number of its solutions does not exceed k ! ( 1 72) is obtained from the system

Yll ' . .

Xl , . . . , Xk

pk .

}

�l � : : : .: . �� �. �� . : � � � � .�k. . . . , x f + . . . + x i: = yf + . . . + Y i: by omitting the last n - k equations, then Nk (P ) does not exceed the number of solutions of the system (172) and, therefore, Nk (P) � k ! Thus, under k � n the estimate (171) has the precise order with respect to P. Under k � n it is easy to show that Nk (P ) � n! Indeed, using the trivial estimation of the sum we get

pk . p2 k - n .

S( a 1 , " " a n ) 1 Nk ( P ) = J . . . J I S( a1, ' " , a n W k da 1 . . . da n o 0 1 1 2k � p - 2 n J . . . J I S ( 0' 1 , . . . , a n W n da 1 . . . da n o 0 2 2 n k Nn (P) � n! p 2k - n . =p 1

( 1 73)

85

Systems of equations

Ch. /I, § 1 2)

Nk (P),

having under k > n a precise order with A question on upper estimates of is much more difficult. This question, which is referred to as the mean s to t ec e p r value theorem, is main in a method suggested by Vinogradov to estimate Weyl's

P,

sums .

In proving the mean value theorem we shall need two lemmas. LE MMA 1 5 . Under any fixed integer a , the number of solu tions of the system of eq u ations

��� : �� � ::: � �:� : �� � �

},

(X l + a) n + . - ( Yk + a) n = 0 does not depend on a and is equal to Nk (P) . , Y k be an arbitrary solution of the system of equations Proof. Let Xl , .

.

.

.

.

.

•

•

.

.

.

.

.

.

.

•

�.1 .� : : : �.�� .� .� } ,

xf + . .

Then under any

( 1 74)

s

=

1 , 2,

.

.

.

.

- Yk

=

( 1 75)

0

we obtain

, n

B (X j + aY ( Yj + aY = L C:as -lI (xj - yj ) , 11= 0 -

k

L ((Xi + a )S - ( Yi + a) B ] i= l

=

S

k

L C: as -II L (xj yj) 11= 0 i= l -

= 0,

where C: denotes the number of combinations of s objects v at a time. Therefore each solution of the system (175) is a solution of the system ( 1 74) . It is just as easy to verify that in its turn each solution of the system ( 1 74) is a solution of the system ( 1 75). But then these systems of equations have the same number of solutions, and this is what we had to prove. Note. According to Lemma 15

a + P e2'11'i (alx + . . . +a xn ) 2k n L l x =a +

and, therefore, under any integer 1 1

a the equality

a + P e2'11'i (a1x + .. . +an X n ) 2k . . . dX1 . . . dX n J J L o

0 x =a +1 I

=

I

" e211' i (alx+ ... +Q n x n ) 2k dXl . . . dX n p

Jo J0 x =l .

..

L...,;

Weyl's sums

86

[ Ch. II, §

12

holds. Let, as in § 6 (the note of Lemma 7) , Tk (P) be the number of solutions of the system of congruences

.�� .� .. . ... �. �� . � � . . (��� :: .

.

X l + . . . - y;

==

.

.

0 ( mod p n )

},

( 1 76)

We shall show that the number of solutions of this system can be expressed in terms of the quantity Nt) (>' 1 , . . . , A n ) ·

LEMMA 1 6 . We bave tbe equality Tk ( P ) =

L

>'1 , . . .

,An

NiP) ( A lP, . . . , A n pn ) ,

wbere tbe summation is extended over tbe region

( 1 77) Proof . It is easily seen that the congruence system ( 1 76) is equivalent to the totality of the systems of equations

�

�

�l . . . � �� � l � . . .. . . X l + . . . - y; = A npn .

.

.

.

},

arising under all possible n-tuples of integers A I " ' " A n . Since under fixed values A I , . . . , A n the number of solutions of this system is equal to Nk P) (A l P , " " A n pn ), then the sum of the quantities NiP) (AlP " . . , A n pn ) , extended over all possible values AI , . . . , A n is equal to the number of solutions of the system of congruences: Tk( P ) =

L Nt) ( Al P ' " ' ' A npn ). Al , . .. ,A n

It is sufficient to carry out the summation over the region ( 1 77) , because otherwise at least for one value v ( 1 � v � n) the inequality IA"p" l � k P" would be fulfilled and the corresponding summand NiP) ( Al P , " " A n pn ) would vanish .

Ch. 1/, § 13)

Vinogradov's mean value theorem

87

§ 13. Vinogradov's mean value theorem As

it was said in the preceding section, the mean value theorem pursues the aim est ablish an upper estimate for the quantity Nk(P), where Nk (P) is the number of in tegral solutions of the system of equations to

� 1 � ::: :. �� .�. �� . �. :'. : � .Y.k.

} , . x f + . . . + x � = yf + . . + Yk ..

.

1 � Xj , Yj � P.

( 1 78 )

The proof of the mean value theorem, suggested by I. M. Vinogradov, is based on recurrent process reducing the estimation of the quantity 'Nk(P ) to the estimation and PI < P. Two proofs of the mean value theorem are < of Nkl (P1 ) , where sented below. The first of them is simpler, but it leads to a result , which is valid repre only under an over-abundant number of variables in the system ( 1 78). The second proof is a bit more complicated, but it enables us to obtain results which are close to final ones. Both proofs are carried out with the help of different variants of the p-adic approach suggested by Yu. V. Linnik [34] for this problem. a

kl k

PI

=

estim ate

holds.

1

1

n � 2, = n2 , P > n n , p be a prime, p 7i � 2P 7i , and . l n p - Then under k > n 2 for the number of solutions of the system ( 1 78) the

L EM MA 1 7 . Let

r

P <

n n+!)

( Nk (P) � 22n k p; rp2 k - -2- Nk_ r (Pd

. f(x) = a l x . . + a n x n and the sums S and S( z ) be determined with

Proof. Let + the help of the equalities

p+pPl

S = L e2 1fi z=p+1

J( z )

P1

S( z ) = L e2 1fi

,

J ( z+px)

.

x=1

Then, obviously,

P P1 2 i J( %+P x ) L e 1f = L S(z) , %=1 %=1

S=L

P

x= 1

%=1

Since the number of solutions of the system (178) grows as P grows , then using the equality ( 16 1 ) and the note of Lemma 15, we obtain 1 1 k Nk (P) � Nk (pPI ) I S12

= J...J o

0

da l . . . da n

Weyl's sums

88

J . . . J ISI 2r IS( z ) 1 2k-2r da l . . . da n . 0

0

Let the maximal value of summands in the sum ( 179 ) be attained at we obtain

J...J I

Nk (P) � p2k-2r

I P+PPl

o

L

x=p+ l

0

13

I

P I 2k-2r-1 �p L z= l

[ Ch. II, §

e2 7ri f( x )

2r

Pl

L e2 7r i f( zo+px)

(179)

z = zo o Then

2k-2r

x= l

It is easily seen that the integral in this estimate is equal to the number of solutio ns of the system of equations

o

( zo

�� : : : : � ,� , � , , , , � :�l,), � .. :: ,� , �� , � ������ , r

,

,

,

xl + . . ,

- Y�

=

(zo + pXI ) n + . .

P < x j , Yj

� P + PPI ,

- ( zo + PYk_r ) n

,

1 � x j , Yj �

PI ,

,}

, or, this is just the same, to the number of solutions of the system

.. . , ,:�� ,� ,��� � . . . ,�, :�� ,� ,��), � ,(,�o, � :,� l,� � .. .. .. ,� ,��o, � ����� �" (zo + XI ) n + ' " - (zo + Yr ) n (zo + pXI ) n + ' " - ( zo + PY k _ r ) n } , P - Zo x Y � P - Zo + pPI , 1 � , Yj � PI , ,

=

<

xj

j, j

In its turn this system is equivalent (see Lemma 15) to the system

P - Zo

<

x 1 > Yj � P - Zo + PPI ,

Let us replace the interval of variation of x j and (j

1 � x j , Yj �

PI ,

Yj by wider one:

=

a

1 , 2, . ,

)

, , r ,

Then, collecting solutions with fixed values of the sums X l + " and using the estimate ( 1 62), we get

'-Y k-r ( v

=

1 , 2" " , n)

Nk (P ) � p2 k-2r L N� �V (A I " ' " A n ) N� 2 P Pd (A I P , " ' , A n p n ) Al , ... ,A"

./

::::::: P

2k-2r Nk-r (PI ) " ) (2 L.J Nr P Pd ( AlP, , , , , A n p , \

Al , ... ,A"

\

n

Vinogradov's mean value theorem

Ch. /1, § 13]

89

where the summation is extended over the region 1 ..\ ,, 1 < r{2P1 )" ( v = 1 , 2, . . . , n). lIenee, since P1 = p n - 1 , using Lemma 16 and the note (90), we get the lemma assert ion:

--

THEOREM 1 5 . Let

n

--

+ 2(n-1} r- n(n2+ 1 } 2k 2 r � (2n) p Nk -r (Pt ) n(n 1} + � 22nk p12 rP2 k- 2 Nk -r ( P1 ) � 2, r � 0, k > n 2 r ,

(

)

1 n 1+n-1 P>n , r

Gr

_ -

•

( .!.)T

n(n + 1 ) 1 -n 2

•

Then for the number of solutions of the system (178) we have the estimate

( 180) Proof. The assertion of the theorem can be obtained with the help of rather simple i n duction. Indeed, under r = 0 the estimate ( 180) is trivial. Let it be true under a certain r � O. Choose

k > n2 ( r + 1 ) Determine positive integers

and

( -)

n 1 1 P > n + n-1

r+l

r, P1 , and a prime p as in Lemma 17: 1 1 pn � p < 2 pn ,

(

Then

)

1 n -1 n 1+_ n-1_ r P1 > p ---n > n

and by the induction hypothesis (181 ) But according to Lemma

17

n( n +1} 2k-2nk 2 r 2 Nk -r ( P1 ) Nk ( P ) � 2 P1 P

Weyl's sums

90

[ Ch. II, §

13

and, therefore, by (181)

Hence, since PP1

=

pn

<

2 n P, (n - l )e.,.

=

ne.,. + 1 ! and

we get Nk ( P )

� 24 nk .,. +2nk ( 2 n P )

n ( n+ l ) + 1 n( n+l ) 2k- -+e r � 4 n k ( .,. +1 ) 2k- --+e: r+ 1 . 2 2 2 P

The theorem is proved. Now we shall consider the question about the precision of the obtained results. Since . then under the growth of T and the corresponding growth of k , the quantity e .,. tends to zero. So under T > 2 n log (n + 1 ) we get

Respectively, under T > 3n log ( n + 1) we have e.,. < 2 ( n� 1) ' Hence it follows that for any e > 0 under k > 3n3 log (n + 1 ) and n � 21., we have the estimate n(n+l )

_ 2 k- -- +e 2 Nk ( P ) � 2 n k p 2 2

=

(

0 P

n(n+1» 2k- -- +e 2

)

•

(182)

On the other hand, by (171)

It is seen from comparison of this estimate and the estimate ( 182), that the order of the estimate ( 180) is almost best possible. A question about the least value of k, under which the estimate ( 180) is fulfilled, is much more difficult. This question is important in connection with the following circumstance: estimates of Weyl's sums obtained by the help of the mean value theorem are, as a rule, more precise, if one succeeds to establish an estimate of the form ( 1 80) under lesser values of k, i.e. , the lesser the better.

Ch. /I, § 13 )

Vinogradov's mean value theorem

91

Let us show that the estimate ( 183)

n ( n2+ 1 ) .

Indeed, according to (171) cannot be fulfilled under k < e, in order to satisfy the estimate (183), the estimate for re the

pk

=

0

(p2 k--n( n+2-l» +e)

Nk ( P)

�

Pk

and,

n( n+ 1 ) . 2

should be fulfilled, but that is possible only under k � Thus the best result which might be expected to obtain is getting a precise estimate (with respect to 1 the order) under k = ('; ) . The estimate (183) following from Theorem 15 was ob tained under k � 3n3 log (n + 1). Using the Linnik lemma (Lemma 9) instead of Lemma 7, we get now this estimate under k � 3n 2 1og (n + 1 ).

n +

p i1

LEM MA 18. Let n � 2, P � (2n) 2 n , p be a prime, � [pp -l] + 1 . Then under k � we have the estimate

n( n2+1 )

�

P <

p i1 ,

and

PI

=

(184) Proof. As in Lemma 17, we introduce the notation f(x) S

+pPI 2 x L e 1
=

Then we get

S(z)

=

=

QI X + . . . + Qn Xn ,

PI x=1

L e2 1
2 P L S ( ZI) " , S( Z k ) . %1, ,%1:: = 1 •••

ZI ,

We shall say that a k-tuple . . . , Z k belongs to the first class, if it is possible to find distinct quantities in it. All remaining k-tuples are to be of the second class. Since

Zj

n

2

%1 , .

