Number Theory: Tradition and Modernization (Developments in Mathematics)

( mod r 7 mod q

max I W ( h rq116 IXl
41:

k s L 2 B 7 mod k

T-R k < P B 71 mod r q2 mod k

T N RT ~ #kTs L 2 B E mod r 7 mod k

Here and in the sequel C*x primitive characters x mod q.

77'

mod r f

means the sum taken over all the

On the major arcs.

4.

In this section, we will calculate the integrals on the major arcs

!Dl1,!Dl2and !Dl;. Let us begin with !Dl1.We have

x e (-N(a/q where

+ A ) ) d A =: Iio + Ill + 1 1 2 + 1 1 3 ,

(4.1)

51

The Goldbach-Vinogradov Theorem in arithmetic progressions

say, and

Substituting ( 3 . 6 ) and ( 3 . 8 ) into 11: 3 , we have

IX x /IiqQ -

l/qQ

N

1 v2( A )1 dA 1

<

<

7-17-2 q 5 p

mod q

~ max y

P(V)

Ihl41/~Q

a=l (a,q)=l

< 1, J l i q Q

Iv2(.\dh << -l/qQ Now applying ( 3 . 3 ) and the estimate p ( q ) primitive characters, we obtain on using

1131

J G ~ ( ~ Oa)I ,

N

<< -LC 'lr2

C x C*q3"1

Hence, summing over

7-1

mod q

and

7-2,

Similarly as above, we have

v 2 ( ~ ) l d . i<< N .

>> q L P C and

reducing t o

N

max I W ( x , A ) / << Jo-LC 7'17-2 I451/qQ

-

q
1'

we have

52

Z.Cui We also have by (3.6) and (3.8)

Reducing characters J j mod rj, rlj mod q and x mod q t o primitive characters [j mod r;, rlj mod qj ( j = 1 , 2 ) and ~3 mod q3, we have divisibility conditions r;lrj and qjlq, where the former reduces to r; = 1 or r; = rj and the latter, to qo = [ql, q2, qs]lq, [a, b, c] denoting the least common multiple of a , b and c. Hence the right-side of the above inequality is majorized by the sums over r * < R, J j mod rj, qj 3 T P, 17j mod qj ( j = 1 , 2 ) ,qs 5 P, x3 mod q3 and qo/ q wlth summands

<

Distributing the factor suitably (collecting those with the same suffices into one), we have

Similarly as above, we have, on noting IV(X)I _< N ,

Now let us go on t'o the estimation on 9&, the estimation on 331; being similar.

The Goldbach-Vznogradov Theorem in arithmetic progressions

53

where

4

55

'22 = TI

l/qQ LlIqQ

S2(7-1,a1 bl1 q, X)U2(r21 a , '521 41 A ) S ( a / q f

f9,7-2/4 ( a , q ) = l

x e (- N ( a l q

+ A))

and

B y (3.3) and ( 3 . 4 ) )arguing as in the proof of (4.2), we have

on replacing q-l by 7-TI. Hence

'1

54

2.Cui

Substituting (3.6) and (3.7), we deduce that 11221

C

=

1

Is(a/u

-

+ I /'IqQ

-'/qQ

q 5 p TI Y ~ , Q I Q

(a,q)=l

1

2:

el mod rl

F(b)

+

Instead of (2.9) and (3.4) we use the fact that for rnlq, S(a/q A) << L ~ N / ~ ' /and ~ , C(7, q, b, 7-2, a) << q/r2, respectively as well as (3.3) to obtain LCN

1'221

<<--

'Ir2

C C C C

q S P El r1 k I r 2 1 4

mod

r l q 1

mod q q 2 mod q

/'IqQ W ( h , X)W(772, A)/ dh -1IqQ

on decomposing q = r2k, k 5 L ~rlj ~= 17jl77j2, , ( j = 1 , 2 ) . Reducing to primitive characters mod kj, mod r;, j = 1 , 2 , we have

As in the proof of (4.4), we note that r; = 1 or r,* = rj for j = 1 , 2 and that if rl # 7-2, then (rT,r;) = 1. Hence for primitive characters (1 mod rT and 711 mod r;, < 1 ~ 1 1is also a primitive character rnodryr;. Hence, by the Schwarz inequality,

55

The Goldbach- Vinogradov Theorem in arithmetic progressions

7)11

mod r;

712

mod k~

Similarly, we have

and

Now, collecting the formulas (4.1)-(4.10), we have

from the integral over Dl;. where I;o is defined in a similar way to We will show that the estimates of J and K are admissible in the next section.

5. M

The estimates of J and K . We will need some lemmas in the following discussion. Let u satisfy < u 5 N , and let M I , . . . , M1o be positive integers such that hi11 . . . Mlo < u , 2 - l ' ~I

and

2

,, 2 M 15 u

We write M = ( M I , M 2 , .. - , Mlo). For j = 1,..., 10, let aj (m) =

logm, if j = 1, 1, if j = 2 , 3 , 4 , 5 , ( m ) , if j = 6, 7!8 , 9 , 1 0 .

5

(5.1)

56

Z.Cui

We define the following functions of a complex variable s:

Lemma 5.1 ( [ 8 ] ,Lemma 5.2). Let F(s,X) be defined as above. Then for any S 2 1 and 0 < T3 5 N ,

lT3 (1+ x) 1

Z*

kwS

x mod k

where k

/F

it,

S means S/2 < k

dt

<< (s2T3+ S

T

~

+~N'/~)L', ~ N (5.2)

< S.

>

Lemma 5.2. For T 2, let N * ( a , q, T) denote the number of zeros of all the L-functions L(s, X) with primitive characters x mod q in the region Re s a , IIm sl 5 T. Then

>

)/~ N * (a,q, T) << ( q ~ ) ' ~ ( l - " logC(qT)

Lemma 5.3. Let T 2 2. There is an absolute constant cl mod q L ( s , x ) has no zeros in the region

nx

Res

2 1 - cl/ maxilog q, log4/5T), I m s /

> 0 such that

< T,

except a possible Siege1 zero. Lemmas 5.2 and 5.3 are well-known results in number theory. For the proof of Lemma 5.2, see for example pp. 640 and 642, 634, and 669 in Pan and Pan [Ill. For the proof of Lemma 5.3, see Satz VIII.6.2 in Prachar [12]. In order to approximate W(X,A) by a sum over integers, we introduce

Since W(X,A) we have

-

w ( ~A), is a sum of logp over 9, M < 9 5 N, j 2 2, W(X,A) = W(X,A)

+o(N'/~).

Hence in what follows we will use w ( ~A), in place of W ( X ,A). Our task is to prove Lemma 5.4 and Lemma 5.5 which give the estimates for Jj,Kj ; the estimates for Jo,KO over S 5 L~~ are given in the former, and those over L~~ < S 5 P are given in the latter. The proofs depend on Lemmas 5.1-5.3, Gallagher's lemma plus the explicit formula in the former, and Heath-Brown's identity in the latter.

~

~

The Goldbach-Vinogradov Theorem in arithmetic progressions

57

It is customary and convenient to work with the truncated Chebyshev function with x

where

Nx,X ) = @(x,0,

(5.5).

Then we have

We now prove

Lemma 5.4. Let A1 > 0 be arbitrary. Then for any B1 > 0, we have

max

2

1

sriBl

max iw/ka

C* IW(X, A ) I <<

NL-"1,

mod

(53)

and

. (5.9) S'LB1

k-S

x mod k

Proof. We use the explicit formula (see 131, p. 109 and 120, or [ I l l , p. 313).

+

where the sum is over non-trivial zeros p = p ir of the L-function L(s, X ) such that 171 T, and T is a parameter, 2 T u. Substituting (5.10) with T = N1I3 in (5.6)) we have

<

n,

< <

>

I, L ( s , X) has no zero in the region c7 Now, by Lemma 5.3, 1 - q ( T ) , It1 T except for a possible Siege1 zero, where we write q ( T ) = cl log-4/5 T. But in our present situation, moduli k do not + ~ Siegel's ~ , theorem (see e.g. [3], 521) asserts that exceed L ~ ~whence

<

58 there is no such zero, and so Hence

2.cui

p

< 1 - T ~ ( N ' / ~and ) , a fortiori /3 - 1 < 0.

by Lemma 5.2. Summing over k

x mod k , leads to (5.8).

<

~

~

and the ~ primitive + ~characters l

On the other hand, by Gallagher's lemma [4] it follows that I2

I

on writing X = m a x ( t , M ) and Y integrand on the right of (5.12) is

=

min(t

+ Q, N ) .

By (5.10)) the

Hence, applying (5.l l ) , we obtain

Summing over primitive characters x mod k and k 5 L (5.8) and (5.10).

~

~we +deduce ~ ~ 0

,

The Goldbach- Vinogradov Theorem in arithmetic progressions

Lemma 5.5. For an arbitrary constant A1 B1 = B1(A) > 0 large enough, we have

max LBl 5 5 ' 9

~~1

2

'&

< S S P k_S

and

1

59

> 0 and for a constant

C*

max I W ( ~A , ) << NLP*', Ti;;'45l/kQ mod

(5.14)

l/kQ

*'

x mod k

{llIkQ K5 << N112Lc.

Proof. We only give proofs of (5.14) and (5.17), other cases being similar. We apply Heath-Brown's identity ( Lemma 1 [5] with k = 5)

A(m)x(m) to find that it is a linear to the sum $(x, Ad, u) = combination of O ( L ~ ' ) terms, each of which is of the form

By Perron's formula (see e.g. p. 60, Lemma 3.12 in [14])

Shifting the line of integration to s = 112

+ it, we obtain

U1/2+it - n/fl/2+it

~ ( uM; ) = For S > R

27T

-N

F(Z + i t , X)

112

+ it

dt

> NE we have x # X0 and (5.3) becomes

+ 0(L2).

(5.18)

and so W(X, A) is a linear combination of O(LlO)terms, each of which is of the form

whence

<< L

~rnax J iW

-N

F

(f+ ;1

( 4log u + AU

it, X)

2~

>

dildi

1

In view of

Lemmas 4.4 and 4.3 in [14, p.711 show that the inner integral in (5.19) can be estimated as

<

min

(

t

2 min

M
N It+2nAul

+

where To is a parameter such that J t 2nAuJ > It112 whenever It1 > To. It is clear that (5.14) is a consequence of max kwS

~l/i/(x, A ) / << N S ~ I ~ L L - ~ ' .