�2

.. , .%"

L l S ( ZI ) " , S( Zk) 1

ZI , • • • ,Z.\:

Z I , · . · , %.

2

+2

1

2 2: 2 S( ZI) " , S( Z k ) ,

%1 ,

•••

, Zk

Weyt's sums

92

L:l

L:2

[ Ch. II, §

13

are over k-tuples of the first and the second classes , where the sums and respectively, then observing that � and using the note of Lemma 15, We obtain

P PP1 1

1

Nk (P) � Nk (pP1 ) = / . 0 . / ISI 2k dal dan 1 1 2 � 2 / . . . / L l S(zt) " , S(Zk ) da l . . . da n 1 1 2 + 2 / . . . / L 2 S(zt ) " , S(Zk ) da l . . . da n . 0

Nk

Z I , . . . , Z.

0

( 185 )

Z l " " , Z' k

0

o

'

0

o

o

.

Nf

Denote by and the integrals of the right-hand side of this inequality. Here­ after we shall designate the number of combinations of k objects n at a time by Since n distinct quantities can be arranged on k places in ways, then

Cr .

Cr

1 1 2 2 / NL � ( Cr ) . . / L ; S(zt ) . . . S(Zk) da l . . . da n , .

o

0

'" 1 , . . . , Z k

where in the sum L: ; distinct occupy the first n places and the variables independently run over the interval Hence, observing that

Zj

2

[l, pj .

S(Zl) ' " S(zn)

L;

Zn+l , . . . , Zk

2

z=1

...., p2(k - n) -1 �

z=1

Z l " " , %n

we get

1 1 k n 1 NL � (Cr) 2p2( - ) - L / . 0 . / L 1 S(Z l ) . S(Zn ) IS(z)1 2 ( k - n ) da 1 da n . 2

P

o

%=1 0

0

ZI

,•••

.

0

0

'

, %n

Z

It is easily seen that under a fixed the integral in this estimate is equal to the number of solutions of the system of equations

l ( k . . ��� .: .����.:""".� ���.:.���!.�. ��.:.��. :.:. : : : � . �.� �. .� �� ( Z1 + pa:D n + . - (tn + py�) n ( z + pxt} n + . - (z + pYk_n) n •

.

.

=

.

.

•

},

( 1 86)

Vinogradov's mean value theorem

Ch. /1, § 13 )

w here 1

::;;; xj , yj , Xj , Yj

::;;; Pl , 1

::;;; zj , t j ::;;;

93

p and under

i f.

Zi =f Zj , t i f. tj are fulfilled. Introduce new variables Xj and Yj

(j = 1 , 2,

Zj + pxj = Z + X j ,

..

J

the conditions

. , n

).

Then the system (186) takes on the form

. . ��.:.���.:. '' '. '. � ��.:.���.�. ��.:.����.:. '. '. '. � .( �.� :�.k.� �� . . ( z + Xl ) n + . . . - (z + Y n ) n = (z + pXl ) n + . . - (z + PYk_ n ) n .

}

( 187)

,

,

and the region of variables variation is determined by the conditions 1 ::;;; x j , Yj ::;;; Pt , p - Z < Xj , Yj ::;;; P - Z + PPl and X i ¢ Xj , Y i ¢ Yj ( mod p ) under i f. j . By ( 1 74) the number of solutions of the system ( 187) does not exceed the number of solutions of the system of equations

PPl + p , i f. j ::;. Xi ¢ Xj , Yi ¢ Yj ( mod p ) , 1 ::;;; Xj , Yj ::;;; Pl . 1

::;;; x j , Yj ::;;;

Collecting solutions with fixed values of sums Xl + obtain

.

.

. - yr- n (

v

=

1 , 2,

..

.

, n ) we

::;;; (Cr) 2 p2 ( k -n ) Nk_ n ( Pd L N� ( A Ip, . . . , A npn ) , >' 1 , . . . ,>' n

where

N: ( AI P , . . . , A n p n ) is the number of solutions of the system

� l � " " " � �� � � l �

. . . . . xf + . - y� = A npn ..

},

. .

and the summation is over the region according to Lemma 16

1

::;;; Xi , Yi ::;;;

PPI + p ,

i f. j ::;. X i ¢ X j , Yi ¢ Yj ( mod p ) ,

IA"I

<

n ( PI

+ 1 )" (v

=

1 , 2 , . . . , n ) . But

L N� ( AIP, . . . , A npn ) = T; (pPI + p) ::;;; T� ( mp n ) , >' 1 , . . .

where

, >' n

Weyt's sums

94

[ Ch. II, §

13

(mpn ) is the number of solutions of the system of congruences

and T:

,

�� .� . . �. ��. � �. . . �����:. .

.

xf +

..

.

.

.

- Y�

==

( mod

0

} , n p)

1 � X j , Yj � mp n i i= j =} Xi ¢ Xj , Yi ¢ Yj ( mod p ) .

Therefore, using the estimate (93), we obtain

(188) Now we shall estimate the quantity Nf . Observing that the number of k-tuples of we get the second class does not exceed

n kp n - l ,

2

p

p

L 2 S( Z I ) ' " S( Zk ) � n 2kp2n-2 L I S(z) 1 2k � n 2kp2n - 2 pm L I S(z) 1 2 k - 2n , %= 1 %=1 1 1 N;" � n 2k pmp2n - 2 L f · · · J I S(z)1 2 k-2n da l ' " da n n2k pmp2n - 1 Nk _ n ( P1 ) . z=l o 0 Since by the lemma conditions k � n ( n2+ 1 ) and p > n 2 , then % l , . " , %k

p

=

and, therefore, " 1 Nk � 2' ( 2

n (n+ l ) 2 Nk - n ( PI ) . k)2n PI2 n 2k - -P

(189)

Now we obtain the lemma assertion from (185), ( 188), and ( 189)

n ( n + t) 2 Nk-n ( PI ) . Nk ( P ) � 2Nk, + 2Nk" � 2 ( 2 k ) 2n PI2n 2 k - -P

The recurrent inequality (184) enables us to make the statement of the mean value theorem essentially stronger, because this inequality reduces the estimation of t o the estimation for k Pd ( but not t o Nk- n 2 (Pd as i t was obtained earlier in Lemma 17).

N -n (

Nk ( P)

Vinogradov's mean value theorem

Ch. /I, § 1 3)

THEOREM 16. Let n � 2 , T � 0, k

eT

-

=

95

n (�+ I ) + nT, and

(

)

_ n(n - l ) I _ ! T n 2

•

Then for the number of solutions of the system ( 1 78) the estimate ( 1 90) holds under any P � 1 . Proof. Since, obviously,

then to prove the estimate ( 190) it suffices to show that ( 19 1 )

If T

=

0, then this estimate takes on the form

and i s fulfilled by ( 1 73) under any P � 1 . Apply the induction. Let under a certain T � 0 and k = n ( n2+ 1 ) + nT the estimate (191) be fulfilled under any P � 1 . Prove it for T + 1, i.e. , under k = n ( n2+ 1 ) + n ( T + 1) . We shall consider the cases P � ( 2n ) 2 n k 2 and P < ( 2n ) 2 n k 2 separately. If P � (2n ) 2n k2 , then by Lemma 18 Nk ( P )

� 2 ( 2 k) 2 n PI2n P2

where i p7i � P < p 7i and PI = [pp- I ] the induction hypothesis, we obtain I

1

+

n ( n+ 1 ) k- -2 Nk- n ( PI ) ,

1 . Since k - n

=

n( n + 1 ) + nT, then using 2

[ Ch. 1/, § 13

Weyt's sums

96

Observing that

P > 4 k 2 and, therefore,

P1 we get

< 2P

n(n+1)

(pP1 ) 2 k - -2 p{ r

1_1n + 1 < 2 P 1-1n ( 1 +

< 2" T p

n(n+1) 2 k - -- + 2

< 3 . 2" T P

2k-

1

2k '

( 1 - -1 ) € r n

n ( n+ 1 )

-2

)

( 1 + -1 ) 2 k - n 2k

+€r+l

But then it follows from ( 1 92 ) that

Let now

P<

(2n ) 2 n k 2 . By the induction hypothesis

and, therefore,

Hence, observing that

for values P less than k 2 ( 2n ) 2 n , we get the estimate ( 1 9 1 ) too. Thus the estimate is fulfilled for any P � 1 , and the theorem is proved completely. 1 Let now k � ko , where ko = n ( n2+ ) + nT. It is easy to verify that the estimate ( 190) proved in Theorem 16 for k = ko holds under k > ko as well. It suffices to use the evident inequality

Nk (P ) � p 2 k - 2 k o Nko ( P ) and apply the estimate ( 190) to Nk o (P ) .

Estimates

Ch. 11. § 1 4)

of

Weyl's sums

97

Note . With the help of more complicated considerations [44] the mean value theo­ rem can be improved by removing the factor p e and so it is possible to get under n � 1 , k > cn 2 10g n, p � 1 the estimate ..

Nk (P)

k n(n+t) � C(n)P 2 - 2 ,

( 1 94 )

--

where c is an absolute constant and C(n) is a constant depending on n only. An elementary proof of the mean value theorem in the form ( 1 90) is obtained in the article [37] . §

14 .

Estimates of Weyl's sums

To obtain estimates of Weyl's sums by the Vinogradov method besides the mean value theorem we need two comparatively simple lemmas. L EMMA 1 9 . Let f(x) be an arbitrary function taking on real values. Then under any

positive integers P, PI , a,

an d

k we have:

P1 - l P i I(x) 21f e L L e21fi I(x+y) + P1 1 � � L P I '" O x=1 y= x=1 P P1 1 P 1fi I(x) 2 2 1fi I( x+ayz) + 2 aPl , e L 21 L L e p x=1 x=1 y,z =1 k+t 2 2k P P-l P i i k+1 /(x) /(x+y) 21f 2 21f �2 L e Le x=1 y= O :1: =1 P

10 .

-

2° .

,

�

3° .

L

Proof. Under any integer y � 0

y

P

L e 21fi 1(:1:) x=1

=

=

e21f i 1(:1:) +

y+ P

e 2 1fi 1(:1:)

L L :1:=1 :I:=y+l LP e21fi I(z+y) + 20yY,

( 1 95)

P+y

-

e 21f i 1 (:1:) L z=P+l ( 1 96)

:1: = 1

where I Oy l

� 1 . Hence, carrying out the summation over P P /(x) � L e21f i /( :I: +Y) 2 , 21fi e L + y z=1 z=1 PI

P

L e 2 1ri 1 ( :1: ) :1: = 1

Pl - l P

� yL=o

i

e21f I( :I: + Y) L :1:=1

y,

we get the assertion 1 ° :

+ PI (PI - 1 ) .

Weyl's sums

98

To prove the estimate summing with respect to

[ Ch. II, §

2 ° we replace y by ayz in the equality (196) and carry out y and z:

I: e27ri f ( x ) = L e 27ri f( x +a yz ) + 20(y, z) ayz, P

P

x=1

x=1

P; I: e27ri f ( x ) = I: I: e27ri f( x +a yz ) + 2a I: O(y, z) yz . x=1 y,z=1 x=1 y,z=1 Hence, because IO(y, z) 1 ::::; 1 and P

P

the assertion

PI

PI

2 ° follows:

x=1 y,z=1 x=1 Determine SI and PI with the help of the equalities 2k PI mm. ( [SI2 k� l ] + 1 , P ) . SI I: I: e27ri f( x + y) y= o x=1 P

P-l

=

=

1

SI2 k+ l ::::; PI ::::; P and, therefore, __

Then

1- " e27ri f( x + y) ( 1 S -21k S 2 k1+ l " "" P1 1 ) "" 1 ' p1 y=O L.J x=1 L.J Hence, using the estimate 1 we get the assertion 3°: P

1

I

P

__

�

�

_

0,

I ::::; ;1 L L e27ri f ( x + y) + PI - 1 ::::; 2 S12k+ 1 , x=1 y= o x=1 2 k+ l 2k L e27ri f( x ) ::::; 22k+ 1 SI 22 k+ 1 I: I: e27ri f( x + y) x=1 y=O x=1 P

14

I: e 27ri f ( x )

PI - l

P

P

__

P- l

=

P

Ch. 1/, § 1 4 ]

L EMMA

Estimates of Weyl's sums

20.

If a function

99

F(a l , ' " , a n ) is given by the multiple Fourier expansion 00

L and

satisfying the condition

F ( a 1 , . . . , a n) � O, , qn we have th en under any positive integers Q1 ,

00

•

•

•

L

Proof. Since

F(a 1 , . . . , a n ) � 0, then F (a t , . . . , an )

�

91-1 '" I=O XL

9n -1 F (a 1 + X l , . . . , a n + X n ) L Qn Q1 x n =O

( 1 97)

00

=

L

By Lemma

X l=O x n =O 2

Using this equality, we obtain the lemma assertion from ( 197 ) : F ( at ,

.

. . , an )

00

L 00

L COROLLARY . Let f(x)

= al X + . . . + a n x n and S (a 1 , . . . , a n ) � n( n+1)

Then under any positive in tegers r

n

e2 11" i /(x ) . L x =l p

=

and k we have the estimate

I S(al , . . . , a n ) 1 2 k � k n - 1 P-2- -r

L N�P\O, . .

i'>'. i < k p r

.

, }. r ,

.

.

.

, 0) e211"iar.>.r .

Weyl's sums

100

[Ch. II, §

14

Proof . Let us consider the function

F(

a t . . . . , a n) IS( at. . . . , a n) 1 2k . =

By ( 1 59)

where the range of summation is

IA " I

( v = I , 2, . . . , n ) . kP" Since F( � 0 , the lemma conditions are satisfied. Choose I if v = r , = if v t= r . kP" Then using the lemma we get

aI , . . . , an )

<

( 1 9 8)

q" {

( 1 99)

IS (al , . . . , a n ) 1 2k ( � Nk P } ( q :::::: ql . . · qn � ./

AI , ... ,A n

\

1 "1 1 ,

.. , ·

IA " I < { k{r

qnAn )e21fi ("'lql Al+ ... +a n qn A n} , \

(2 0 0 )

where by ( 198) and ( 199) the range of summation may be written in the form

!! � ; �:

or, that is just the same, in the form

Al =

.

.

.

=

A r- l

I A rl

Ar+l n( n+t-} -r n-l -2 ql . . . qn k P ,

=

Observing that

0,

<

k pr,

=

...

An

=

=

=

O.

) IS(a t , . . . , a n) 1 2k � ql . . . qn L N� P} (0 , . . . , A r , . . . , 0) e21fi "'r A r IA r l 2, f( x ) = a lx + . . . + a n x n , and a n -aq + 2q0 ' ( a , q ) = 1 , 1 01 � 1 . H P � � p n - l then the estimate l-��-n 1fi 9n2 n f(x e2 e } 3 P � L

from (2 00 we obtain the corollary assertion:

=

•

n

=

q

p

holds.

x= l

log

•

•

\

•

•

•

Estimates of Weyl's sums

Ch. /I, § 1 4) Proof .

Using Lemma 19, we get

P

L e21ri f( x)

2k+l

x=l

::;; 22k+1 = 2 2k+1

where

101

P-l P

L x=l L y=o P-l P

L L y=O x=l

e21ri f ( x+y )

2k

e21ri (al( y ) x+ ... +a n ( y)x n )

2k (2 01)

all ( Y ) = � f ( II ) ( y) and, in particular , v

a n- l ( Y ) - (n 1 I ) ! f ( n - l ) ( Y ) - n a n Y + a n -I ' _

_

_

By virtue of the corollary of Lemma 2 0 we have

P

L

x=l

n e2 1r i (al (y)x+ ... +a n (y)x ) n( n -l) +1 ::;; k n - 1 p -2

L

e2 11'i f ( x )

x=l x

Choose e = Then we get

2k + l

Nk( P) ( O , . . . , A n -l 0 ) e21rj (na n y+a n -dA n- 1 . n- 1 \

IA n _ t !
Substituting this estimate into obtain

P

2k

(20 1 ) , after interchanging the order of summation we

n ( n -l) ::;; kn - 1 2 2k + 1 P-2- + 1

Nk(P) ( O, . . . , A n -l O )

P-l

e211'i na n A n _ I Y L y=O

2 ! 2 in the note of Lemma 14 and use the condition P

::;;

q

::;;

p n -l .

and, therefore,

2k+ 1

P

e 21ti f( x ) L x=1 Choose

[ Ch. II, § 1 4

Weyl's sums

102

r =

� n 3 k n 2 2k+7 P

n(n+ t ) +1 -2 2 n2 Nk (

P

).

(20 2)

[3n log nJ + 1 in Theorem 16. Since, obviously, n(n

- 1 ) ( 1 _ .!.)r

2 then by ( 190) under k

.

n

<

-

=�

n(n 1 ) 2n 3

_

2n

1 _, 2n 2

_

= n ( n2+ 1 ) + nr we have

Substituting this estimate into (202) and using 3n 2 log n � k � 4 n 2 log n we obtain the theorem assertion:

x=1 p

L x=1

e 2 11'i

f( x ) � e3

1 - ..!... 1 � n P 2k+ 1

� 3nP e

1

1

9 n 2 Jog n

.

The Weyl's sums estimate obtained in Theorem 17 depends on rational approxi­ mations for the leading coefficient of the polynomial f(x) = a 1 X + . . + a n x n . Let us show that a similar estimate is valid in the case, when rational approximations of an arbitrary coefficient a r (2 � r � n) are given. LEMMA 21 . Let Q, t be positive integers,

a 8 a= -+-, q q2

(a, q ) = 1 ,

and T b e the number of solutions of the inequality

I l ax l l Then we have the estimate

<

-t , q

181 � 1,

Ch.

Estimates of Weyl's sums

/I, § 1 4)

103

Proof. Denote by T(fJ) the number of solutions of the inequality

By

l I ax + fJ lI < -qt , 1 � x � q. (14 9 ) under a certain integer b only depending on fJ, the estimate 1 ax + I l -q-b l � l Iax + fJll + q

holds . Hence it follows that inequality

T ( fJ )

does not exceed the number of solutions of the

a + 1 . 1 , the number of solutions of this inequality is equal to 2 t + 1 and Consider now the inequality

Since ( , q )

T(fJ) � 2 t

=

t

where

Ql = [�]

Since

Ql q >

Il ax ll < -q ,

(203)

+ 1 . Replacing x by q +

Q, then

(203) and, therefore,

Xl

T

X2 ,

we rewrite this inequality in the form

does not exceed the number of solutions of the inequality

T�

Ql -l

L T( aq xt ) .

+ 1 , we get the. lemma assertion: T � 2(2 t + I )Q l � 6 t ( 1 + �). LEMMA 22 . Let n > 2, f ( x) = a l x + . . . + a n x n , and fJ s ( X ) - 1 f (8) ( x ) - as + Cls + l as + l x + . . . + Cnn -s a n x n -s ( � 1 ) .

Hence using the estimate

a

_

I"

_

s.

Iffor some integers y,

T(a qx d � 2 t

z,

s

and t (0 � y, z < P) under s = 1 , 2 , . . . , n - l the inequalities (204)

hold, then the inequalities (s

are valid also.

= 1 , 2,

. . . , n

- 1)

Weyl's sums

104

Proof. Determine quantities

ha =

n! (n - s + 1 ) !

hs

and

[ Ch. II, § 1 4

'Ya by means of the equalities (1 �

,

s

� n - 1)

.

Then, obviously,

f3n-s ( x ) -

O!n-s

.- 1

- i +1 . s i +l , + ( n - s + 1 ) O!n-.+l X + "' L...t Cn· - i +l O!n-J+I X -

i =1 s-1 s- i +1 hs s- i + l h f3n-s ( X ) = h sO!n-. + 'Ys X + '" L...t Cn-j+l -;;:-- 'YiX J+l j=1 s-1 s - j+l , = h sO!n-s + 'Ys X + "' L...t Hs i 'YiX j=1 •

where the quantities

Hs i

(205)

are determined by the equalities

C s- i +1 ( n - j ) ! Cns--ii +1 (1 � j � +l � hj+l - n- i + l ( n - + I ) ! It is seen from the definition of hs and Hsj , that the estimates H . � n 2 s- 2 j HSJ. -

_

s

.J

s

- 1).

""

hold, and that h. and iI. i are integers. Using the fact that under an integer m and an arbitrary Im l lhll is valid, we obtain from (205)

'Y

the estimate

(206)

IIm'Y ll �

s-1 h. (f3n-.( Y) - f3n-.(Z ») = 'Y.( Y - z) + L H. i 'Yj (y · - i + 1 - z ·- i + 1 ) , i =l s-1 Ih .( y - z)1 I � Il h. (f3n - .( Y) - f3n -. ( z») II + L II H.i 'Yi (Y - z) (y s -i + . . . + zS - i ) 11 i= 1 s-1 (207) � hs llf3n-s( Y) - f3n - s ( z) 11 + L Hs i (y s - i + . . . + zS -i ) lhiCy - z) lI . i= 1 Now we shall show that under s = 1 , 2, . . , n - 1 the inequality .

(208) holds.

Estimates

Ch. 1/, § 1 4)

of

Weyfs sums

lOS

Indeed, under s = 1 this inequality coincides with the last of the inequalities

(204):

2 � s � n - 1 and the inequality (208) be fulfilled under (j 1, 2, . . . , s - 1 ) . (209) If follows from (207) by virtue of (206) that Apply the induction. Let

j�

s - 1:

=

s-1 + L n2 s - 2j(S - j + l) p s -j ll·Yj ( y - z ) l I · j =1 Hence, using the induction hypothesis and the estimate (204), we get s-1 s1 s l I'Ys ( Y - z )1 I � n - p !-s + L n 2 2 j (s - j + l) p s - j (2 n) 2 j - 2 p !- j j=1s - 1 t . t ( (2 n ) 2 s - 2 _ + 1 ) 22 j -2 ) _ '" n s - 1 + n 2 s - 2 '"' n-s n-s p p � ( J =1 Thus, if the inequalities (209) hold under j � s - 1 , then they hold under j � s too. Therefore these inequalities hold under any j � n - 1 . This coincides with the assertion (208). Now, observing that n ! a s+1 = S ! ')'n -s and using the estimate (208), we get the �

s

_

' J

_

<

_

lemma assertion:

s ! (2 n ) 2 n - 2 s-2 � ps (2 n )2 n � p a (S 1 , 2, . . . - 1 ). LEMMA 2 3 . Let n 2, f( x) = alx + . . . + a nx n , f3s ( x ) = � f ( 8 ) ( X ) , and under a certain from the interval 2 � � n a r = -aq + 2q() ' (a,i q ) = I , 1()1 � 1 . � �

<

>

r

r

FUrther let P � q � p r-l and the sum

=

, n

[ Ch. II, § 1 4

Weyl's sums

106

be extended over those values of y and z (0 � y, z < P) , which under a cert ain positive integer t satisfy the inequalities

t

1I.B. (y) - .B. (z) 1 I < p .

(s

=

1 , 2, . . . , n - 1).

(21 0)

Then we have the estimate

L1

n(n- 1 ) 1 n � (2n)3 P -2- + t.

y, z

Proof. Denote by T1 the number of summands in the sum L 1 ' Then estimating all

the summands trivially, we obtain

L1

n(n - l ) � P -2- Tl '

(21 1)

y,z

Since Tl is the number of those values of y and z, which satisfy the conditions (210), 'then by Lemma 22, T1 does not exceed the number of solutions of the system of inequalities

O � y , z < P (s = 1 , 2, . . , n - 1 ) , .

and, therefore, does not exceed the number of solutions of the inequality

o � y, z < P. Replace n! (y - z) by x. Then, obviously, Ixl < nIP and the quantity takes on each integer value at most P times . Therefore T1 � PT, where T is the number of solutions of the inequality

x

I x l < nIP. Since

q

� p r- 1 , then T does not exceed the number of solutions of the inequality I x l < nIP,

and by Lemma 21

Estimates

Ch. (I. § 14)

But then, observing that

of

Weyl's sums

107

q � P, we get

The lemma assertion follows by substituting this estimate into (21 1 ) :

TH EOREM 18 . Let n

If

P � q � p r- l ,

n(n-l ) n(n - l ) I: 1� P-2- T1 � (2n) 3 n P -2- + 1 t . Y,:e > 2, f( x ) = (t1 X + . . . + a n xn , 2 � , � n, and () a r = -qa + (a, q) = 1 , I()I � 1 . q2 , r

then

P 1 I: e 27ri f(X) � e 3 n p 24 n2 )og n x=1 f3s(Y ) with the help of the equalities f3s (Y) = � f (s ) (y) ( = 0 , 1 , . . . , n).