(5.21)

x mod k

Substituting (5.20) in (5.19) with To = 8nN/SQ, we see that (5.21) is a consequence of the following two estimates: For 0 < TI 5 To, we have

zz*lT1 (f

k-S

x mod k

JF

+

The Goldbach- Vinogradov Theorem in arithmetic progressions

61

while for To < T2 5 N , we have

< <

Both (5.22) and (5.23) follow from lemma 5.1 provided LB1 5 S P N ~ / ~ This ~ - completes ~ . the proof of (5.14). Now we turn to the proof of (5.17), which is rather parallel to that of Lemma 5.4 and (5.14). We therefore suppress the details and indicate necessary modifications of the proof of (5.14). Correspondingly t o (5.12), we have

where X = m a x ( t , M ) and Y = min(t (5.18) then reads

where

+ R Q / 2 , N ) . The counterpart of

yi+it - x;+it

~ ( t=)r ( t ;Y,X )

=

4 +it

Similarly as in the proof of (5.18), we have

<

since ~ , hir - R Q / 2 t _< N . Using the the last term being << t - 1 / 2 ~ - ~M ~/ R in the above and the trivial estimate data R Q / 2 ~ N / R L 5

<

we deduce that

EI )<< L~ min

M - ~ I ~ R QI , ~

(- T) R Q N1I2 N1/2 '

62

Z.Cui

Equating R Q / N ~ = / ~i ~ l / ~ / l twe l , have jtl = To = N / ( R Q ) , so that

Substituting these O ( L l O )terms in ( 5 . 2 4 ) , we obtain

Thus, as in the proof of ( 5 . 1 4 ) ) it suffices for the proof of (5.17) to

holds for R / 2

<S
L and ~ 0~< TI

< To,and <

S R L and ~ To ~ < T2 N . (5.25) and ( 5 . 2 6 ) again holds for R / 2 follows from Lemma 5.1 provided R N2/15-&. This completes the 0 proof of ( 5 . 1 7 ) . Collecting the estimates (5.7)-(5.9) and (5.13)-(5.17)) it follow from (4.11) that

6.

<

The main terms and the singular series

Now, it remains to compute I l o and 120,Ibo. Recalling the definition of I l o (given right after ( 4 . 1 ) ) and using ( 3 . 6 ) and ( 3 . 8 ) , we have

The Goldbach- Vinogradov Theorem in arithmetic progressions

63

Similarly, (recalling the definition right after (4.6)), we have

and

Now we may give the postponed definition of the singular series

From the well-known result IG(XO)1 checked that

< 1 and JC(q,b, r , a ) 1 < 1, it is easily

Here the implied constants are absolute. Collecting (6.1)-(6.3), we have

Now, our Theorem 2 follows from (5.27) and (6.5).

Acknowledgments T h e author would like to thank Professor Liu Jianya for constant discussions and valuable advices. He also thanks to Professor Shigeru Kanemitsu and Yoshio Tanigawa for careful reading of the manuscript.

References [l] R. Ayoub, On Rademacher's extension on the Goldbach-Vinogradov theorem,

Trans. Amer. Math. Soc. 74 (1953), 482-491. [2] Z. Cui, The ternary Goldbach problem in arithmetic progressions, to appear. [3] H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, Berlin 2000. [4] P. X. Gallagher, A large sieve density estimate near u = 1, Invent. Math. 11 (1970), 329-339. [5] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan's identity, Canad. J . Math. 34 (1982), 1365-1377. [6] J.-Y. Liu, The Goldbach-Vznogradov theorem with primes in a thin subset, Chinese Ann. Math. 19B:4 (1998), 479-488. [7] J.-Y. Liu and T . Zhan, The ternary Goldbach problem in arithmetic progressions, Acta Arith. 82 (1997), 197-227. [8] J.-Y. Liu and M.-C. Liu, The exceptional set in the four prime squares problem, Illinois J . Math. 44 (2000), 272-293. [9] M.-C. Liu and K.-M. Tsang, Small prime solutions of linear equations, in: ThCorie des Nombres, J. M. De Koninck and C. Levesque (Cd. J. M. Dekoninck and C. Levesque), 595-624. Walter de Gruyter, Berlin 1989. [lo] M. C. Liu and T. Zhan, The Goldbach problem with primes zn arithmetic progressions, in: Analytic Number Theory (Kyoto, 1996) (London Math. Soc. Lecture Note ser. 247, ed. Y. Motohashi), 227-251. Cambridge University Press, 1997. [ l l ] C.-D. Pan and C.-B. Pan, Fundamentals of analytzc number theory (in Chinese),

Science Press, Beijing, 1991. [12] K . Prachar, Primzahlverteilung, Springer, Berlin 1957. [13] H. Rademacher, ~ b e reine Erweiterung des Goldbachschen Problems, Math. Z. 25 (1926), 627-660. [14] E. C . Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., University Press, Oxford 1986. [I51 R. C . Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts Math., vol. 125, Cambridge University Press, Cambridge, 1997.

The Goldbach- Vinogrudov Theorem in arithmetic progressions

65

[16] I. h l . Vinogradov, The representation of an odd number as a sum of three primes, Dokl. Akad. Nauk. SSSR 16 (1937), 291-294. [17] I. M.Vinogradov, Some theorems concerning the theory of primes, Math. Sb. N. S., 2 (1937), 179-195. [18] D. Wolke, Some applications to zero-density theorems for L-functions, Acta Math. Hungar. 61 (1993), 241-258. [19] T. Zhan and J.-Y. Liu, A Bombieri type mean-value theorem concerning exponential sums over primes, Chinese Sci. Bull. 41 (1990), 363-366. [20] A. Zulauf, Beweis einer Erweiterung des Stazes Von Goldbach-Vinogradov. J . Reine Angew. Math. 190 (1952), 169-198.

DENSITIES OF SETS OF PRIMES RELATED TO DECIMAL EXPANSION OF RATIONAL NUMBERS Toshihiro ~ a d a n o l Yoshiyuki , ~ i t a o k a Tomio ~, Kubota and Michihiro ~ o z a k i ~ Department of Mathematics, Meijo University Tenpaku, Nagoya 468-8502, Japan Department of Mathematics, Meijo University Tenpaku, Nagoya 468-8502, Japan [email protected]

Chitahigashi High School Dougashima, Yawata, Chita 4 78-0001, Japan

Abstract

For a prime p and a given integer a # 0, k l , not divisible by p, we suppose the order of a mod p is composite n k , n > 1 . Let ri be the least non-negative residue mod p of a", 0 5 I 5 n - 1 (which has and let s,(p) = rz a relationship to a-adic expansion of =

{$I,

i),

with {a) denoting the fractional part of n

xyc;

We are

interested in the set of primes for which ~ , ( p )takes a prescribed value s for a given n, and in particular in the limit of the relative frequency (density) P , ( x , n , s ) . We propose several new conjectures on them and n,S ) constructed in the same way from the subgroup G,,, also on P,,(x, of ( Z I p Z ) ' of order n, where a is taken to be a primitive root mod p. We also refer to rather general probabilistic phenomena. Keywords: decimal expansion, density, uniform distribution, normal distribution

2000 Mathematics Subject Classification: 11Ji'l, 1 lKO6

Number Theory: T ~ a d i t i o nand Modernization, pp. 67-80 W. Zhang and Y . Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

68

1.

T . Hadano, Y . Kztaoka, T . Kubota and M . Nozaki

Introduction

Among a number of numerical data which have been incentive to our research, the following is typical:

where 142857 signifies the period of the purely periodic decimal (10-adic) expansion of 117. For (1.1)we have:

Our starting point was the following Theorem 1 which elucidates the above phenomenon. Before stating Theorem 1 we introduce notation which we will use throughout the paper Notation p signifies a prime : the integral part of x = the largest integer 5 x {x) = x : the fractional part of x a : an integer # 0, *I, not divisible by p e = e,(p) : the order of a mod p e = nk, n > 1 (we so suppose in anticipation of application to the group (ZlpZ) r 2. p[$] = p : the least non-negative residue mod p of -

[XI

[XI

{$}

,

=

(modp), 0 < r i < p , 0 5 i 5 n i ~ ( p= ) ~a (p) = Gn,p: the subgroup of (ZlpZ) of order n . -Ti

We are now in a position to state Theorem 1. (i) As in Notation suppose the order e of a mod p (p 1. a ) is composite, e = nk, n > 1. Let ri denote the least non-negative residue mod p of aki :

Then

69

Densities of sets of primes

is an integer such that s(p) = if n is even and 1 5 s(p) 5 n - 2 if n is odd. 3, (ii) If in (i), we assume n further decomposes as n = nln2, n1 then

>

Before giving the proof, we state some observations which lead t o plausible conjectures on the set of primes for which s(p) takes the prescribed value s for given n / e . In tabulating the numerical data, we will restrict ourselves to the case n (25) an odd natural number, 1 5 s 5 n - 2 and t o the base 10 (a = 10). We introduce the relative frequency (density)

under the notation of Theorem 1. We note that P,(n, s , x) = 0 or 1 if n is even or n = 3.

Conjecture 1. lim P,(n, s , x) exists and the value P ( n , s ) is indepen2"oo dent of a. Hence, in particular, lim Plo(n, s , x) = P ( n , s ) is expected, and so 2'00 we shall give tables with a = 10 in the following, and in $3 we shall give those for a = 2 , 5 and Pg,(n, s , x), which is to be introduced in $2 and is also expected to converge to the same P ( n , s ) . Table 1 is the table for P ( n , s , lo9), in which 0.0000 in the column for n = I 1 means that primes which take values s = 1 , 9 are very rare and the figure 0 for n = 9 means that there is no such prime (Theorem l,(ii)). Table 1

Table 1 (Table 6) suggests that the values P ( n , s , x) are symmetrically distributed relative to s = ( n - 1)/2, and in the first instance, that

Conjecture 2. P(n,s)=P(n,n-1-s)

for

1 i s s n - 2

,

70

T . Hadano, Y . Kitaoka, T . Kubota and M . Nozaki

Secondly,

Conjecture 3. P ( n ,s ) > 0 holds for 1 5 s

< n-

2 if n is an odd prime.

And thirdly, the distribution of P ( n ,s ) should be the normal distribution. To make the last statement more precise, we recall standard notation of statistics. Given a table of frequency distribution Table 2 value relative freauencv

x1

52

1 r , I r? I

"

'

..,

xm Ir,

1

sum 1

we compute the mean p = p ( x ) = p(n, x ) = Cz1xiri and the standard deviation a = a ( x ) = a ( n ,r ) =

n

-

Jr < x . r , - p2 with xi

=

i, 1

i 5

2 , and ri = P ( n ,i, 10') to obtain Table 3

Table 3 now suggests the first half of

Conjecture 4.

n-1 lim p = X-+W 2 '

lim a = X+OO

( n - 1)/12 if 3 { n , ( n - 3)/12 zf 3 1 n.

The last half is not apparent from Table 1 , but in accord with data (Table 7) and intuitively supported by Theorem 2. We denote the density function of the normal distribution with mean p and standard deviation a by fp,,(x) and compare the above data with its values whose tabulation is Table 4

Table 4 suggests

Conjecture 5. lim

max

n+co l<s
I P ( n ,s ) - f,,, ( s )1 = 0.