Proal. Determine quantities

8

Then, obviously,

f( x + y ) = f3o(Y) + f31 (Y ) X + . . . + f3n(y )x n and according to Lemma 19

P

L e27ri f( x ) x=1

2 k+ l

� 22k+1

2k x + y) 7ri 2 f( e L x=l 2k P I: 2 7r i (Pl(Y) X + ... + Pn (y) x n ) x=1

P-1 P

L

y=o P-1 = 2 2 k + 1 I: y=o

e

Further, using the equality ( 159) , we obtain

P

L e27ri f( x ) x=l

2 k+ 1

P-1 � 22 k+1 I: L N� P ) ( >' 1 , " " y=O Al , ... , A n � 2 2 k+1 L N� P) ( >' 1 , , >' n) Al , ... ,An •

•

•

>' n ) e 27ri (A I Pl(y) + . .. + AnPn(Y» P-l

L e 27r i (A I PI (y) + .. . + An Pn(Y» ,

y=O

[ Ch. II, § 14

Weyl's sums

108

where the range of summation is

( = 1 , 2, . . . , n). v

Hence, using the Cauchy inequality (1 43 ) and the relation (165), we get

4k + 2 P e271'i /(':) L .:=1

2 P-1 e271'i (AIPI ( y )+ ... +AnPn (y» L y =O (212)

where

P-1

V(P)

=

L L AI , ... ,An y= O

e271'i (AIPI (y)+ ... +AnPn(y»

2 •

Now we shall estimate the magnitude of V(P). Observing that f3n ( Y ) does not depend on y , we obtain V(P) � 2 k p n

.; 2 k P '

P-1 L

,�o � (

� (2 kt p n

m;n 2 kP ' ,

( Ly, z 1 + Ly, z � '

2I 1 p.(y)

� ,1, ( ' ) 11 )

(213)

where the sum

is extended over those values of y and inequalities

z,

which under a certain t � 1 satisfy the

( s = 1 , 2 , . . . , n - 1). Respectively, the sum

(214)

Estimates of Weyl's sums

Ch. /I, § 1 4] is

over those values of y and z, for which there is

s

109

(1 �

s

� n - 1 ) such that (215)

By Lemma 23 for the sum E 1 the estimate n(n- 1 )

L 1 � (2 n) 3 n P -2- + 1 t y,%

holds. Applying the estimate (215) for one of factors in (214� and estimating all the other factors trivially, we get

2: 2 � y , ::

P-1

2:

P

y , z= O

n(n - 1 ) 1 n(n-1) 2 - = -1 p 2 + 2 t t -

•

-

Since by virtue of (213) V(P) � (2k) n p n

(L y,%

)

1+L2 ' y , ::

then choosing t = [ vIP] + 1 we obtain

(

V(P) � (2kt p n (2n) 3 n P � 3 (2k) n (2n) 3 n p

n(n- 1 ) -- + 1 VP 2 + 2

n(n +t) 3 2 - + '2 .

)p P -2- +2 ) n(n-1)

Substituting this estimate into (212) , we get P

L e 211'i /( x )

x=1

4 k+ 2

� 3 (2kt2 4 k + 2 (2n) 3 n N2 k (P)P

[i + � n log n] + 1 , n(n + 1 ) k = [� + ] + nrl ' 4

Choose

ri =

n(n + 1 ) 3 -2 + '2

r = 2r1 ,

It is easy to verify that the estimates

2k

� ?'

n(n + 1 ) + nr, 2

r > 3 n log n + n,

(216)

110

Weyl 's sums

(

)

r n(n - 1 ) n(n - 1) 1 _ .!. < n �� 2 hold. Therefore, using Theorem 16, we obtain

[ Ch. II, § <

�

1 4 ··

U

The theorem follows by substituting this estimate into (216):

P

L e2 '11'i /( x ) x=1 P

L e2

x=1

'11'

4 k +2

37

� ( 8 k ) 4 k +2 ( 2n t 3+ a n (2 k ) n p 4 k+ 2 4

i /( x ) �

e

3n

p

1

The estimates of the form

P 1 - --�-L e2 '11' i /( x) � C (n)P n 2 Jog n x=1

(217)

obtained in Theorems 17 and 18 are established on the assumption that P � q � p r-l , where q is the denominator of rational approximations of the r-th coefficient of the polynomial f(x) = a l X + . . . + a n x n :

ar =

-aq + q()

2'

(a, q) = 1 ,

1()1 � 1 (2 � r � n).

I t can be shown that the estimate (217) holds under p E � q � p r- e with an arbitrary c > 0 too, but it leads, as in the Weyl method ( see the note of Theorem 14), to worsening the constant 'Y. § 1 5 . Repeat ed application of the mean value theorem Let f(x)

=

a x + , . . + a n + 1 X n + 1 and S ( P) be Weyl's sum l

S(P) =

p

L e 2 '11' i f( x),

x= 1

(218)

We shall write the estimate for this sum in the form

I S( p ) 1 � C (n) p l -P

(219)

Repeated application of the mean value theorem

Ch. II, § 15)

111

and call p P a reducing factor. A generic peculiarity of different

methods of the estimation of Weyl's sums consists the fact that a reducing factor becomes smaller as in the process of obtaining the estim ate (219) the sum S( P ) is raised to a greater power. So in the methods of Mordell, Vinogradov, and Weyl the sum is raised to a power having the order n , n 2 10g n and 2 n , respectively, ultimately it leads to estimates with reducing factors in

1

pi ,

"1'1

p n 2 log n and p 2 n , where 71 and 72 are certain positive constants.

"1'2 -

---

Results exposed in this section are of another character. Here Weyl's sum is raised

to a comparatively large power having the order up to n 4 , however this does not

lead to worsening estimates. On the contrary, it becomes possible to improve the reducing factor and besides to decrease (or even replace by ah absolute constant) the coefficient C(n) in the estimate (219). The last circumstance is of great importance in those cases, when under the growth of P the degree of the polynomial f ( x ) grows also. These results are based upon the following lemma. LEMMA

estimate where

24. Under any positive integers k 1 and

V

1l = "'Nk1( P) ( A\ 1 , · · · , A\ n ) Nk2( P) ( r1"

k2 for the sum (218) we have the

" , rnI l ) e21fi (P11'1+ ... +Pnl'n ) ,

£J

the range of summation is

(v an d

the quantities f3 11 are determined under 1 �

v

= 1 , 2, . . . , n)

� n by the equality

Proof. Consider the sum

Sl =

P

P

L L e2 1f i f ( 2: + Y) y=l 2:=1

2k1

Define quantities all ( Y ) and f30 with the help of the equalities

a ll ( Y ) = f30

�v. f( II) (y )

(v

= 0, 1 , . . . , n + 1 ) ,

= a1 A1 + . . . + a n A n

and write the polynomial f(x + y) in the form

(220)

Weyl's sums

112

[ Ch. II. §

15

By ( 159) 2kl

� e21ri ( al (y)x+ ... +an + l (y)X n+ 1 ) P

x= 1

=

Al , ... ,An+l

and, therefore, n 1 81 = � � e 2 1ri ( at {y) x+ ... +an+l (y) X + ) P

P

2kl

y = 1 x=1

Since by ( 163)

then observing that 81 �

=

a n + l {Y) does not depend on Y, we get

"

�

AI , . .. ,An

P

21r i (al(y)AI+ ... +an ( y)An ) " �e y=1

\

\

AI , . . . ,An+ 1

�

Nk( lP) ( " 1 , · · · , "n+l )

� e 21r i (al(Y) AI + ... +an ( Y) An ) P

N! �) ( ,xl , . . . ' '\ n )

y=1

.

Applying the inequality (141) and using the relation

al {Y),\1 + . . . + a n {y),x n = /30 + /31-Y + . . . + /3n y n , which follows from the definition of the quantities /31/ and al/{Y) we obtain '

51"

.; x

C�'"

N! ;) (�" . . . , � n )

" ( P) ) � Nk l ( "I ! . . . , /In \

AI ,. " ,An

\

)

,

" -,

P

2 (Pl Y+". +Pny n ) " � e 71'i y =1

( P) " � Nk l ( "\ 1 1 . . . , "n ) \

Al t". ,An 1 2kl k2 (2 ) =p V,

P

2 k2

21r + " .+Pn y n ) " � e i (PI Y y= l

2k2

(221)

Repeated application of the mean value theorem

Ch. /I, § 1 5)

113

wher e V is determined by the equality (220). Let us show that V � p2k \ . Indeed, it is seen from the determination of the quantities f3v that in the equality (220) the sum f31 J1. 1 + . . . + f3n J1. n is a homogeneous line ar function of the quantities

Al l " . , An :

B ut t h en by ( 159 )

P " L...J N(kl ) ( A\ I ,

.

.

•

\ ) e2 1l'i ( P\I'\ + ... + Pnl'n ) , An

=

(222)

and the quantity V can be estimated by the summand obtained under /Ln =

0:

V=

" Nk2( P) (II.,.. 1

L...J

,

•

•

.

, 11.,.. n )

" L...J

Nkl( P) ( AI\ ,

•

•

•

,

\ )e An

J1. 1

=

.

. =

2 1l' i (Pl l'1 + ... + Pn l'n )

Nk2 ( P) L N� �) ( AI , " An) = p 2 kl Nk2 (P) � p2 kl . >.\ . . . ,>'n Now, using the inequality ( 195 ) , we get 2kl -1 P x+y) 21l'i I S(P) 1 2 kl +1 � 2 2 kl+1 PL L /( e y=O x =1 P L e 21l'i /(x+ y) x= 1 �

. .

.

( 22 3 )

,

Hence, using the inequalities ( 221 ) and (223), we obtain the lemma assertion:

I S(P ) 1 4kl k2+2k2 � 24k lk2+ 4 k 2- 1 ( p4kl k 2 + S: k2 ) � 24klk2+4k2-1 ( p4k 1 k2 + p4kl k2-2kl V) � 2 4kl k2+4k2 p4k\ k2- 2kl V. COROLLARY . Under any positive integers k l , k 2 ' and m ( 1 � m � n ) for the sum ( 2 18) we have the estimate ( 224 )

[ Ch. I/, § 1 5

Weyl's sums

114

where

(2 25 ) the summation range is 1 . \ 1 equalities

<

k l P j and the quantities

/3"

are determined by the

(1 � Proof.

Since by (222)

" � Nk( P , ) ( A\ I , . , . , An\ ) e2 1\'i (�'I" + .. . +�nl'n )

'- 0

,::;

v

� n) .

,

then, obviously,

\ , , . . , An \ ) e 2 1\'i (�'1'1+ ... + �n l'n ) ( P) ( ,-u 1 , . . , , ,-1/. n ) � " Nk,( P ) ( AI " � Nk2

v=

Al , ... ,A n

Il l , . . . , It o

/' �

\ , · · " An \ ) e 21\'i (�II" + ... +�n l'n ) . " � Nk( ,P) ( Al

Nk2 ,n (P) " �

1'1 , .. ·,l'n A, , ... ,An

Interchanging the order of summation, we obtain v � Nk , n (P) X L N! �) ( Al , . " ' An ) min 2 kl p, 2 11 1 . . · min 2 kl p n , 2 ll 11 11 n A , , ... ,A n m ( m-l ) � (2 k I ) n P 2 Nk 2, n ( P ) L N! �) ( Al , . " ' An ) min pm , I · . . min p n , l I X 11/3 1 n ll . A, ... ,A n 2

,

(

�)

(

(

�)

( ;)

�)

Hence, observing that under m � 2 the quantities /31 , " " /3m do not depend on An +2- m , ' . . , An and using the equality L N!�) ( AI " ' " An) = Nk�) (AI " ' " An+l -m) ,

A n + 2 - m , ... ,A n

we get the corollary assertion: m( m -l ) V � (2 kl r p 2 Nk 2 n ( P ) ,

(

�)

( ;)

N! �) (Al , . " ' An+l _ m) min p m , I /3 II ' " min p n , I l n ll 1 A, , ... ,A n +' - m m ( m-l ) � (2 kl r P 2 Nk" n+l - m (P)Nk2 ,n( P) a, X

L

Repeated application of the mean value theorem

Ch. II, § 15 ]

where 0' =

� L....J

.