71

Densities of sets of primes

It may be more appropriate to compare distribution functions instead of density functions. We now turn to the proof of Theorem 1.

Proof of Theorem 1. (i) First we prove the identity

Since the right-hand side of ( 1 . 6 ) is an integer and p { ak - 1 , it follows ri and, a f o r t i o r i , that s ( p ) E Z. that p I To prove ( 1 . 6 ) , we substitute ( 1 . 3 ) in

t o write

Then substitute ( 1 . 3 ) in the form ak('+') = ri+l + p

[?]

.k(t+l)

to obtain

the second term on the right being ( r , - r o ) / p = 0 , whence ( 1 . 6 ) follows. To prove the range for s ( p ) , we first note that ( 1 . 3 ) , or its equivalent

>

implies p { ri in view of p { a , so that ri 1 . Secondly, since ki < e, all aki are distinct mod p, and so are T i , 0 5 i 5 n - 1 w i t h r o = 1. We conclude that r,, 0 5 i 5 n - 1 are distinct integers between 1 and p. Hence s ( p ) is at most

which is less than ( n - 1 ) p for n

2 3.

72

T. Hadano, Y . Kitaoka, T . Kubota and M . Nozaki

Hence, we always have, for n

Now suppose n is even, so that

and p { a k ?

-

> 3, is an integer. Then, since

1, we must have

whence rn+i 2

+ ri r O

yielding s(p) = n/2. (ii) We divide the sum s(p) into partial sum by st:

722

mod p.

equal parts and denote the !-th

nl-1

sf =

C Tf+in2

P i=O we then claim that sf E Z.Indeed, by (1.7)

and sf E Z follows. p - 1, we have nl 5 s f p (p - l ) n l and hence In view of 1 ri 1 5 sf 5 n l - 1. Noting ro = 1, we may improve the bound for so as follows.

< <

whence so 5 n1 - 2. Therefore, summing over conclude that

<

e

= 0 , . . . , n 2 - 1, we

Densities

of

73

sets of primes

and the proof is complete.

["*'a1'][$I

Remark 1. Each summand - ak on the right of (1.6) admits the following familiar interpretation. We restrict to the case a = 10. Let dl . . . d, be the period of the purely periodic decimal expansion of l l p :

Then, since cl . . . cri =

[TI, it follows that

viewed as a k-digit integer. Thus s(p) is the sum of these integers divided by l o k - 1, and Theorem 1 explain (1.2) with e = 6, ( k ,n , s ( p ) ) = (3,2, I ) , (2,3, I ) , (1,6,3), where for computation of ri, formula (1.7) is most feasible.

2.

Generalization And Stochastic Discussion

In the setting of Theorem 1 we suppose a is a primitive root mod p. Then the order e of a is p - 1, which is a composite number, e = p - 1 = nk, n > 1. Let Gn,pdenote the subgroup of (ZlpZ)' of order n. Then it consists of akifor i = 0,1, . . . , n - 1. Hence in view of (1.3) and (1.4),

where the residue class g is identified with an integer in the class. The relative frequency Pg,(n, s , x) is defined in the same way as (1.5), and with the same limit density function P ( n , s ) , we propose

Conjecture 6 . lim Pgr(n, s , x) = P ( n , s ) .

xi00

The following conjecture, also numerically supported, lays a basis for stochastic discussion on P ( n , s ) .

Conjecture 7. For an odd natural number n (> 2) and a prime number satisfying p E I mod n, we set

74

T. Hadano, Y . Kztaoka, T . Kubota and M . Nozaki

Then the points in tends to infinity.

Up,,

S ( p ) are uniformly distributed in ( O , l ] as x

Remark 2. Suitably generalized , this conjecture leads to the following plausible conjecture: Let F = Q ( a ) ( # Q) be the algebraic number field generated by an algebraic integer a , and let k be a non-negative integer. For a prime number p which decomposes fully in F and a prime ideal p lying above p, we write in Fp = Qp Then the points (cp(O)/p, c, ( l ) / p , . . . , cp(k)/p) (E (0, l]k + l ) are uniformly distributed as p and p run through primes and prime ideals described above. This conjecture is true [DFI], [TI when F is quadratic and k = 0. The next theorem with m = n- 1 tells intuitively, under Conjecture 7, why the distribution P ( n ,s) is approximated by the normal distribution by neglecting the small difference l / p between s ( p ) and the sum over elements of G,,p. It also gives an intuitive evidence for the variance part of Conjecture 4 in the case 3 n on noting the ratio of the standard deviations of JmXm and Xm is

+

Jm.

Theorem 2. Let X I , 2 2 , . . . , x, be random variables on R obeying the uniform distribution I ( O , l ) , or what amounts to the same, their distribution function are all equal to the (set-theoretic) characteristic function fi(x) of [0,11. Then, putting

X

=

lim Xm determines the normal distribution on R with mean p = 0

nicO

and standard deviation a

=

1 rn'

Proof. Since the mean and standard deviation of z i are 112 and 1, m Xm has the same mean and standard deviation. The theorem is then a special case of the central limit theorem. For curiosity's sake, we shall prove below that the value -2- of a is a rn natural consequence of the theory of Fourier transforms, with p given. We may suppose p = 0 instead, i.e. we may take the characteristic function fo of [-1/2,1/2] instead of f l . We recall the standard notions of Fourier transform theory. The convolution f * g of two function f and g is defined by

Densities of sets of primes

and the Fourier transform f * of a function f is defined by

Starting from Fl = fo, we define the m-ple convolution F, inductively by F, = Fm-1* fo. It is a basic fact in probability theory that the distribution density of x l x2 . . . + x, = fix, is F,, whence that of X, is f i F , ( f i x ) . sin .iry Now the Fourier transform fo* of fo is -, and so that of F, is

+ +

7rU u

(%) m

. Denote by cp,

density f i F m ( f i x ) that

the Fourier transform of the distribution

of X,.

Then, it follows from the above argument

Since the distribution density of X can be determined as the Fourier transform cp* of cp, it suffices to find p(y) = lim cp,(y). For this, we m+w

need to study cpl(y) first. A simple calculation shows

whence by the Taylor expansion, we deduce that

= lim

m

m+w

so that cp = lim cp, m-+w

exists and is a solution t o the differential equation dcp - n 2 - - -dy

Y(P

76

T. Hadano, Y. Kztaoka, T . Kubota and M . Nozaki T 2

Solving the equation, we have y(y) = coe-TY distribution density, we have

712

2

.

But, since cp* is the

2

and cp(y) = e - T Y . Recalling that e - ~ ~is' invariant under the Fourier transformation, we find that the Fourier transform of e p a x 2is h a = 7r2/6, we conclude that

which is f,

I -

T

of the normal distribution, i.e. o =

'C

e

I

- L y 2

. Hence with

1/m. 0

T h e proof is complete. Finally we state

Proposition 1. For n 2 5 an odd integer, Conjectures 1, 6 and 7 yield

Proof. We have, by definition,

where ~ ( xk,; t) is the number of primes less than x lying in the residue class l mod k. Now the unity of Gn,pgives rise to the sum

1 1

C

-1 = o ( log log x ) = . ( I ) ,

p<,

P

x / log x

77

Densities of sets of primes

by Dirichlet's prime number theorem, while the sum over g

which tends to ( n - 1)J : tdt = uniform distribution of points in

# 1 is

9by the defining property of the

Up,, S ( p ) ,whence the result

follows.

0

Numerical data

3.

To see the dependence of numerical data on x, we give the following s , x) in the case of subgroups, where x = 10' table of densities Pg,(n, in the upper row, and x = 10'' in the lower row. We round off to five decimal places. Table 5

n=5

n=7 S

4 5 2 1 3 0.00837 0.21644 0.55012 0.21673 0.00834 0.00834 0.21663 0.55002 0.21669 0.00832

n=9 s

1 0 0

n = 11

2 0 0

3 4 5 6 0.24993 0.50014 0.24993 0 0.25001 0.50000 0.24998 0

7 0 0

T. Hadano, Y . Kztaoka, T . Kubota and &I. Notaki

Table 6

the table of P,,(n, s , l o 9 ) .

In the following table, max of error means max

l<s
and

IP,&, s , 10"

-

fp,&)l,

79

Densities of sets of primes

Table 7

Table 8 the tableof

rnax

l<s
/ P g r ( n , s , 1 0 9 ) -P , ( n , s , 1 0 ~ ) 1for a = 1 0 , 2 , 5 .

80

T. Hadano, Y.Kitaoka, T . Kubota and M . Nozaki

the table of

1-

max

I<s
Table 9 - Pgr( n ,S, 10'; k, 5 ) 1 , where

1 Pgr( n ,S, 10')

Finally, we give expected densities by rationals. Table 10 n=5

n=7 s

1

1 1 11120

1

1 2 261120

1

1 3 1 4 1 5 661120 261120 11120

1

1

References [KN] Y. Kitaoka and M. Nozaki, O n the density of the set of primes which are related t o decimal expansion of rational numbers, RIMS Kokyuroku, to appear. [DFI] W. Duke, J.B. Friedlander and H. Iwaniec, Equidistribution of roots of a quadratic congruence t o p r i m e moduli, Ann. of Math. 1 4 1 (l995), 423-441. [TI A. T 6 t h , Roots of Quadratic congruences, Internat. Math. Res. Notices 2000, 719739.

SPHERICAL FUNCTIONS ON p-ADIC HOMOGENEOUS SPACES Yumiko Hironaka Department of Mathematics, Faculty of Education and Integrated Sciences, Weseda University, Nishi- Waseda, Tokyo 169-8050, Japan [email protected]

Abstract

In 51, after defining spherical functions on homogeneous spaces, we examine the case of symmetric forms as a n enlightening example. In 52, we introduce a general formula of spherical functions using functional equations under suitable assumption. In 53, we study a certain mechanism of functional equations of spherical functions.

Keywords: spherical functions, p a d i c homogeneous spaces, relative invariants, prehomogeneous vector spaces 2000 Mathematics Subject Classification: Primary 1lF85; Secondly 1lE95, llF70,22350

1.

Introduction

31.1. Definition Let G be a reductive linear algebraic group and X be a G-homogeneous affine algebraic variety, where everything is assumed to be defined over a non-archimedian local filed k of characteristic 0. We write the G-action on W by (g, x) g . x. For an algebraic set, we use the same ordinary letters for the set of k-rational points, e.g. G = G ( k ) , X = X(k). Taking a maximal compact subgroup K , we denote by E ( G , K ) the Hecke algebra of G with respect to K

-

$(kgkf) = $(g), vk, kf E K, vg E G compactly supported

Number Theory: Tradition and Modernization, pp. 81-95 W. Zhang and Y.Tanigawa, eds. 02006 Springer Science Business Media, Inc.