>' 1 , ... , >' n + 1 - m

115

. (pm l I ,8m1 ll ) . (pn l I ,8n ll ) .

mm

1

. . . mm

,

,

I t is seen from the inequality (224) that Lemma 24 reduces the estimation of the sum S( P) to the estimation of the product of the quantities Nk 1 , n + 1 -m ( P ) and Nk , n (P). In estimating of this product one has to ap p,ly the Vinogradov mean 2 value theorem twice. That is why the use of Lemma 24 in estimating Weyl 's sums is referred to as the repeated application of the mean value theorem. Let us show that the repeated application of the mean value theorem enables us to strengthen the estimates of Weyl's sums obtained in Theorem 1 7. TH EOREM 1 9 . Let P

>

1, n

ll!n + 1

=

>

2, f(x) = ll!l X + . . . + ll! n + 1 X n + 1 ,

a 8 + -2 , q q

-

(a, q ) = 1 ,

181 � 1 ,

and r be determined by the equality q = pr. Then under any r from the interval 1 � r � n we have the estimate p

2n log n

1 ----''"---

L e2 ".i /(x ) � 2 e 82 (log 3n -Iog 8) P 95n2 (log 3 n -Iog 8) ,

(226)

x=l where s

=

min ( [r] , [ n + 1 - rD .

Proof. According to the corollary of Lemma 24 the estimation of the sum (226) is reduced to the estimation of the magnitude of

where the summation is extended over the region 1. \ 1 < k1 Pi ( 1 � j � and ,8" = C:+1 ll!"H A1 + . . . + C�H ll! n + 1 A n + 1 - " . Determine ,8� by means of the equality ,8� = C:+ 1 ll! ,,+ l A1 write ,8" in the form (m �

n

+

1 - m)

+ . . . + C� ll! n A n - "

v � n) .

and

Weyt's sums

116

[ Ch. II, §

Since fJ� does not depend on An+l-v, then by the note of Lemma 14 under € we have the estimate

Denote by 0"1 the sum

0" 1

=

(

1 . p m+1 mm , I fJ m l + 1 11 '\l '''','\n - m

" �

)

. . .

(

=

2

15

�2

)

mm pn , fJ1 . Il n ll .

Observing that among the quantities fJm , . . . , fJn only fJm depends on An + 1 - m , we get

(227)

Further let the sum 0"2 be determined by the equality

0"2 =

(

"

1 . pm + 2 mm Il fJ m + 2 11 Al , ... ,'\n- m - l �

,

)

.

.

.

(

1)

. mm pn , fJ . Il n ll

Since among the quantities fJm + b ' " , fJn only fJm +1 depends on A n - m , then similar to (227) obtain we

Continuing this process, finally we arrive at the estimate

Choose m = n + 1 -

s.

Then =n+

min ( [rJ , [n + 1 r]) ;:: n + 1 m = min ([r] , [n + 1 1']) � m

-

1

-

-

-

r, r,

Repeated application of the mean value theorem

Ch. II. § 1 5]

and under

pn+ 1 - V �

q,

v

�

pv

a

m

�

q,

the inequality and

n+1

- v

�

r

�

117

is satisfied. Therefore,

v

n ( n1 n + 1 ) - m(m-l ) + 1 i. 2 � k f24n 2 p i II 4 p v � (4k 1 r24n 2 P 2

v=m But then according to the corollary of Lemma 24 P

L e211'i f ( x ) x=1

4 kt k2 + 2 k2

(228)

Now we use the mean value theorem in the form indicated in Theorem 16:

T � 0, k = n ( n2+ 1) + nT and n(n 1 ) (1 - .!. ) r Cr n . 2 Determine T1 and T2 by the equalities 3n] T1 = 28 , T2 = 1• + [2n log -:;-

where P � 1,

_

Then, obviously, C rt

_

-

C r2 � '"

-

( 1 - -;1 ) 28 8 (8 e2- 1 ) < 82 2e2 ' 2 3 1) ( 1 2n (Hlog :) n( n 1 - -)

8 (8 1 ) -

2

n( n

-

2

�

�

'"

n

Therefore, choosing ( k 1 = s s + 1) + 2 2

we obtain

s

.

2

an d

k

2=

-

18

1 ) 82

n2

2

8 < -. 18

n(n + 1) n n [2 n log 3n ] + + --; 2

,

[ Ch. II, § 1 5

Weyt's sums

118

Now, observing that 1 we get P

L

e2rri /(x)

�

8 � n + t , 2 k l + 1 � 78 2 and ETI + ET2

� � 82 ,

from (228)

4 kl k2+ 2k 2

x=1

where C(8, n) = (2k l ) 2 kl (2 k2 ) 2k 2 (28 ) 83 (2n t3 ( 8 k 1 k2 r24k l k2 +4k 2 +4 n2 Hence, because under n > 2 k2

<

2 . 5 n 2 log

p

x= 1

24 kl k 2+ 2 k 2 n 4 0n 3 .

3n

-

8

the theorem follows:

L e21fi I( x)

<

20n3

� 2 n 2kl k2+k2 P

1

_ _1_ 38 k2

2n log n

� 2 e 8 2 (log 3n-Iog 8) p

1-

----''--__

__

95n 2 (log 3n-Iog 8)

(229)

Note, that the strongest estimates in Theorem 19 are obtained under large values of 8. So, for instance, under even n and 8 = i it follows from (229) that p

L e2rri I( x )

�

1

1

_ 11 P 1 7 2n 2 _ __

x=1 Under an arbitrary E (0 < E � t) for any > En we have the estimate p 1 - ..2.. e 2 1fi I(x) � C P n 2 x= 1 with constants C and 'Y depending only on E. Finally, the estimates of the form p 1e hi I ( x ) � C P n 2 log log n x=1 8

L

'

L

and

P

L e 2rr i /(x)

x=1

� CP

1 - --'-

n2 10g n ,

where C and 'Y are absolute constants, follow from Theorem 19 under 8 ;? Vii , respectively.

8

;?

n og n and

I

Sums arising in zeta-function theory

Ch. II, § 1 6]

119

§ 1 6. S ums arising in zeta-function theory

In investigating a problem on a bound of zeros of the Riemann zeta-function there arises necessity to obtain nontrivial estimates for sums of the form

Q+ Q1 S( t, Q) = L z i t

(230)

The strongest estimates of such sums are obtained with the help of the repeated appli cation of the mean value theorem, which was suggested in [24]-[26] and applied later in [46] , [27] , [28] , [47] , and in a series of other papers. ' At first we shall prove a lemma similar to Lemma 24. LE MM A

25. Under any positive integers P, n, and k for the sum

p

p

S = L L e211'i (a1'l: y+ .. . +a n X n y n ) x=1 y=1 we have the estimate where

p

n V = L N� P) (>. 1 , . . . , A n ) L e2 11'i (a l ,x1 x + ... +a n ,x n x ) x=1 >'l , ... ,>' n

2k

(23 1 )

and the summation is extended over the region

1J.l ,, 1 < kP " Proof.

(v = 1 , 2, . . . , n ) .

Using the inequality ( 142) , we obtain

I S I 2k �

t e2 11' i (alxy+ . . .+an x n yn ) )

(tx=1 y=1

2k

P P 211'i (a1x y+ ... +a x n y n ) 2k 1 p2ke n � L L x=1 y=1 Since by (159) P

e 2 11'i (a1 xy+ . . . +a n x n y n ) L y=1 =

2k

(232)

[ Ch. II, §

Weyl's sums

120

where the summation is extended over the region

I A1 1

<

then it follows from (232) that

kP, . . . , I A n l < kp n ,

P n I SI 2 k :::;; p 2 k- 1 L N�P) ( A 1 , . . . , A n ) L e27ri ( 0'1 ,xl X+ ... +O'n ,xn x ) . x=l ,xl , ... ,,xn Raise this inequality to the power 2 k and use the inequality ( 141): n 2k I S I 4 k 2 :::;; p 4 k 2 _ 2 k L N�P) ( A l l " " A n ) t 2 7ri ( 0'1 A 1 X+" '+O'nAnX ) x=l ,xl , ... ,,xn 2 k -1 -2k k ) L �P) 4 N 2 " A p (A 1 , " n :::;; ,xl , ... ,,xn 2k P L N�P) ( A 1 , . . . , A n ) L e27ri (0'1A1X+ ... +O'n,xn x n ) X x=l ,xl , ... ,,xn . Hence, since by ( 164)

( (

)

e

)

we obtain the assertion of the lemma:

I S I 4 k 2 :::;; p 8 k 2 - 4 k L N�P) ( A l l " " A n ) I::e 2 7r i (O'l ,xlX+" ' +O'nAnX n ) x=l ,xb ... ,,xn = p 8 k 2 -4 k V. P

COROLLARY . If under

v

= 1 , 2, . . . , n

-q"

a " = a ll

Oil +2 q ' "

(hen under any positive integer

k�

�

for the sum

P S = L e27ri (O'lXY+"' +O'n x n y n ) x, y=l

we have the estimate

ISI 2 k 2

n ( n + 1) 2

:::;;

(2 k) 2

(

-) .

n n p4 k 2 _ 2k+ ...!. 2 k . Nk (P ) II min p" , .;q; + P " ,,=1

�

2k

16

Ch.

Sums arising in zeta-function theory

1/, § 1 6)

Proof .

For the quantity V determined by the equality V=

L �l ..... �n

N!P)(>-l, . . . , >- n ) x=l

121

(231 ) we get

P

L e2 11"i (
� Nk( P ) �l J .... �n x=1 L

P

L e 2 11" i (
2k

2k

Hence 'i t follows by Lemma 1 that

(J.t l , . . . ,J.t n ) L e2 11"i
.

.

1

c

Since the trivial estimate

1J.t I ) < 2kp2 11 , , i (pII I a II 1l 1I' . I
,

Weyl's sums

122

[ Ch. II, §

16

Now, using Lemma 25 , we arrive at the corollary assertion:

To estimate the sum (230), we need two simple lemmas more . Denote by the sum extended over integers x belonging to a certain finite set M.

L XE

M

LEMMA 26 . Let functions /t (x) and h (x ) be defined under x E M. Then

L e 2 11'i (ft (x )+f2 (x »

=

L e2 11'i jt { x ) + 211'8 L I h (x) l ,

xEM

xEM

xEM

where 1 8 1 � 1 . Proof .

Since, obviously,

then

L e211'i ( h ( x ) +h(x» - L e211'i h (x )

xEM

L

�

xEM

l e 211'i f2 (x )

xEM

- 11

� 211'

and, therefore,

L e211'i (Jt {x )+f2 (x »

xEM

where 1 8 1

=

=

L

xEM

e211'i h(x) ( e 211'i h (x) - 1 ) L xEM

I h ( x) \

L e2 11'i h( x ) + 211' 8 L

zEM

zEM

� 1.

LEMMA 2 7. Let n, under any t >

where and

q

0

P, Q, Ql be positive in tegers, Ql � Q, and we have the estimate

(v = I , 2, . . . , n ) is

a

\ h(x) \ ,

certain integer from the interval (Q, 2Q] .

P<

../Q. Then

Ch.

Sums arising in zeta-function theory

II, § 1 6]

Proof .

123

Let function J(z) be determined by the equality J(z) =

t 2 7r log ( Q + z).

Then using Lemma 19 we obtain p

::;;;

where

p

L

e 2 1r i J( zo+xy)

=

�2

e21ri J ( z+ xy) L x y =1

+ 2p 2

,

p

e21ri J (zo + xy) L , x,y=1

( 2 3 4)

+ 2p 2 ,

p

L

max I � Z�Q

l x ,y=1 x, y=1 Denote by q the s m Q + zo . Then, obviously, Q < q ::;;; 2 Q and u

J ( zo + x y )

� log q 2� log ( 1 + x:)

= 2

t = 2 7r

t = 2 7r

+

(

)

( )

X y x 2 y2 x n y n + ( x y) t p 2 n + l log q + 2t7r q + 8 , 27r Q - 2q 2 ± n n q p2 n+ l log q + Q 1 X Y + . . . + Q n x n y n + 8 (x , y ) 2t Q ' 7r .

.

.

( )

where 0 ::;;; 8 ( x, y) ::;;; 1 . Hence, by Lemma 26 it follows that

We obtain the lemma assertion by substituting this estimate into ( 2 34 ) . THEOREM 2 0 . Let n , Q, Q l be positive integers, even n � 1 2 and Q l n-l n belongs to the interval Q -3- ::;;; t < Q"3, then 1

1 L z it ::;;; 3 Q 2800 n2

Q+Q l

z=Q + l

_ ____

::;;; Q. If t

Weyt's sums

124

Proof.

[ Ch. II, §

16

1

Choose P

=

[ Q 3" J . Then

2 p 2 + Qt

( P2 )

n+ l

n+l 2 2 � 2 Q 3" + t Q - -3- < 3 Q 3" 1

Q and the assertion of Lemma 26 may be written in the form

(235)

where

S

p

=

L e 2 '11"i ( O t x y+ . . . + O R X R y R ) ,

x , y=1

(v = I , 2, . . . , n)

and q is a certain integer from the interval ( Q , 2 QJ . Write the quantities a " i n the form (236 )

and use the coroll ary of Lemma 2 5:

(

n k+ "!" P" I S I 2 k � (2k) 2 n p4k3 -2 2 k Nk ( P ) II min P " , ,jq; + r;;,,= 1 y q" Applying the trivial estimate under < I and = we get 2

n

II min

(P " ,

,, = 1

)

v

v

=

! and (3 � 1 . Then

Since under v � I

.

..ffv ) ( r;;- ) II - + � . n,

(237)

(

l) n P " = P n ( n+ 1 2 II min 1 , r;;+ � y q" ,,= 1 y q" P

,jq; + ;;;;

�P

Let a

)

1

(3

n ( n +1 )

a = (3J - (3{ (3J} = ( J [ [ (3 1

1

2

n .... '2 ",, " < n

+

fJ

((3] 2 '

..ffv

IfJ l

< 1.

P"

(23 8 )

(239)

Ch. /I, § 1 6)

Sums arising in zeta-function theory

125

then by (239) _ -1 ( _ 1 ),, - 1 0" a " = 211"( v1),, = 1 0,, 1 < 1 , q "t- 1 [211" vq "t - 1 J + [211" vq"t - 1 ) 2 , and, therefore, under v � � we may choose q" = [211" vq" t - 1 J in the equalities (236). B ut then for every v from the interval � � < n the following estimates are satisfied n n-1 Q ,, - -3 < q " < 211" v2 " Q ,, - 3 , n-1 In ,, - n+1 " n " " q 1 1 Q 31 - + 2 n Q -6vq; + � : � Q 2 6" + v'211" v 2 2 Q 2 - -62 < , 2" v

(

In) < 22 n2

1 q IT -+ yP : vq;

( )

,, - n + 1 Q -an 2 �"
IT

=

-II

, n2 - 2 n 22 n2 Q - 4"8 .

n 2 �"
(

y

,,= 1

) q"

n { n + 1) n2- 2 n 2 - 2k + 1 + k 4 2 n n2 2k 2 2 P 2 - Nk ( P ) Q 4 8 ISI � (2k) 2 2k Choose 'T 3 n and k = n { n2+ 1) + n'T. Then by Theorem 16 n{n+1) + n{n+1) Nk (P) � (2 k ) 2k (2nt 3 p 2k - -2- eT � ( 2n) 3 n 3 p 2k- -2- + eT , _

=

where

er

n(n - l)

- 2 Therefore, we obtain from (240) _

(1 - .!.n ) 3 n < n240- n .

_ 2 1 n2 n n2 -2 n ISI 2k 2 � (2 k ) 2 n 22 n2 (2n) 3 n 3 p 4 k + 2k + """40 Q - 4"8 - n2 -3 n � (2n ) 4 n 3 p4k 2 Q 80 . Since n � 12, then n 2 - 3n � �n 2 .and 9n4 < k 2 < 13n4 . But then 2 nD _ _1_ _ K.. lSI � (2n) k 2 p 2 Q 6 4 0 k2 < 2 p2 Q 2 800 n2 • Substituting this estimate into ( 235), we arrive at the theorem assertion: 1_ Q+ Ql 1_ 1 1 1 800 < 3Q 00 zit � 2 Q n2 8 n8 Q 2 + 2 3 3 L _

:z: = Q + 1

_

_ _

( 24 0)

(241 )

[ Ch. II, § 1 7

Weyt 's sums

126

§ 1 7 . Incomplete rational

s ums

=

Let n > 2 and f ( x ) a l x + . . . + a n xn be a polynomial with integral coefficients . Consider the rational exponential sum

P

If ( an, q) = 1

.

f( x )

q . S(P) = I:> 2 71" x=1 and q = pr, then under 1 � � n - 1 the estimate r

1 - --"---

I S ( P) I � e3 n P 9 n 2 1og n follows from Theorem 17. Using the repeated application of the mean value theorem, this result can be slightly strengthened. So it follows from Theorem 1 9 that for a certain interval of values we have the estimate 1 - 2.. ( 2 42 ) I S ( P) I � CP n 2 , . where C and 'Y are absolute constants. Under an arbitrary positive integer q the estimate (242) is the best among known ones and no approaches to the problem of its essential improvement are seen for the present. But this estimate can be strengthened under a special choice of the denominator q. Let us show how it can be done under q being equal to a power of.a prime. LEMMA 28 . Let (x, n, P be positive integers, p > n 2 be a prime, F (x) bo + b 1 x + . . . + bn xn , and Ta (F, P] be the number of solutions of the congruence F( x) == 0 ( mod pa ), O � x < P. ( 24 3 ) r

=

If (bo , . . . , bn , p) = 1 and

holds.

P � pn , a

then the estimate

a Ta [F, P] � 2n Pp - 2 n

=

Proof. At first we shall show that under P ap8 with 1 � a < p and � 0 we have the estimate Ta (F, ap8] � nap 2 n . (244) Under = 0 this estimate is trivial, because a p n � P = a < p, s

s

Or 8- -

Incomplete rational sums

Ch. /I, § 1 7]

and, therefore, nap - 2 n '"

>

n,

127

but the congruence o�x

<

a,

has at most n solutions. Let s � 1 . Write the congruence (243) in the form F(y + px) == 0 (mod p"' ),

and consider the case 1 � a �

n.

o�y

<

p, 0 � x

<

aps- l ,

(245)

Passing to the congruence to modulus p, we get

F(y) == 0 (mod p) ,

O � y < p.

Hence it is seen that y in (245) may attain at most n distinct values and, therefore, Now we shall apply the induction . Let us assume that the estimate (244) holds under a certain "a � n and all smaller values a. We should show that the estimation is fulfilled under a + 1 too. Indeed, denote by YI that solution of the congruence (245), for which the congruence F(y + px)

==

0 (mod p",+ l ) ,

< aps- l ,

o�x

has the most number of solutions, and determine the polynomial FI (x) by the equality where p"'l is the largest power of p dividing all coefficients of the polynomial F(YI + px ). Note that a l � beca';1se otherwise from the equality n,

it would follow that F(yd == . . . == F( n )(YI ) == 0 (mod p) and bo bn == 0 (mod P), which contradicts the hypothesis of the lemma. Reducing the congruence F(YI + PX ) == O ( mod p"'+ 1 ) , o � < aps- l , by p"'l , we obtain _

_

x

o�

x

<

ap s -l .

Since a l � n, the number of solutions of this congruence does not exceed the number of solutions of the congruence o�x

<

aps-l .

Weyl 's sums

128

[ Ch. II, § 1 7

Therefore +1 Using ap8 � p""'it , we get 0'

a+1 - n +1 ap8 - 1 � p 0'n _ 1 p n But then by the induction hypothesis 1 a+ 1 - n 8 1. 0'+ 1 nap - 2 - 2n . Ta + 1 - n[F1 , ap8 - 1 ] � nap8 - - 2n Substituting this estimate into (246 ) and using the condition < -/p, we obtain 0'+ 1 1 0'+ 1 8 S 2 "'2;; "'2;; "2 ap ap8 nap < n Ta+ 1 [Fl , ] � The proof of the estimate ( 244) is completed. The lemma assertion for an arbitrary P follows immediately from the estimate 2 ( 44 ) . Indeed, determine integers s and with the help of the conditions =

=

n

a

Here � 0 and 1 < a < p. Since s

then by ( 244)

a s a -a Ta [F, P] � Ta [F, apS] � na p - 2n n [( a - l )pS + pS] p - 2n � 2 nP p 2 n . THEOREM 21 . Let and a be positive integers , f(x) a x + . + a n n � 35, a � 4n 2 , p > 4n 2 be prime, q pO' , p r � q < p r+1 , and TI/ ( P ) be the number of =

=

r

1

. .

=

solutions of the congruence

Then under any r from the interval 2

< � r

� we have the estimate

p ' /( x ) 1 2. l: e 2 71' 1 -q- � 3 P - r 2 + nT( P ) ,

x=1

where 'Y is an absolute constant and

T( P)

=

max T,, ( P). 2 r+3< ,, �3r+3

x

, n

Incomplete rational sums

Ch. /1, § 1 7)

Proof .

129

Determine an integer 8 with the help of the inequality 4(r

(247)

+ 1)8 � 0' < 4 ( r + 1)(8 + 1) .

It is easy to verify that the following estimates hold: ( 4r + 8 )

8 � r + 1,

> 0',

In fact, if 8 � r, then we arrive at a contradiction: 4 n2

� 0' < 4 ( r + 1)(8 + 1) � 4 (r + 1 ) 2 < 4 11. 2 ,

and, therefore, 8 � r + 1 . Further, it is obvious 8(4r

+ 8) � 4r8 + 48 + 4 ( r + 1 ) = 4(r + 1)(8 + 1 )

>

0' .

�

Finally, P > p r+ 1 � p4 s . In Lemma 19 we choose PI

= a =

. f(x )

P

pS .

Then we obtain p'

P

1 L e 271" -q- � 2s L

P

x=l

Le

2

x=l y , z= 1

' f(x +p ' y z) p o. + 2p3 s . 71'1

Denote by M a set of those x from the interval 1 � single congruence of the form

x

� P, which do not satisfy a

2r + 3 <

l/

� 3r + 3 .

For the remaining x E [ 1 , P] at least one of the congruences is satisfied and, therefore, 3 the number of such does not exceed rT(P). Hence, observing that p3 s < p 4. , we get x

P

' f( x )

1 L e 2 7I" -q- � p2s L

x=1

xEM

where Sx

p'

=

Le

2 71'i

I Sx l

3

+ rT(P) + 2P 4. ,

f(x + p' yz) p"

y,z=1

Since by ( 247 ) f (x

+ pS yz ) == f (x) + f' (x)pS yz + . . . +

1

( 4 r + 7)!

J<4r+7) (x)(pS yz ) 4r+ 7

( mod pO ) ,

(2 4 8 )

[ Ch. II, §

Weyl's sums

130

then setting

n1

=

17

4r + 7 we obtain I Sx l

=

where bJJ and qJJ are determined by the equalities � f ( JJ }(x) v!

b pot-SJJ qJJJJ ' =

Let as show that for the quantities qJJ the estimates (2r + 3

<

v � 3r + 3)

(249 )

are valid. , Indeed, since in the sum Sx the quantity belongs to the set M, for every from the interval 2r + 3 < v � 3r + 3 the greatest common divisor of the numbers r ex) and pot is less than p [�J and, therefore, x

-

-

v

Hence, observing that � a 8V � 8 (3r + 3 v) and a

- 8V

=

8V + a

- 28V � 8V + 8 (4r - 28 (2r + 8)

+

4)

=

8V,

we obtain the estimate (249) . Now we shall estimate the sum Sx . According to the corollary of Lemma 25 I Sx l 2 k2

� (2 k ) 2 nl pB (4

k 2 - 2 k+ 21k ) Nk .n l (p8 ) IT min (pSJJ , ..;q;; + pS JJ ) . y'q; JJ=l

Since by (249) under 2r + 3 < v � 3r + 3 we have min

pS JJ ) � 2 y'q; pSJJ , p8 JJ , ..;q;; + y'q;

(

then applying the trivial estimation under remaining

v,

we get

(250)

Incomplete rational sums

Ch. /I, § 1 7]

By

131

( 249 ) qv > p8 (3rH - v ) and, therefore, 3r+ 3

IT

v=2r+ 4

_1 2 qv

3r+3

< IT p -

s (3r+3-v)

2

v=2r+ 4

4 = p- --sr ( r- l )

sr (r-l)

4

Substituting this estimate into (250), we obtain

Now we shall use Theorem 1 6:

where under

= 6n l and r > 2

T

k cr

=

_

-

n l ( n � + 1 ) + 6n� = 2( 4r + 7)( 1 3r + 23) < 270r 2 ,

nl ( nl + 1) (1 2

( �)6nl < 71 1 n8 l 0 1) n 0

l

-

<

� r2 . 20

Since the estimates rs �

3(r + l)( s 5

take place, then using the determination of the quantity

s,

+

1)

'

we get

r2 2s- 9 2 8 - r(r+ l)( s + l ) k 4k n nlk n� 2k 2 71 l k k 15 / ) ) ( ( ( 2 ) 2 1 p � 4 < n� +l) 8 -2 8 - (r+l)(s 5 r8 < 2p2 - 9 · 1 0 6 r3 ·10 4 22 I Sx l < 2p Hence the theorem follows by (248 ) : P _ P 2 1' /( x ) Sx l + rT(P) + 2p 43 e I � 71' -q- � p2 s l max E;; x E;; P 2:=1

2 ISx l 2k

s

a

--

where 'Y = 9 .f0 6 .