+

which is generated by the characteristic functions chKgK of K g K (g E G) over C , and becomes a C-algebra by the convolution product

where dg is the normalized Haar measure on G. We consider the following X ( G , K)-modules

1

C ~ ( K \ X ) = ( ~ : X - CQ~( ~ . x ) = ~ ( x ) , ~, ~ E K S ( K \ X ) = { Q E CCO(K\X) I compactly supported), where the action is given by

A nonzero function Q E CCO(K\X) is called a spherical function on X if it is a common X ( G , K)-eigen function, i.e, there exists a C-algebra C such that 4* Q = A(+)* for every 4 E X ( G , K ) . map X : X ( G , K )

-

Typical spherical functions are constructed in the following manner. We denote by / 1 the absolute value on k normalized by 1571 = q-l, where n is a prime element of k and q is the cardinal number of the residue class field of k. Let IP be a closed subgroup of G defined over k satisfying G = K P = P K . Let fi(x), 1 5 i 5 n be nonzero regular functions on X defined over k which are relative P-invariants; namely, 1 5 i 5 n of IP satisfying there are rational characters $i,

Here n is arbitrary and the characters $i become automatically k-rational. Consider the following integral for x E X and s = ( s l , . . . , s,) E Cn w(x; s) =

:i,

I f

x)ISzdk,

<

The right hand side is absolutely convergent if Re(si) 2 0, 1 i 5 n , and has an analytic continuation to a rational function of qSl,. . . , qS71,

Proposition 1.1. If w(x; s ) is a nonzero function on X, then it is a spherical function on X . Indeed

Spherical functions on p-adic homogeneous spaces

where d r p is the normalized right invariant Haar measure on P . Spherical functions on homogeneous spaces are an interesting object to investigate and a basic tool to study harmonic analysis on G-space X . They have been studied also as spherical vectors of distinguished models, Shalika functions and Whittaker-Shintani functions, and have a close relation to the theory of automorphic functions and representation theory. When G and X are defined over Q,spherical functions appear in local factor of global objects, e.g. Rankin-Selberg convolutions and Eisenstein series (e.g. [CS], [Fl], [HS3], [Jac], [KMS], [Sfl]). To obtain explicit expressions of spherical functions is one of basic problems. For the group case, it has been done by I. G , Macdonald and afterwards by W. Casselman by a representation theoretical method (cf. [Ma], [Cas]). There are some results on homogeneous space cases mainly for the case that the space of spherical functions for each Satake parameter is of dimension one (e.g. [CS], [KMS], [Of]). On the other hand, the author has given an expression of them independent of the dimension, based on the data of the group G and functional equations of them, which we show in $2. Hence the knowledge of functional equations is important to obtain explicit expressions of them. In $3, we show a unified method to obtain them and explain they are reduced to those of p-adic local zeta functions of small prehomogeneous vector spaces of limited type. Some specific X such as the space of symmetric forms, hermitian forms, or alternating forms, there is a close relation to classical problem in Number Theory. In the latter half of this section, we will explain this situation by taking the space of symmetric forms. Some more comment about this wil be given a t the end of $1. 51.2. The space of symmetric forms For a symmetric matrix A of size m and a matrix X E Mmn, we denote A[X] = t X . A = t X A X , which is a symmetric matrix of size n. One of the fundamental problems in the theory of symmetric forms is to find solutions X E A&,(Z) of the matrix equation

which is difficult in general. Even the problem "Characterize prime numbers for which x2

+ ny2= p is solvable"

84

Y. Hzronaka

is not completely solved. The cases n = 1 , 2 , 3 are classically known (stated by P. Fermat, proved by L. Euler, p = 1 (mod 4); p = 2 or p = 1 , 3 (mod 8) ; p = 3 or p 2 1 (mod 3)) in this order). In general, it is impossible to characterize p in a similar style. D. Cox's book [Cox] is an interesting modern approach to this problem, where Hilbert class field theory and complex multiplication are exploited. For A E Symm(Z) and B E Symn(Z), it is easy to see that A[X] = B has a Z-solution + A[X] = B (mod N) has a Z-solution for any N E W ==+A[X] = B has Zp-solution for any prime p. The implication for the opposite directions is known as follows. Theorem (Minkowski-Hasse). Let A E Symzd(Q), B E ~ y r n (; Q~ ) with m n 2 1, where nd indicates nondegenerateness. Then

>

A[X] = B has a Q-solution ++A[X] = B has an R-solution and a Qp-solutionfor each prime p. In general, this result cannot be extended to the case of Z-solutions. There are equations which have R-solutions and Zp-solutions for each p, but no Z-solutions; for example

While Hasse-Minkowski theorem is a qualitative theorem for Q-solutions, Siegel's main theorem given below shows us a quantitative results for Z-solutions. To describe the theorem, we need some more notations. Z ) B E ~ y m ; ~ ( Zwith ) m n 2 1 and R = For A E ~ y m & ~ ( and Z, ZplR,. . ., set

>

We define t h e local density of integral representations of B by A pp(B, A) by the limit

where the inside term of lim is independent of 1 if 1 is sufficiently large, and it is interpreted as the volume of X B , ~ ( Z p )For the cases A is unimodular, the size of B is 1, or A = B , good explicit formulas are known. There is a formula for general A and B when p # 2, but it is too complicated (cf. [SH]).

85

Spherical functions on p-adic homogeneous spaces

For simplicity, assume that A and B are positive definite. Then one ~ . can consider the volume p, (B,A) of R-solutions X B ,(R) Now let A1, . . . , Ah be a set of complete representatives of the SL,(Z)equivalent classes within the genus containing A, i. e. within

{A'

E

(

A and A' are SL,(Zp)-equivalent for 'dp and sYmkd(~) SL, (R)-equivalent.

Theorem (Siegel). Under the notations above

where

Now we go back t o our original theme, spherical functions on Q ~ group ). G, = GLm(Qp) acts on X, by g .x = X, = ~ ~ r n & ~ ( The gxtg = zltg]. For simplicity, we will write ,u( , ) in stead of ,up(, ). Set K m = GL,(Z,) and consider the following integral w(x; S) =

jd,(k . x)1:

dlc,

x E X,,

s 6 Cm,

where dk is the Haar measure on K , I l p is the normalized p-adic value on Qp and di(y) is the determinant of upper left i by i block of y. Here d,(x), 1 5 i 5 m , are relative B,-invariant on X m , where Bm is the Bore1 subgroup of G, consisting of lower triangular matrices, and G , = KmBm= B,Km as is well known. 0, 1 5 i 5 The above integral is absolutely convergent if Re(s,) m - 1, and analytically continued to a rational function of pS1,. . . , pSm. As a function on X,, w(x; s ) is an element of Cm(K,\Xm) and we obtain a typical example of spherical functions on X,. The following theorem shows that spherical functions can be regarded as generating functions of local densities.

>

86

Y.Hzronaka

Theorem ([HI-I]).Let m

>n

and x E X,.

Then we have

w ( x ;31, . . . , S,,, 0 , . . . , 0) n

m-n

where y runs over the representatives of Kn-orbits in X,. By the above theorem, we can expect to extract the information on local densities from that of spherical functions when the latter is well analysed, and conversely. Unfortunately this approach has not been successful yet. Similar consideration is valid for alternating forms and hermitian forms, and in those cases we have obtained good results (cf. [HSll, [HS21, [H21, [H31).

2.

Expressions of spherical functions

In this section we introduce (formal) explicit expressions of spherical functions after [H2, $11,while some notations are changed from there. For a connected linear algebraic group W. We denote by X(W) the group of k-rational characters of W, which is a free abelian group of finite rank, and by Xo(W) the subgroup consisting of characters corresponding to some relative W-invariants. A set of relative W-invariants is called basic if the set of the corresponding characters forms a basis for Xo(W). In 52 and $3, let G be a connected reductive linear algebraic group defined over k, I5 a minimal parabolic subgroup of G defined over k, and K a maximal compact subgroup of G for which G = B K = K B . The group I5 is not necessarily a Bore1 group. Denote by dg and dk the normalized invariant Haar measure on G and K , respectively, by dp the left invariant Haar measure on B normalized by dg = dpdk, and by 6 the modulus character of B (d(pq) = S-' (q)dP, p, q E B). In $2, let K be a special good maximal bounded subgroup of G, U the Iwahori subgroup in K which is compatible with B, and assume the following. ( A l ) X has only a finite number of I5-orbit. (A2) The group Xo(15) has the same rank with X(B), and a basic set of relative B-invariants on X can be taken from regular functions on X.

87

Spherical functions on p-adic homogeneous spaces

(A3) For any x E X , t'here exists $ E Xo(15) whose restriction to the identity component of the stabilizer IB, is nontrivial. By ( A l ) , there exists unique open B-orbit XOp in X,and XoP decomposes into a finite numbers of open B-orbits, which we write

Let { fi(x) / 1 5 i 5 n ) a basic set of regular relative B-invariants, and Xo(15) the character corresponding to fi(x) for each i, where n = rank(X(IB)). For each s E Cn and v E J ( X ) , we set

gi E

and determine

E E

Qnby

Now we consider the integral

>

The right hand side of (2.1) is absolutely convergent if Re(si) -Ei, 1 5 i 5 n , analytically continued to a rational function of q l , . . . , q,, and we get a spherical function on X . Let W be the restricted Weyl group of G with respect to a maximal k-split torus T contained in I5, then W r NG(T)/ZG(T),and NG(T) is a Levi subgroup of IB. Thus W acts on s E Cn through the canonical (conjugate) action on X(I5) and the identification X(IB) @z C Cn. Then we have

Theorem 2.1. Let x E X , s be generic, and M = { v E J ( X ) / X U n G s ~ j : O )Then .

Y.Hzronaka where

and A(s, w) is the invertible matrix determined by

Remark 1. In the above theorem, Q and y ( s ) depend only on the groups G and I5. C+ denotes the set of positive roots with respect to B in the set C of roots of G with respect to T.For the definition of a, E T and numbers q,, q,12 for a E C, we refer to [Cas, (9), Remark 1.1 and (12)l. If G is split, each q, = q and gal2 = 1.

3.

Functional equations of spherical functions

Under the assumption (AF) below we show the existence of a certain functional equation of spherical functions and it is reduced to those of p a d i c local zeta functions of small prehomogeneous vector spaces, for details see [H5]. 53.1. Type (F) Let W be a connected linear algebraic group W and Y an affine algebraic variety on which W acts, where everything is assumed to be defined over k . We say (W, Y) is of type (F)if it satisfies the following conditions:

( F l ) Y has only a finite number of W-orbits. (Then n7 has only one open W-orbit YOp.) (F2) For y E Y\Yop, there exists some $ in Xo(W) whose restriction to the identity component of the stabilizer Wy is not trivial.