[ Ch. II, § 1 7

Weyl's sums

132

[ �] . Iffor a polynomial f ( x ) = a 1 x + . . . + an x n the hypoth eses of Theorem 21 are satisfied and under v � 4r at least one of the coefficien ts a v is n ot COROLLARY . Let fi

=

divisible by pP , then we have the estimate

p

L: e

2 7r i

z= l

where p = Proof.

min ( fi , ;2 ) By Theorem 21 'Y

and 'Y

P

L: e

p c.

2 1 :::;; 3n p - p ,

> 0 is an absolute constant.

2 7r i

f( x )

p c.

z= l

where

fe z)

T(P) =

1 - ..2.

:::;; 3P r 2 + n T(P) ,

max

2 r+ 3 < v � 3 r + 3

(251)

TII (P)

and Tv(P) is the number of solutions of the congruence (252)

Denote by fi ll the highest power of p dividing every coefficient a v , . . . , a n and de­ termine polynomial Fv ( x ) by the equality Since the congruence (252) is equivalent to the congruence

and at least one of the coefficients of the polynomial FII ( x ) is not divisible by p , then by Lemma 28 Tv(P) :::;; 2nP p

[ � ] -P. [� ] -P+ 1 2 ( n - v) 2n :::;; 2nP p :::;; 2nP p 40n a

But then the estimate _ ..!L

T(P) :::;; 2 n P p 4 0n

holds and the corollary follows from (251).

:::;;

1 - -L 2nP 4 0n

Double exponen tial sums

Ch. 11. § 1 8)

§

18.

Do uble exp onential

133

s u ms

In § 14 in estimating Weyl's sums P S(P) = L e2 11'i f ( x ) x= l

(253)

polynomial f{ x) = al X + . . . + a n x n was replaced by polynomial f(x + y) depending on two variables and the estimation for the sum (253) was reduced to the estimation of double exponential sum P P S = L L e2 11' F( x , y) x=l y=l

with polynomial

F(x, y) = a l e x + y) + . . . + O! n (x + y) n .

Another important particular case of double exponential sums with polynomial F( x , y) = O!l X y + . . . + O! n x n y n

was considered in § 16. We shall show that using the repeated application of the mean value theorem it is easy to obtain ( [40] , Appendix II) estimates for double exponential sums of a general form P P 2

I

S(Pl , P2 ) = L L e2 11'i F( x,y) , x=l y=l

(25 4 )

nl n2 F(x, y) = L L a jk x j y k .

(255)

where THEOREM 22.

j=O k=O Under any positive integers n l , n2, k l , k2, PI , and P2 for the double

exponen tial sum ( 25 4 ) we have the estimate

(256) where

(257) the summation is extended over the region I>' j l < quantities {3k are determined by the equalities

{3k

nl

=

L O!jk>'j

j= l

kIP!

(j

(k = 1 , 2, . . . , n2)'

=

1 , 2, . . . , n l ) and the (258)

Proof.

[ Ch. II, §

Weyt's sums

134

18

Determine quantities <Xj (Y ) by means of the equalities n2

<X j (Y ) = L <Xjky k k= O

(j

= 0, 1 ,

.

.

.

, nd

and write the polynomial (255) in the form

Then, obviously, P2

PI

2k l xn1 . I S(P1 , P2 ) 1 l � pi k l - 1 L L e 2 rr i (ll' l (Y)X+ . . + ll' n (y) ) j=l x=l

-- p22kl - l � N ( Pl l ( A 1 , L...J kl AI , ,Anl ...

V sing the equalities (258),

.

.

.

,

2k l

P2 e2rri ( ll' l (y) A I + ... +ll'n l (Y) An l ) . Anl ) � L...J y=1

we get

and, therefore, �

L...J

AI , ... ,An l

( d Nk Pl ( A 1 , .

.

· ,

Anl )

I j�= 1 e2 rri L...J

(PIy+

...

+ Pn 2 y n 2 )

1

.

Hence reasoning as in the proof of Lemma 25 , we obtain the inequality similar to the inequalities established earlier in Lemmas 24 and 25 (259)

where

V=

Ch. I/, § 1 8]

Double exponential sums

135

and the summation is extended over the region Determine quantities Pj with the help of equalities n2

pj = L O!jkJ.Lk k=l

(j

=

1 , 2, . . . , nl ) '

Since by (258)

then by (159)

Therefore

Hence using Lemma 1 we obtain

where the quantity (7 is defined by the equality (257). Substituting this estimate into (259), we get the theorem assertion.

[ Ch. II, § 18

Weyts sums

136

Let n = 1 + max (n1 ' n2 ) , P1 � P2 , and O' r s is an arbitrary coefficient of the polynomial F( x , y ) . If r and s are not equal to zero,

THEOREM 23.

a B O'r8 = -q + 2 q '

(a, q) = 1 , 1 -

I BI � 1 , 1 8- -

2 2 and the quantity q lies in the interval P2 � q � P[ P2 , then for the sum S (P1 , P2) determined by the equality ( 25 4 ) we have the estimate 'Y 1n l n 2 ) 2 \og2 n ( I S(P1 , P2 ) 1 � C (n ) P1 P2 , where 'Y is an absolu te constant and

C(n) is a constant depending only on n. Proof. Theorem 22 reduces the estimation of a double exponential sum to the esti­ mation of the quantity

where f3k are determined by the equalities (258). Here we estimate all minima except for the s-th trivially. Then we obtain

By (258) f38 is a linear function of the quantities A1 " ' " Ani and, in particular, a linear function of A r:

Therefore under c we get

=

1 4 n ' using the theorem conditions and the note of Lemma 14,

(260)

Ch. II, § 1 8]

S et Tl

=

Double exponential sums

[ en 1 Iog

137

n 1 1 , T2 = [ en 2 Iog n 2 1 , k 1 = n d nl2 +1 ) + n 1T1 , k 2 = n 2 ( n22 + 1 ) + n 2 T2 ,

where is an absolute constant under which by (194) we have the estimates e

Then we get the theorem assertion from (256) and (260):

1'Y:----:-­ 2 -2 I S( Pb P2 ) 1 :;;; C ( n )P1 P2 ( n l n 2 ) Jog n __

where 'Y = 1 61c2 ' The results obtained for double exponential sums are without difficulty extended to a case of Weyl ' s sums of an arbitrary multiplicity m :

S( P1 , . . , Pm ) --

Pm

PI

"

"

L.J . . . L.J

xl=1

x m =1

e 2 1r

i

F(X l , . . . ,Xm) '

where F(X1 " ' " Xm) is a polynomial of m variables nl

nm

111 =0

II m =O

F(X 1 , " " Xm ) = L . . . L a(vb " " Vm)x �l

. . . x �m .

Let m � 2, PI � . . . � Pm , n = 1 + max ( n 1 , . . . , nm), and a = a(rl , ' " , rm ) is an arbitrary coefficient of the polynomial F( x l , . . . , xm ) . If the product . . . rm I- 0, r1

a

fJ

a = - + -2 ,

q

q

(a, q) = 1 ,

I fJ l :;;; 1 ,

and the quantity q lies in the interval

-1 rm - -21 1 p pm2 ....::::..: q ....::::..: p1rl ' " pmrm-1 m , then we have the estimate

138

Weyt's sums

[ Ch. 11, § 18

where C and 'Y are constants depending on and m, respectively. A proof of this estimate is obtained with the help of the successive m-fold application of the mean value theorem and in its nature is close to the proof of the analogous estimate for double sums , which was presented in Theorem 23. Another approach to the estimation of multiple sums is based on a multidimensional generalization of the mean value theorem [1] . This approach allowed to strengthen the sum estimates, which follow from Theorem 22 . The strongest estimates of mul­ tiple exponential sums are obtained in the article [41] . In the last years numerous publications deal with multiple sums, but interesting applications are known only for double sums arising in estimating ordinary Weyl ' s sums. n

CHAPTER

III

F RACTIONAL PARTS D ISTRIBUTION, NORMAL NUMBERS, AND Q UAD RATURE FORMULAS

§

19.

Uniform distrib utio n of fractional parts

The notion of uniform distribution in a general form was introduced by H. Weyl

[49] . He obtained also fundamental results concerning functions, whose fractional

parts are uniformly distributed. Let a function f (x) be defined for positive integral values of sequence of its fractional parts

x.

Let us consider a

(26 1 ) { f ( l ) } , { f (2) } , . . . , {f (P) } , . . Denote by Np('Y) the number of x ( x = 1 , 2, . . . , P ) , which satisfy the inequality { f (x) } < 'Y, where 'Y is an arbitrary number from the interval (0,1]. The sequence (261) is called uniformly distribu ted, if .

lim

P -co

pI Np ('Y)

=

'Y .

(262)

If fractional parts of a function f(x) are uniformly distributed , then this function is said to be uniformly distributed too . Rewrite the relation (262) in the form Np ("'{ )

=

'YP + o(P).

(263)

The equality (263) shows that for uniformly distributed functions under an arbitrary 'Y E (0 , 1] the number of these fractional parts among the first P from the sequence (261), which fall on the interval [0, 'Y), is asymptotically proportional to the length of the interval . If 0 < 'Yl < 'Y2 � 1 , then the number of fractional parts, which fall on the interval ['Yl , 'Y2 ) , is equal to Np('Y2 ) - Np("'{l ) .and by (263), evidently, is also asymptotically proportional to the length of this interval: Np ("'{ 2 ) - Np ("'{l ) =

("'{2 - 'Yl ) P + o(P) .

[ Ch. III, §

Fractional parts distribution

140

19

Let us consider, for example, the function f(x) = y'X and show that its fracti on al parts are uniformly distributed . In fact, denote by Tk ( 'Y) the munber of satisfact ions of the inequality { y'X } < ",,{ , when x runs through integers from the interval k 2 � x < (k + 1)2 . Then under n = [ JP) we get

n-l n Tk ('Y), ("{) � T Np('Y) < L L k k=l k=l and, therefore,

n -l Np("{) = L Tk ("{) + O( yIp). k=l Since, obviously, k = [y'X ) , then under = k 2 + y we obtain x

{ y'X}

=

y'k 2 + y - k

=

k+

(264 )

� , k2 + y

But then n(n

n -l

- l h � L Tk ( 'Y ) � ( n 2 - l h,

k=l n-l L Tk ("{) = 'Y P + O ( v'P) , k=l

and by (264)

Np ('Y) = 'Y P + 0(v'P) . Hence it follows by the definition, that fractional parts { y'X } are uniformly dis­ tributed. In the same way it is easy to investigate other monotonic functions f ( ) satisfying the condition f (x ) = o. lim ...... x

X

oo

X

In particular, it can be shown that under 0 < < 1 fractional parts of the function xQ are uniformly distributed, but fractional parts of the function 10g.8 x are distributed uniformly or not, depending on whether f3 > 1 or f3 � 1 . The investigation of functions growing as polynomials and especially functions growing faster is much more difficult . So, for example, it is not known whether fractional parts of the functions eX and (!) X or of the functions (�) X under n > 1 , 01

Uniform distribution of fractiona l parts

Ch. Iff, § 1 9 ]

141

rn > n , and coprime m and n are uniformly distributed. At the same time it is easy to p resent nontrivial examples of exponential functions, whose fractional parts are

no t uniformly distributed . In deed, let ). and () be roots of the quadratic equation Z 2 + pz + q = 0, with integral coefficients, such that ). > 1 and 0 < () < 1 ( it is so, for instance, under p = -3 and q = 1 ) . Since the symmetric function ). x + () X takes on positive integral values u nder x = 1 , 2, , then {>. X } = 1 Therefore fractional parts of the function ). x monotonically grow, approaching unity, and, evidently, are not uniformly distributed. We will consider a general criterion for uniform distribution, connecting problems of distribution of fractional parts with estimations of exponential sums. Let 0 < 'Y < 1 and 0 < c < min ( 'Y, 1 'Y ) . Determine functions .,pi ( x ) and .,p2 ( X ) by means of the equalities .