(F3) The index of Xo(W) in X(W) is finite. (F4) A basic set of relative W-invariants on Y can be taken from regular functions on Y.

Spherical functions on p-adic homogeneous spaces

89

For a simple root a , let P be the standard parabolic subgroup ID{,). We consider a k-rational representation p : P ---t RkrIk(GL2)satisfying

where k' is a finite unramified extension of k, Rkllk is the restriction functor of base field, w, E N G ( T ) is a representative of the reflection in W attached to a , and B2 is the Borel subgroup of p(P) consisting of upper triangular matrices. Now we assume that

(AF) (B, X) is of type (F) and there is a k-rational representation p satisfying (3.1) for a simple root a. Remark 2. Chevalley groups are typical examples which have p as above for k' = k, so Rlcrlb(GL2)= GL2. If G is k-split, then lB is a Borel subgroup and P = B U BwaB. The assumption in $2 is the same that (B, X) is of type (F). Set % = X x V and = P x RklIr(GLI), where V = Rk~lt(M2,1) defined over k, and consider the following @-action on %:

Here we identify kt with its image by the regular representation in Md(k) and realize R k ~ I(GL2) k (resp. V) in G L ~ ~ (resp. ( ~ ) M~,,,(,)), where d = [kt : k] and is the algebraic closure of k. We note here that we may identify as P = P x GLa(kt) and V = kt2. We regard B as a subgroup of by the embedding

z

where ~ ( b )E ~Rk~lb(GL1)is the upper left d by d block of p(b) E Rkllk(GL2) Then, one can identify lB with the stabilizer of P at vo =

There is an isomorphism

90

Y . Hzronaka

And we have Proposition 3.1. (i) The space (F, 5) is of type ( F ) . (ii) The set of open B(k)-orbits in X ( k ) corresponds bzjectively to the set of open F(k)-orbits in % ( k ) b y the naup B . x . ( z ,v o ) .

-

53.2. Spherical functions and zeta distributions Because of the assumption of type ( F ) for (I5,X),there exists a basic set { f i ( x ) I 1 i n) of regular relative I5-invariants, where n = be a basic set of regular relative

< <

-

- IB

@-invariants satisfying - f i ( x ) = f i ( z , v o ) . We denote by Qi the character corresponding to f i ( x , v ) , then gi = $i is the character corresponding to f i ( x ) for each i. We may write

is the open P-orbit corresponding to X , by Proposition 3.1. where Denote by S ( X ) and ~ ( 2 the)spaces of Schwartz-Bruhat functions on X and 2, respectively. For s E Cn and u E J ( X ) , we have spherical function w u ( x ;s ) defined by (2.1),further we consider the following integrals

where dx is a G-invariant measure on X and dv is a Haar measure on V. The above integrals are absolutely convergent for R e ( s i ) 0 , 1 i n, and continued to rational functions of q S l , . . . , q s n . It is easy t o see that

>

< <

where ch, is the characteristic function of K . x and v ( K . 2 ) is the volume with respect t o the above measure dx. Since x ( x , v ) is a relative Rt,lr(GLI)-invariant with respect t o the action on v, it is homogeneous in the coordinates of v over k , and we set

ei = deg,

-

fi

( x ,v )

(1 I

2

I n).

Then we have the following relation between

flu($;

s) and

6,($; s).

91

Spherical functions on p-adic homogen,eous spaces

-

Proposition 3.2. Let q5 = q 5 @ ~ h ~ ( ~where r n ) >q5 E S ( K \ X ) and chv(pm) is the characteristic function of V ( p m )with pm = 'irmOk. Then for every uE J(X),

where c is a constant depending only on the normalization of measures. By (3.3) and the above proposition, we expect to get functional equations of wu(x;s ) from those of fi,($; s ) . 33.3. Functional equations Take an additive character rl on kt of conductor 'irlOk~, and define the by ) partial Fourier transform F($)for $ E ~ ( 2

We consider the following distributions on and s E Cn

By the action of

=P

~ ( 2for )each

u E J(X)

x G L l ( k t )on S ( X ) given by

we get

On the other hand, if (W,Y) is of type (F), it satisfies the following property (F5) (See [Sf2, Lemma 2.3, Corollary 2.41). (F5) There is a finite set (L) of linear congruences of type n

CmisiEX i=l

(mod

2~

Z

1% q

)

mi E Z,X E C

92

Y . Hironaka

which satisfies the following: If T is a nonzero distribution whose support is contained in Y\YOP and satisfies

then s satisfies some relation in (L) By the uniqueness of relatively invariant distributions on homogeneous spaces (cf. [Ig, Proposition 7.2.11) and the property (F5), we obtain the following proposition. Let e be the group index [X(I5) : Xo(B)],and Ju = {u E J ( X ) / P . X u = P . X u ) for each u E J ( X ) .

Proposition 3.3. There exist rational functions $,(s) which satisfy the following identity.

51

of q e , . . . , q

sa e

Rewriting by terms of the zeta integrals, we obtain

Theorem 3.4. There exist rational functions " i U ( s ) of q?, . . . , q which satisfy t h e following identity.

-

a e

Let normalize the measure dv on V to be self dual with respect to the r l ( t v a w ) .Then, by Theorem 3.4 and Proposiinner product (v, w) tion 3.2. we obtain

Corollary 3.5. For a n y

4 E S ( K \ X ) , we have

which i s independent of the choice of the character 7 o n k'. By Corollary 3.5 and the relation (3.3),we get

Theorem 3.6. For a n y x E X , we have

Spherical functions on p-adic homogeneous spaces

93

33.4. Small prehomogeneous vector spaces In this subsection, we look at the prehomogeneous vector space (p(IP,) x R k ~ I k ( G L Vl )), ,where the action comes from the one of on %. For each u E J ( X ) ,fix an xu E Xu and denote by P, the stabilizer of xu in IP.

Lemma 3.7. ( i ) For anyu, v E J ( X ) , takep, E IB satisfyingp,~x, = xu. Then the map

gives an isomorphism of prehomogeneous vector spaces (p(P,) x R k ~ I k ( G L 1 ) , (p(P,) x & I ~ ~ ( G L ~ ) ,IfV v) .E J,, we may take p, E P, and then the above isomorphism is defined over k . (ii) The set of k-rational points of the open orbit in (p(Bu)x R ~ I ~ , : ( GVL)~ ) , decomposes as

V) and

where p, E P satisfying p i 1 . xu For

4'

E S ( X ) and

4

E

E

Xu

S ( V ) ,we have

Then, applying Theorem 3.4, we obtain

hence we obtain Theorem 3.8. The prehomogeneous vector space (P(P,) x G L 1 ,V) has the functional equation:

94

Y . Hironaka

where the g a m m a factors T9,,(s) are the s a m e as those for T h e o r e m 3.4.

&(?; S )

in

The existence of the functional equation of the above type gives the following.

Theorem 3.9. For the prehomogeneous vector space ( p ( P u ) X R ~ I / ~ ( G L ~ ) , V), the identity component of p(P,) x Rk~/k(GL1)is isomorphic t o RpIk(GL1 x G L l , V ) over the algebraic closure % of k . Remark 3. The gamma factors yu,(s)'s of the functional equation of spherical functions w,(x;s ) on X are reduced to those for the small prehomogeneous vector spaces (p(Pu) x RklIk(GL1),V), for which the identity component is isomorphic to R k ~ l k ( G LX1 G L d over .. The set of isomorphism classes of k-forms of GL1 x GL1 corresponds , (cf. [PR, 52.2.41). bijectively to ~ o m ( G a l ( x / k )GL2(Z)) Some explicit examples are found in [H5, 541.

References W. Casselman, T h e unramzfied principal series of p-adic groups I. T h e spherical functions, Compositio Math. 40 (1980), 387-406. W . Casselman and J. Shalika, T h e unramified principal series of p-adic groups II. T h e Whittaker function, Compositio Math. 41 (1980), 207-231.

+

D. A. Cox, Primes of the forms z2 n y 2 ,Wiley Interscience, 1989. Y. Z. Flicker, O n distinguished representations, J , reine angew. hlathematik 418 (1991), 139-172.

Y. Hironaka, Spherical functions of hermztian and symmetric forms I, 11, 111, Japan. J . Math. 14 (1988), 203-223; Japan. J . Math. 15 (1989), 15-51; T6hoku Math. J. 40 (1988), 651-671. Y. Hironaka, Spherical functions and local densities o n hermitian forms, J . Math. Soc. Japan 51 (1999), 553-581. Y. Hironaka, Local zeta functions o n hermitian forms and its application to local densities, J . Number Theory 71 (1998), 40-64. Y. Hironaka, Spherical functions o n Sp2 as a spherical homogeneous space Sp2 x (Sp1)2-space,J . Number Theory 112 (2005), 238-286. Y. Hironaka, Functional equations of spherical functions o n p-adzc homogeneous spaces, to appear in Abh. Math. Sem. Univ. Hamburg. Y. Hironaka and F . Sato, Spherical functions and local densities of alternating forms, Amer. J . Math. 110 (1988), 473-512. Y. Hironaka and I?. Sato, Local densities of alternating forms, J . Number Theory 33 (1989), 32-52. Y. Hironaka and F. Sato, Eisenstezn series o n reductive symmetrzc spaces and representations of Hecke algebras, J . Reine Angew. Math. 445 (1993), 45-108.

Spherical functions o n p-adic homogeneous spaces

95

J . Igusa, A n introduction t o Theory o f Local Z e t a Functions, AMS/IP Studies in Advanced Mathematics, vo1.14, 2000.

H. Jacquet, A u t o m o r p h i c spectrum o f s y m m e t r c spaces, Proc. Sym. Pure Math. 61 (1997), 431-455. S. Kato, A. Murase and T . Sugano, Whittaker-Shintani functzons for orthogonal groups, Tohoku Math. J . 55 (2003), 1-64. I. G. Macdonald, Spherical functions o n a group of p-adic type, Univ. Madras, 1971. 0. Offen, Relative spherical functions o n p-adic s y m m e t r i c spaces, Pacific J . Math. 215 (2004), 97-149. V.Platonov and A. Rapinchuk, Algebraic groups and n u m b e r theory, Academic Press, 1994. F. Sato, Eisenstein series o n semzsimple s y m m e t r i c spaces of Chevalley groups, Advanced Studies in Pure Math. 7 (1985), 295-332. F.Sato, O n functional equations of zeta distributions, Advanced Studies in Pure Math. 15 (1989), 465-508. F.Sato and Y. Hironaka, Local densities of representations of quadratic forms over p-adic integers 'the non-dyadic case, J . Number Theory 83 (2000), 106-136.