.

.

-

() x .

-

�, ( X ) �

11

f ( I' c; 0

X

-

-

if If if if

x)

o � x < c;,

c; � x < 'Y - c , I' - C; � x < I', I' � x < 1 ,

(265) .,p 2 ( x ) =

1

� c;

(

if O � x < I', if I' � x < I' + c; , if I' + c � x < 1 - c , if l - c; � x < 1 .

X)

� 0

1 x + c; - I ) - (7 .c;

LEMMA 29 . Let .,p( x ) be the characteristic function of the interval [0, 1' ) . Denote by C1 (m) and C2 (m) the Fourier coefficients of the functions .,p 1 ( X ) and .,p 2 ( X ). Then the relations

.,p 1 ( X ) � .,p (x) � .,p2 ( X ) (0 � x � 1 ) , C1 ( 0) = I' c;, C2 ( 0) = I' + c; , l ' �) (m = ±I , ± 2, . . . ) max ( I C1 (m) l , IC2 (m) ) � min (m c; m l l l -

71'

(266)

71'

hold.

Proof . Since

.,p(x ) =

{�

if 0 � x < 1' , if I' � x < 1 ,

the first of the relations (266) follows directly from the determination of the functions

1P1 ( X ) and .,p2 ( X ) .

Fractional parts distribution

142

[ Ch. III. §

19

Further, obviously, 1

C1 (O) =

J tP1 (x) dx = � J x dx + J dx - � J (-y - x ) dx -y - e

I!

0

o

= 2 + l' - 2t: + 2 = t:

t:

Analogously we get

/I

-y - e

e

l'

- t:.

1

C2 (O) =

J tP2 (X) dx = o

l'

+ t: ,

the second assertion of the lemma is proved. Finally, under m i= 0 we have

Hence, since

J e -27ri mx dx e

� min

o

(t: , 7r1� 1) '

it follows that In the same way we obtain the estimate

the lemma is proved completely. THEOREM 24 ( the Weyl criterion) . distribution of a function I( x) is

A

necessary and sufficient condition for unifonn P

.

1 lim - � e 27rt mf(x ) = 0 R-oo P L..J x=l for

any

integer

m

i=

O.

(267)

Uniform distribution of fractional parts

Ch. III. § 1 9 )

143

Proof. Let 0 < 'Y < I, .,p ( x) be the characteristic function of the interval [0, 'Y ) and the functions .,pI ( x ) and .,p2 ( x) be determined by the equalities (265). Then, evidently

= x=l L .,p ( {f( x ) }) . p

Np C'Y )

Since by the lemma

(268)

then, carrying out the summation over x, by (268) we get p

p

L .,p 1 ( {f ( x) }) - 'Y P � Np C'Y ) - 'YP � L .,p2 ( { i( x ) }) - 'Y P. ",=1 x=l

(269)

Let us suggest that the condition (267) is satisfied. Choose P > Po (e) in such a way that under n = [ ;] + 1 the estimate p

L e21fi m/( x) 1� m � n 2 x=l max

7r

( 27 0 )

eP � 4 ( 1 + 2 10g n )

would be satisfied. Then, using the expansion of the function .,pI ( { f( x) } ) into the Fourier series, we obtain

p

L .,pl ({f(x)} ) - 'Y P x=l

=

m= - oo p e2 1fi m/ (x ) ' C (m e ) P L L 1 + x=l m =-oo 00

+ L I C1 (m) 1

Iml>n 2

p

L e21fi m J (x) x=l

=

(the sign in the sum indicates the deletion of the summand with m 0). Hence applying the estimate (270) under I ml � n 2 and the trivial one under I ml > n 2 , by virtue of the lemma we get I

p L.J '" .,pl ( {f( x) } ) - 'YP

x=l

� �

� eP + 4 + P L.J e P L.J ' 7r 2 2 ( 1 + 2 log n) ml Iml > n 2. 7r m e m= -n 2 l 2P 1 - < 2eP. eP + - e P + -2 2 7r n 2 e 7r

�

",

1

1

Fractional parts distribution

144

[ Ch. 11/, §

19

In the same way the estimate p

L 7fJ ({ f(X) } ) - "IP x =1 2

:::;; 2eP

is obtained. But then it follows from (269) that

-2eP :::;; Np C'Y) - "I P :::;; 2eP

and, because

e

can be

small as we please, we get

as

lim N ) = "I. P -oo p pC'Y

I

The sufficiency of the condition (267) is proved. Now we shall prove the necessity of that condition. Indeed, let the function f( x) be uniformly distributed. Take m f= 0 and choose an integer q > Im l . Denote by Mk a set of those x from the interval [1 , Pj , which satisfy

�q :::;; { f (x) }

<

k+l . q

(271 )

Tk the number of satisfactions of this inequality. Then, obviously, p q- 1 m f ( x) = m 2 11"i e L L L e211" i f( x ) . x=1 k=O xEMk It follows from (271 ) that for x E Mk

Denote by

{ f ( x) }

=

1: + Ox , q q

Using Lemma 26, we obtain

L xEMk

e 2 11"i mf( x) =

L xEMk

- .L k e _ 'T'I

where I O' ( k ) 1

k m /1I"i ( � + :", )

211" i �k

:::;; 1 . But then p

e211"i mf( x ) L x=1

+

k

=

e 2 11"i � L xEMk

+ 27rO( k)

27r l m l O'(k)'T'I .10 k , q

I m l Ox L xEMk q

q'""" - 1 'T'I 2 11" i m k 2 7r I m I q'""" -1 q + -- L..J 0'k .L k L..J .L k e q k=O k=o 1 qq-1 . mk :::;; L Tk e 211"1 q + 2 7r l m l L Tk q k=O k=O

=

'T'I

+

27r l m l P. q

(272)

Uniform distribution of fractional parts

Ch. 1/1, § 1 9)

145

Since by the hypothesis fractional parts of the function J( x) are uniformly dis­ tribute d, then 1 P P)

Tk = -q + o( . Take an arbitrary e > 0 and choose P > Po(e;) such that under q

I Tk - !q p l � �2q eP

es tim ate

will hold. Observing that

[ 4 11'�m l ] + 1 the

1 � I m l < q and therefore q- l 2 11' i m k e

L k=O

we obtain

=

1

q

q- l k=O

=

0,

q-

l(

L Tk - :::-q1 P e2 q k=O l q - I Tk - -1 P I � 21 e;P. �L q k=O But then, because of q > 411'�ml , we obtain the equality (267) from (272): = qPLe

p

" e 2 11' i m /(z) � z=1 1·Im

an

2 11' i m k

q +

)

11'i m k

1 2 1rlml ./ "::: 2 eP + -q- p ,,::: e P, ./

1 " e2 11'i m!(z) = p p

�

z=1

0

.

The theorem is proved completely. As an example of application of the Wey1 criterion we shall show that fractional parts of a linear function J(x) = ax + f3 are uniformly distributed under an irrational a and an arbitrary 13. Indeed, let m 1 0 be an integer. Using Lemma 1, we get p

L e2 11'i m( a z+ p)

z=1

Therefore

(273)

Fractional parts distribution

146

-L p

1

lim

P""'oo P :r; = 1

e 211' 1. m ( a :r; + p)

=

[Ch. 111, §

19

0,

and by virtue of the Weyl criterion fractional parts of the function (273) are unifonnly distributed. Now we shall consider a question concerning the distribution of fractional parts of a polynomial of an arbitrary degree n � 1 :

A t first w e shall prove a sufficient condition of uniform distribution of fractional parts which is due to van der Corput [7) . LEMMA 30 . Let 6 /(x) be a finite difference of a function I(x) with step h : h

6 /(x) = I(x + h

h ) - I(x) .

Then under any PI from the interval [1 , P) we have the estimate , P

L e 2 11'i l ( :r;)

:r;=1

2

(

� 2P P P1-1

+ 2P1 + max l � h < Pl

P 211'i tl. /( z)

Le

:r; =1

"

)

•

Proof. By ( 196)

and, therefore, P

L e211'i I(z )

2

:r;=1

�

� LP

Pl -1

L e211'i I(z+y )

1 :r;=1 y=O

2 + 2 pl ·

(274)

Since, obviously, P Pl -1 2: L e21fi I(z+y)

:r;=1 y=O

2 �

=

L 2: e2 1fi ( f (:r;+ Y)- /( :r;+z» =O :r;=1

Pl -1

P

Y,z

PP1 + 2 L

P

L e2 1fi ( f ( :r; +y) -/ ( :r; +z» ,

y>z :r;=1

( 2 75 )

Uniform distribution of fractional parts

Ch. 1/1, § 1 9 )

then

observing that p

L e 21ri (f(x+ y)- /( x+ .:»

x=1 we

147

=

�

2z +

obtain from (275)

p

L e2 1ri (f(X + II-':) -/( x» ,

x

=

1

2 p Pl - 1 L P 2 1ri (f( x+ y-.:)- / ( x » L / + 2 ) (X i II 1r p e 1 t PP � + 2 + Le L x=1 II >': P 21r i a /(x) e L x=1

11=0

x=1

h

Substituting this estimate into (274), we get the assertion of the lemma: P

L e21ri /( x )

2

x=1

� �

THEOREM 25 .

P1-1 2P (PP1-1 2 p2

21ri a le x ) e L � h < Pl 21r i � /( x ) + 2P1 + max t e l �h < Pl x= 1

+ 2P PI + 2P l max

1

P

x=1

h

)

+ 2 pl .

A sufficient condition of uniform distribution of a function f(x) is unifonn distribu tion of its finite difference � f( x) under any integer h � 1 . h

Proof. Let according to the hypothesis of the theorem fractional parts { � f( x ) } be h

uniformly distributed under any positive integer k. Then by the Weyl criterion under eve ry integer m 1= 0

P 21r m a / ( x ) "e i P-+oo P xL.J lim

1

-

=1

h

=

(276)

O.

Apply the inequality of Lemma 30 to the function m f( x ) Then observing that we obtain

Il m f ( x ) = m � f(x)

.

h

h

1 (

P

2 P 21r i m a / (X » P L e 21ri m / ( x ) � 2P P P1- 1 + 2 1 + max L e l �h < Pl x= 1

x=1

h

P

)

•

(27 7)

Let 0 < e < 1 and P1 = [e;62 ] + 1 . If follows from (276) that 2 = P2 (e, m) can be chosen in such a way that under P � max (Pt , P2 ) the inequality

P

Le

x= 1

21r i m a / ( x ) h

Fractional parts distribution

148

t

111.

§ 19

P � max ( !�P1 , P2 ) ' Then we get from (277) 2 e 2 71'i m f ( x ) � 2 P e 2 P + e 2 P + e 2 p e2 p2 ,

will be satisfied. Choose

x=1 p

[ Ch.

(�

'"' e 2 71'i m f ( x )

L.J

., = 1

�

�

)

=

/ """ e P,

and, therefore, 1 P-oo P lim

-

p

.

'"' e 2 mf( ) 71'J .,

L.J

x=1

=

O.

The theorem is proved by virtue of the Weyl criterion. THEOREM 26 ( Weyl's theorem ) . If a polynomial

f(x)

=

a o + at X + . . . + a n x n

(278)

bas at least one non constant term witb an irrational coefficient, tben its fraction al parts are uniformly distributed. Proof. We shall start with a case when the coefficient of the highest degree term is irrational. Under n = 1 the polynomial (278) is in reality a linear function ao + at x with an irrational coefficient at . By (273) fractional parts of such linear functions are uniformly distributed. Apply induction. Let n � 2 and the theorem be proved for polynomials of degree n - 1 , having an irrational coefficient of the highest degr()e term. Choose an arbitrary positive integer h and consider the finite difference

l:::. f(x) h

=

f(x + h ) - f(x)

=

a n [ (x + h ) n - x n ] + . . . + a d(x + h) - xl .

Evidently, l:::. f( x ) is a polynomial of the ( n - l ) -th degree with an irrational coefficient h of the highest degree term. By the induction hypothesis, fractional parts of this polynomial are uniformly distributed. But then by Theorem 25 fractional parts of the initial polynomial are unifonnly distributed also. Thus the theorem is proved for polynomials with the leading coefficient being irrational. Now let 1 � s < n and as be the leading among irrational coefficients of the polynomial f(x) . Denote by q the common denominator of coefficients as + h ' " , a n and write the polynomial (278) in the form

f(x)

=

(x) h e x) +