ON MODULAR FORMS OF WEIGHT ( 6 n 1)/5 SATISFYING A CERTAIN DIFFERENTIAL EQUATION

+

Masanobu Kaneko Graduate School of Mathematics 33, Kyushu Uni.uersity, Fukuoka 812-8581, Japan

Abstract

We study solutions of a differential equation which arose in our previous study of supersingular elliptic curves. By choosing one fifth of an integer k as the parameter involved in the differential equation, we obtain modular forms of weight k as solutions. It is observed that this solution is also related t o supersingular elliptic curves.

K e y w o r d s : modular forms, supersingular elliptic curves, differential equation 2000 M a t h e m a t i c s S u b j e c t C1assification:Primary 11F11, l l G 2 0

Introduction

1.

In our previous work [5], [3], 141,we studied various solutions of the specific differential equation

(to

k

f"(d -

, k+l

E 2 ( d f f ( d+ k(k12

+

')E;(T)

(T)

=0

where T is a variable in the upper half-plane, k a fixed rational number, and E2(r)the "quasimodular" Eisenstein series of weight 2 for the full modular group SL(2, Z) :

In [5], we showed that for even k 2 4 with k $ 2 (mod 3 ) ) this differential equation has a modular solution of weight k on SL(2, Z) explicitly 97 Number Theory: Tradtion and Modernization, p p 97-102 W. Zhang and Y. Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

98

M . Kaneko

describable in terms of the Eisenstein series E4(7)and E6(7),and discussed its connection t o supersingular elliptic curves in characteristic p when k = p- 1. We studied further the modular/quasimodular solutions for other integral or half-integral values of k in [3], [4]. In this paper, we set k = (6n + 1)/5, one fifth of an integer congruent to 1 modulo 6. We then encounter as solutions modular forms of weight k on the principal congruence subgroup r ( 5 ) . Also, modular forms on F1(5) arise naturally. In $2 we describe the solutions in terms of fundamental modular forms of weight 115 which already appeared in works of Klein and Ramanujan. The proof is in essence similar to the one given in [3]. In $3, we discuss a relation between our solutions and supersingular elliptic curves, which is quite analogous to the situation studied in [5] and [8]. The author should like to express his sincere gratitude to Atsushi Matsuo, whose suggestion that the cases k = 715,1315 would provide interesting modular solutions gave impetus t o the present work. The author also learned from him that the differential equation ( # ) k , in its equivalent form, was already appeared in works of physicists, e.g., [6], [7], and its solutions, a t least for small values of k, correspond to particular models in conformal field theory.

Main result

2. Let

and

Here, ~ ( 7is) the Dedekind eta function. These forms are of weight 115 (with a suitable multiplier system) on r ( 5 ) , and the ring of holomorphic modular forms of weight ;Z on r ( 5 ) with this multiplier system (a good reference for this is Ibukiyama is the polynomial ring C [2]). Note that these forms are essentially the famous Rogers-Ramanujan

I,+

On modular forms of weight (6n + 1)/5 functions;

+

Theorem. Assume k = (6n 1)/5, n = 0 , 1 , 2 , . . . , n $ 4 (mod 5). Then the equation ( g ) k has a two dimensional space of solutions in C [ & , q5z]wt=kJthe set of homogeneous polynomials of degree 6n 1 in 41 and 42.

+

Example: Here are a basis of solutions for small k:

In general, we have a basis of the form x (polynomial in 4:) and 4;" x (polynomial in 4: and 4;). Here, we note that 4: become modular forms of weight 1 on

4: 4:

and and

In our previous cases treated in [5], [3], 141, all solutions were explicitly described with the aid of hypergeometric series. In the present case, however, a differential equation with four singularities emerges and we are so far unable to write down the explicit formulas for the solutions in general. We can nevertheless prove the theorem by giving the solutions recursively and by using an inductive structure of solutions of revealed in [3]. To give a recursive description of the solutions, we change the variable by setting

f where

t

=

&/& = q

-

=

W),

+ 1 5 4 ~- 30q4 + 40q5 + .

!jq2

,

is (the reciprocal of) the "Hauptmodul" of I'1(5), and we consider the equation locally as t a local variable. Then by a routine computation we

100

M. Kaneko

see that f ( r ) satisfies (#), if and only if F ( t ) satisfies (b),:

(b),

+

t(t2 l l t

-

+

l)~l'(t)

t2 -

( 7 -I:k

+ 11(1 - k ) t + -5, 6

F'(t)

where ' = dldt. Incidentally, an amusing remark here is that the equation

( b )

+

t(t2 l l t

-

+

+

l ) F t ' ( t ) (3t2 22t

-

+ + 3 ) F ( t )= 0

l)F1(t) (t

obtained by setting k = -1 in ( b ) , is exactly the one used in [ I ] for reconstructing Apkry's irrationality proof of ( ( 2 ) . T h e original equation (#), when k = - 1 becomes the trivial f l' = 0, but the solutions here are 1 and r , "universal periods" of elliptic curves. Hence ( b ) - l is obtained from this trivial equation by rewriting it locally in terms of t-variable. Now we are going to show that ( b ) , has a polynomial solution P ( t ) if k = (6n 1 ) / 5 ( n $ 4 (mod 5 ) ) . If P ( t ) is such a solution, then and are the solutions to (#),. The second one is a solution because S L ( 2 ,Z) acts on the solution space and the

+ &'k~($i/q5:) $ik~(-&/$z) I

the transformation formulas of Proposition. For 0

I n I 8,

and n

see [2]).

# 4, put

For n 2 10, n $ 4 (mod 5 ) , define P,(t) recursively b y

On modular forms of weight (6n + 1 ) / 5

101

9 ) (6n - 49) + 12 (6n( n--24) t(1 ( n- 9 )

-

llt

Then P,(t) is a solution of (b)(6n+1)/5 for all n

-

t 2 ) 5 ~ n - l o ( t ) . (1)

> 0, n $ 4 (mod 5).

Proof. We prove Proposition by induction on n . One may notice that the proof is essentially a translation of those of Proposition 1 and Lemma in [3]. It is straightforward t o check that each P,(t) for n 5 8, n # 4 satisfies (b)(6,+1)/5. S ~ P P Othat S ~ P7~-5(~) and P n - ~ ~ ( t ) (b)(6(n-5)+1)/5 and (b)(6(n-10)+1)/5 respectively. If we compute the left-hand side of ( b ) (6n+1)/5for F ( t ) = Pn(t) by substituting the definition ( 1 ) of Pn(t) in terms of Pn-5(t) and P,-lo(t), and using the induction hypothesis (b)(6(n-5)+1)/5 and (b)(6(n-10)+1)/5, we See that P,(t) satisfies (b)(6n+1)/5 if and only if the identity 12(36n2- 468n

+ 1421)(t2+ l l t

-

~)~~,-lo(t)

+ 5(n - 9 ) ( t 4- 228t3 + 494t2 + 2282 + l ) ~ ; - , ( t ) holds. We prove ( 2 ) also by induction on n. Suppose Pn_5(t) and Pn-lo(t) satisfy ( 2 ) . We want to show the corresponding identity for n being replaced by n 5. By replacing P,(t) by the right-hand side of ( 1 ) and then replacing P,-lo(t) by the right-hand side of ( 2 ) divided by the coefficient 12(36n2- 468n 1421)(t2 l l t - 1 ) 4 (thus expressing everything by Pn_5(t) and its derivatives), we obtain a multiple of the left-hand side of the differential equation (b)(6(n-5)+1)/5 satisfied by Pn-5(t), which vanishes by the induction hypothesis. This concludes the proof of the proposition and hence the theorem is proved. 0

+

+

3.

+

Reduction modulo prime

In this section, we present some observation about reduction modulo a prime p of our polynomials P,(t) as a conjecture. Let

be the elliptic modular j-invariant expressed in terms of t = &/&

Conjecture. 1 ) Let p # 5 be a prime. Then Pp-1(t) mod p is a "superszngular t-polynomzal', i e . , it zs equal to ~ i o c p(,t - t o ) where t o runs through those values for which the corresponding ellzptzc curve with j-invariant j ( t o ) is supersingular.

102

M. K a n e k o

2) For p as follows:

2 7,

the degrees of irreducible factors of P p - ~ ( tmod ) p are

= 1 mod 5, all irreducible factors have degree 2. If p = 3 , 7 mod 20, one factor has degree 2 and all the others have

(i) If p (ii)

-

degree 4. (iii) If p

13,17 mod 20, all irreducible factors have degree 4.

(iv) If p = 4 mod 5, t h e n there are h linear factors and ( p - 1 - h)/2 quadratic factors, where h =the class n u m b e r of the imaginary quadratic field Q ( 6 )

At least the first part of the conjecture should be proven by looking at the Hasse invariant of a family of elliptic curves corresponding to r1(5), but we have not worked out this.

References [I] F. Beukers, Irrationality of rr2, periods of a n elliptic curve and r 1 ( 5 ) , Approximations diophantiennes et nombres transcendants, Colloq. Luminy/FY. 1982, Prog. M a t h . 31, 47-66 (1983). [2] T . Ibukiyama, Modular f o r m s of rational weights and modular varieties, Abh. Math. Sem. Univ. Hamburg 70 (2000), 315-339. [3]

M.Kaneko and M.Koike, O n modular f o r m s arising from a differentzal equation

o f hypergeometric type, The Ramanujan J . 7 (2003), 145-164. [4] M. Kaneko and M. Koike, Quasimodular forms as solutions t o a differential equation of hypergeometric type, Galois Theory and Modular Forms, (ed. K. Hashimoto, K . Miyake and H . Nakamura), Kluwer Academic Publishers, 329-336, (2003). [5]

M.Kaneko and D. Zagier, Supersingular j-invariants, Hypergeometric series, and Atkin's orthogonal polynomials, AhIS/IP Studies in Advanced Mathematics, vol. 7 (1998) 97-126.

[6] E. B. Kiritsis, Fuchsian differential equations for characters o n the torus: a classificatzon, Nuclear Phys. B 324 (1989), no. 2, 475-494. [7] S. D. Mathur, S. Mukhi and A. Sen, O n the classification of rational conformal field theories, Phys. Lett. B 213 (1988), no. 3, 303-308. [8] H. Tsutsumi, T h e A t k i n orthogonal polynomials for congruence subgroups of low levels, The Ramanujan J . (to appear).

SOME ASPECTS OF THE MODULAR RELATION Shigeru ~ a n e m i t s u lYoshio , ~ a n i ~ a wHaruo a~, ~sukada~ and Masami ~ o s h i m o t o ~ Kinki University-School of Humanity-Oriented Science and Engineering, Iizuka, Fukuoka, 820-8555, Japan [email protected]

Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan 3 ~ i n k iUniversity-School of Humanity-Oriented Science and Engineering, Iizuka, Fukuoka, 820-8555, Japan Graduate School of Science and Technology, 4 ~ i n k Unversity-Interdiscriplinary i Higashiosaka, Japan [email protected]

Abstract

This paper is a companion to the forthcoming paper [19] and exhibits various manifestations of the modular relation, equivalent to the functional equation. We shall give a somewhat new proof of the functinal equation for the Hurwitz-Lerch Dirichlet L-functions in $1, elucidation of Chan's result relating the functional equation to the q-series (or vice versa) in §2, while $3 and 54 are devoted to elucidate the location of the partial fraction expansion of the coth (cot, respectively) in the modular relation framework.

Keywords: Modular relation, functional equation, Ramanujan identities, Ramanujan formula for zeta values 2000 Mathematics Subject Classification: Primary 11M35; Secondly llM06, 33B20

Dedicated to Professor Yasuo Morita on his sixtieth birthday, with great respect T h e first, second and fourth authors are supported by Grant-in-Aid for Scientific Research No. 14540051,14540021 and 14005245 respectively.

103 Number Theory: Tradition and Modernization, pp. 103-118 W. Zhang and Y.Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

104

S. Kanemitsu, Y . Tanigawa, H. Tsukada and M. Yoshimoto

In this paper we are concerned with some remote-looking manifestations of the modular relation, which is equivalent to the functional equation for the corresponding zeta-functions (here we confine ourselves to the Hecke type functional equation) ([3, 13, 181). In $1 we shall give a somewhat new proof of the functional equation for the Hurwitz-Lerch type Dirichlet L-function by looking at the global meromorphic function in two different regions where the Dirichlet series are absolutely convergent. Then we go on to $2 t o elucidate Chan's method of proof of the equivalence of two q-series identities of Ramanujan as the modular relation. Chan's method depends directly on Hecke's lemma and ours is just transforming his material in the upper half-plane into the right half-plane (or simply the positive real axis). In the upper half-plane, the complex exponential function arises naturally in the Fourier expansion (of automorphic function) and one is restricted to this; on the other hand, in the right half-plane, one has more freedom of choice of weights (as will be developed in [12]). In 53 we use an equivalent form of the modular relation for the Riemann zeta-function as given by Koshlyakov [22] and elucidate the results of Bradley [4] and in particular point out that one always tacitly or explicitly uses the functional equation. Notation. s = a + it, a, t E IR - the complex variable, - the incomplete gamma function of the second F(S,Z) = SpO kind, r ( s ) = r ( s , 0) = ~ U~' a >e0 - the ~ gamma & function, ?(s, Z ) = r ( s ) - r ( s , 2) - the incomplete gamma function of the first kind, (a > 1, 0 < a 5 1) - the Hurwitz zeta-function, C b , a, = C",==o

Som

-.

&-, (a > 1)

( ( s ) = ( ( s , l ) = Crz1 00

$(w, s, a ) =

m, (a > e2~zwn

the Riemann zeta-function, 1, 0 < a 5 1 , O < w 1) - the

-

<

Hurwitz-Lerch zeta-function, e2irzwn l,(a) = e - 2 T z w @ (S,~1) , = Cr==l T , (u > 1)- the polylogarithm function, x - Dirichlet character mod rn, x(n) L ( s , x ) = C:=lF, (a > 1) - the associated Dirichlet L-function, T(X) = ~ ~ E ( Z / ~ xZ () "X) ~ - the normalized Gauss' sum, where ( Z l m Z ) signifies the group of reduced residue classes mod m.

2T2z

S o m e aspects of the modular relation

1.

The functional equation for the Hurwitz-Lerch Dirichlet L- functions

Here we use the following additional notation. - the conventional trivial Dirichlet character to the modulus 1, the associate Dirichlet L-functions L(s, x;) being the Riemann zeta-function: L(s, xb) = TZ(X)= C n t ( ~ / m ~(n)1e2rri%z- the generalized Gauss' sum, with r l ( x ) = ~ ( x ) . We are now in a position to state

xT,

Theorem 1. For 0 function G x b , s,a) =

x

nEZ

<

a

< I,

x(-n)e(-2.iri(n

0

<

w

< 1,

the (global) meromorphic

27ri nf" a m

+w))~-s

1 - s , -2ni-

+

m

w,

(1.1)

has the representations: (i) in a > 1

and (ii) in a

<0

implying the functional equation

(1.4)

If in particular,

x is a primitive

character mod m and m

> 1, then

106

S.Kanemitsu, Y . Tanigawa, H. Tsukada and M. Yoshimoto 00

X(n)e-2~iEa

C ( n+ w ) l - ~

eY('-s)X(-l)

co

+

e - y ( ~ - s ) x ( n )e27riE a

n=O

Corollary 1. For m

> 1 and x primitive

n= 1

(n- w ) ~ - s

mod m,

Also

implying

which i n t u r n implies

Proof of Theorem 1. (i) For a > 1, we substitute the defining integral for I' ( 1 - s , - 2 7 r i e a ) and making the change of variable z = at - a to get

where we inclined the integration path by ,; which is justified by Cauchy's theorem. If we invoke the pulse train (cf. [27, p.441) and its Fourier series, we may rewrite (1.10) as

(1.11) where 6 ( z ) signifies the Dirac delta-function. Then each term of the $ ~ ( and k the one sum corresponding to k E N gives ~ k ( ~ ) e ~ " ~ a)-'

+

107

Some aspects of the modular relation

with k = 0 is to be halved, whence we conclude that the right-side of (1.11)coincides with that of (1.2). (ii) For a < 0, we complete the incomplete gamma function by expressing it as F ( l - s ) - y ( l - s , z ) to get nfu'

X(-n)e-2n~~a

G,(w, s , a ) = r(1- s ) n€Z

(-27ri(n

+ w))

Making the same change of variable as above, we see that the second term on the right of (1.12) becomes

to which we apply the pulse train again. But this time there is only one non-zero term corresponding to n = 0, which must be halved, giving

Hence, separating the first sum on the right of (1.12) into two, we conclude that the right side of (1.12) is equal to that of (1.3). To prove (1.5) it is enough t o note that x is a primitive character mod 0 m, m > 1, then r k ( x )= r ( x ) ~ ( kand ) ~ ( 0 =) 0. '

Proof of Corollary 1. (1.7) is the special case of (1.4) with m = 1, (1.6) is obtained from (1.4) by letting a -+ 0+, w -+ 0+, and (1.8) follows by 0 letting w -+ Of in (1.4). Remark 1. (i) For the history of the proof of (1.7) (and (1.8)) we refer t o [Ill, 54,the book [23], Oberhettinger [26] and Weil [32]. (1.7) is sometimes referred t o as the Lipschitz tranformation formula (Grosswald [8, Chap. 8, Theorem 2, p.951, Rademacher [28, (37.1) p.771). Another proof of (1.7) was given by Knopp and Robins [21], which is in the spirit of Eisenstein (cf. Weil [32]). (ii) In Erddyi [7],p.29, there occurs an expansion ( [ 7 ,(8)])of 4(w, s , a ) into a power series with Hurwitz zeta-value coefficients, whose generalization was obtained by Johnson [9]. Although they deduced it from the functional equation (1.7) and then, after applying the binomial expansion, the functional equation (1.8), we may deduce [7, (8)] from [7, (7))

S. Kanemitsu, Y . Tanigawa, H. Tsukada and M . Yoshimoto

108

p.261 and the functional equation [lo, (2.10*)]and the Taylor expansion [ l o ) 12.7*)]. (iii) A contour integral representation for $(w, s , a) was given in Morita [24] and Naito [25]. Our global gamma series may be thought of as a concrete form of their contour integral representation dating back t o Riemann.

Ramanujan's identities B la H. H. Chan

2.

Let ~ ( zdenote ) the Dedekind eta-function

where q = e2Tizwith Im z part of q ( z ):

> 0 and let f ( - q )

denote the infinite product

00

n= 1

Raghavan [29]stated his guess about a possible relationship between two remarkable identities of Ramanujan

and

(2)

signifies the Legendre symbol mod 5. where Chan [5] proves that (2.3) and (2.4) are equivalent under Hecke's theory. He uses the transformation formula for f ( - q )

which is a consequence of the most famous transformation formula for the Dedekind eta-function

Using ( 2 . 5 ) , Chan transforms the right-side of (2.3) in the form

Some aspects of the modular relation

109

which is 1 - times the right-side of (2.4), whence he proceeds to prove 5 2 &2 the identity

If we turn the complex plane by

clockwise by writing r = -iz, then from the upper half-plane (Imz > 0) we move into the right half-plane (Re7 > O), and regarding (2.6) as a modular relation, we find that it is a special case of Theorem 2 [14]. But we recover this as an illustrating example. For R e r > 0 we introduce the functions

and

where x may be any Dirichlet character mod m, but we restrict to the case ~ ( n =) ('). Further, as in Chan [5] or more generally as in [14: 151 we form the Dirichlet series

and

With A =

9,CA(X)

Now we not'e that

=

-r(x), they satisfy the functional equation

S. Kanemitsu, Y . Tanigawa, H. Tsukada and M. Yoshimoto

and that

whence we see that g and h are those two Lambert series appearing in the modular relation ([14])

where PA(r) is the residual function given as the sum of the residues

Now we have

which may be evaluated as Hence, (2.15)reads

-

i.

Puttinn -L = iz,we may rewrite (2.16)as &T

which is (2.7).

111

S o m e aspects of the modular relation

Remark 2. (i) Needless t o say, the procedure applies to other moduli, especially the modulus 7 studied by Chan. We will come to the study on this subject elsewhere. (ii) We remark in passing that Weil's argument [31] has been interpreted as the special case n = 0 of Ramanujan's most famous formula for the zeta values C(2n 1) in [18] (cf. (2.14) above); Ramanujan missed this case because he confined himself to the Lambert series.

+

3.

Bradley's results A la N.S. Koshlyakov

Recently, Bradley [4] adopted the partial fraction expansion of the hyperbolic cotangent function to obtain a generalization of Ramanujan's formula for C(2n I ) , already referred to in 52, to the case of Dirichlet series with periodic coefficients. Let g ( n ) be a periodic function with period m and let L ( s , g ) = be absolutely convergent for o > o;, > 1. Then a typical case of Bradley's results is the following: for x > 0, q E R? and g odd (i.e. g(-1) = -1)

+

C;==, 9

In the simplest case of the odd character

xq

mod 4, (3.1) reads

where En is the n-th Euler number defined by

1 coth t

00

- = C zEn. n=O

Bradley claims that he deduces (3.1) by using the partial fraction expansion of coth and that his proof does not appeal t o the functional equation. We shall show that he uses the functional equation as one form of the modular relation. This will defend his method against a possible claim that he appeals t o a stronger result than the functional equation (cf. [30], in this regard, where the functional equation is deduced from the partial fraction expansion, but not conversely; [19] for the converse).

112

S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto

We recall from [18] that the functional equation (cf. (2.13))

is equivalent to the modular relation (cf. (2.14))

and P ( y ) is the residual function given as the sum of residues, and to the K-Bessel expansion (cf. [18])

x 00

A-SF(s)p(s, a ) = 2 a y

+

x

T--

S

bnpn

( 2 A m )

n= 1

~ e (I?s

( r - s)

+ W)~(W)Z'-'-~)

,

(3.5)

, is the perturbed Dirichlet series Cr=l where a > 0 and ~ ( sa) With slight change of notation we may rewrite (3.5) as

where the integrand on the right side is the same as the residual function. in the By applying the operator (- ;&)' and choosing v - p = resulting formula, we deduced in [12]

-;

and that by applying the operator again we successfully deduced Lemma 6 [6]. Therefore, everything boils down to the formula

S o m e aspects of the modular relation

113

Details will appear e.g. in [ll]. We now specialize (3.7) to the case of the Riemann zeta-function:

to obtain

which can be proved to be equivalent to the functional equation of the Riemann zeta-function. (3.8) is the partial fraction expansion for Since c o t h n z = a ( x ) the coth stated by Koshlyakov [22], as one of the forms of the modular relation, who considers also quadratic fields. For a > ag,let

+ i,

Then Bradley applies (3.8) (here the functional equation is used) to deduce that

Then he applies the recurrence

t o reduce (3.9) to Tg(-1, y) or Tg(O,y) according to t'he parity of m and g. But in order to deduce Bradley's results it suffices t o evaluate Tg(m,y) for m E N,which we can do by using the partial fraction expansion

114

S. Kanemitsu, Y . Tanigawa, H. Tsukada and M. Yoshimoto

Using (3.11), we arrive a t ( m = 2q - 1, g odd)

Putting y = nx in (3.12) and summing over n = 1 , 2 , . . ., we obtain

It remains to evaluate C k E 9(k) z which can be done using another result of Koshlyakov (Re z 2 0)

Indeed.

which is

m- 1

m

g(k) coth k=O

T(Y

+ ik) m

Substituting this in (3.13) completes the proof of (3.1).

Remark 3. (i) Bradley's other expressions for the right-side of (3.1) follow by similar argument as in Katsurada [20] and [16]. The special case g = x is already obtained in [17] which is a culmination of the thereto existing results. (ii) For a > 1,

115

Som,e aspects of the modular relation

k)

onto those of L ( s , g ) . so that we may transfer the results on C (s, One may wonder why one can deduce the results on L ( s , g) built on the Hurwitz zeta-function, from functional equation of the Riemann zetafunction. The reason is elucidated in 1121 to the effect that the functional equations for the Riemann zeta-function and the Hurwitz zeta-function are equivalent.

4.

The functional equation for q5 (w , s , a )

In this section we shall use the partial fraction expansion (that of the cot x, slightly more general than (3.8)) of the function

to deduce the functional equation (1.7). Although in our general framework [12] the following procedure may be superfluous, we find it instructive to give the proof, which resembles that of the functional equation for the Riemann zeta-function [30] but is much closer to our standpoint, suggestive of the modular relation. For 0 < w < 1, we contend that

That (4.1) is true up to a constant, say c,is immediate. To determine c, we let x i 2niw in the equality

The left-side is

while the right-side tends to

116

S. Kanemztsu, Y . Tanigawa, H. Tsukada and M . Yoshimoto

Hence we conclude the equality

whence c = 0 and (4.1) follows. Now suppose 0 < a < 1 and substitute (4.1) for the integral in the Mellin transform formula

and integrate term-by-term to obtain

r ( s ) 0 ( w ,s , 4 =

C

e-2wi(w+n)

zs-l

CX)

2-2T'i(W+n)

n EZ

dx.

(4.4)

We now appeal to the well-known formula

dz =

T '

7l

I ~ I S - ~ ~ - ~ Z ( S - ~) S ~ ~ ( ~ )

sin ~s where sgn(w) = 1 for w right of (4.4):

>0

and -1 for w

'

< 0, to the integral on the

for 0 < w < 1, 0 < a < 1 and 0 < a < 1. Separating the sum into two parts n 1 1 and n < 0 (in which case we may write n 1- w for In wI) and appealing to the reciprocity relation for the gamma funct'ion, we conclude (1.7).

+

+

Acknowledgments T h e substantial part of the paper was completed while the authors save for the third were staying in China-Japan Number Theory Institute, North-West University, Xi'an at the end of August, 2004, and §§3-4 arose from the discussion between the first author and Professor H. H. Chan, NUS in February, 2005, who also kindly supplied the reference [21]. The authors would like t o thank these institutes and colleagues for wonderful research environment and fruitful discussions.

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References [l] R . Balasubramanian, S. Kanemitsu and H. Tsukada, Contributions to the theory

of Lerch zeta-functions, to appear. [2] B. C. Berndt, Ramanujan's notebooks. Part 11, Springer-Verlag, New YorkBerlin, 1989. [3] S. Bochner, S o m e properties of modular relations, Ann. of Math. (2) 53 (1951), 332-363. [4] D . Bradley, Series acceleration formulas for Dirichlet series with periodic coefficients, preprint. [5] H. H. Chan, O n the equivalence of Ramanujan's partition identities and a connection with the Rogers-Ramanujan continued fraction, J . Math. Anal. Appl. 1 9 8 (1996), 111-120. [6] K. Chandrasekharan and Raghavan Narasimhan, Hecke's functional equation and arithmetical identities, Ann. of Math. (2) 74 (1961), 1-23. [7] A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of integral transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-TorontoLondon, 1954. [8] E. Grosswald, Representations of integers as sums of two squares, Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1985. [9] B. R. Johnson, Generalized Lerch zeta function, Pacific J . Math. 53 (1974), 191-193. [lo] S. Kanemitsu, M. Katsurada and M. Yoshimoto, O n the Hurwitz-Lerch zetafunction, Aequationes Math. 59 (2000), 1-19.

[ll] S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto, Contributions to the theory of the Hurwitz zeta-function, (submitted for publication).

[12] S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto, Contibutions t o the theory of zeta-functions:The modular relation supremacy, in preparation. [13] S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, O n rapidly convergent series for the Riemann zeta-values via the modular relation, Abh. Math. Sem. Univ. Hamburg 7 2 (2002), 187-206. [14] S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, O n rapidly convergent series for Dirichlet L-function values via the modular relation, Number Theory and Discrete Mathematics (ed, by A. K. Agarwal, B. C. Berndt et al.), Hindustan Book Agency, 2002, pp. 113-133. [15] S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, O n zeta- and L-function values at special rational arguments via the modular relation, Proc. Int. Conf. SSFA, Vol.1 2001, pp.21-42 1161 S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, O n multi-Hurwitz zeta-function values at rational arguments, Acta Arith. 1 0 7 (2003), 45-67. [17] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto, O n the values of the Riemann zetafunction at rational arguments, Hardy-Ramanujan J . 24 (2001), 10-18. [18] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto, Ramanujan's formula and modular forms, Number-theoretic methods - Future trends, Proceedings of a conference held in Iizuka (ed. by S. Kanemitsu and C. Jia) 2002, pp.159-212.

118

S. Kanemitsu, Y . Tanigawa, H. Tsukada and M. Yoshimoto

[19] S. Kanemitsu, Y. Tanigawa and H. Tsukada, Some examples of a variant of the modular relation, preprint. 1201

M .Katsurada, On an asymptotic formula of Ramanujan for a certain theta-type

series, Acta Arith. 97 (2001), 157-172. [21] hl. Knopp and S. Robins, Easy proofs of Riemann's functional equation for ( ( s ) and of Lipschitz summation, Proc. AMS, 129, No. 7 (2001), 1915-1922. 1221 N. Koshlyakov, Investigation of some questions of analytic theory of the rational and quadratic fields, 1-111, Izv. Akad. Nauk SSSR, Ser. Mat. 18 (1954), 113-144, 213-260, 307-326; Errata 19 (1955), 271 (in Russian). [23] A. LaurinEikas and R. GarunkStis, The Lerch zeta-functzon, Kluwer Academic Publ., Dordrecht-Boston-London 2002. [24] Y. Morita, On the Hurwitz-Lerch L-functions, J . Fac. Sci. Univ. Tokyo, Sect IA Math. 24 (1977), 29-43. [25] H. Naito, The p-adic Hurwitz L-functions, TGhoku Math. J . 34 (1982), 553-558. [26] F. Oberhettinger, Note on the Lerch zeta function, Pacific J . Math. 6 (1956), 117-120. 1271 A. Papoulis, The Fourier integral and its applications, McGraw-Hill, 1962. [28] H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag, BerlinHeidelberg-New York, 1973. [29] S. Raghavan, On certain identities due to Ramanujan, Quart. J . Math. (Oxford) (2) 37 (1986), 221-229. [30] E . C . Titchmarsh, The theory of the Riemann zeta-function, (Edited and with a preface by D.R. Heath-Brown), The Clarendon Press Oxford University Press, New York, 1986.

[31] A. Weil, Sur une formule classique, J . Math. Soc. Japan 20 (1968), 400-402 = Scientific works. Collected Papers, Vol. I11 (1964-1978), Springer-Verlag, New York-Heidelberg, 1979, pp. 198-200. [32] A. Weil, On Eisenstein's copy of the Disquisitiones, Advanced Studies in Pure Mathematics 17, 1989 Algebraic Number Theory in honor of K . Iwasawa, pp. 463-469. -

ZEROS OF AUTOMORPHIC L-FUNCTIONS AND NONCYCLIC BASE CHANGE Jianya ~ i u 'and Yangbo ye2 ' D e p a r t m e n t of Mathematics, Shandong University, Jinan, Shandong 2501 00, China. jyIiu@sdu,edu.cn

~ e ~ a r t m e of n t Mathematics, T h e University of Iowa, Iowa City, Iowa 52242-1419, U S A .

Abstract

Let n be an automorphic irreducible cuspidal representation of GL,,, over a Galois (not necessarily cyclic) extension E of Q of degree !. We compute the n-level correlation of normalized nontrivial zeros of L(s, T). Assuming that T is invariant under the action of the Galois group Gal(E/Q), we prove that it is equal to the n-level correlation of normalized nontrivial zeros of a product of !distinct L-functions L ( s , n l ) . . . L(s, i7t) attached to cuspidal representations T I , . . . , .ire of GL, over Q . This is done unconditionally for nz = 1 , 2 and for m = 3 , 4 with the degree !having no prime factor ( m 2 1)/2. In other cases, the computation is made under a conjecture of bounds toward the Ramanujan conjecture over E, and a conjecture on convergence of certain series over prime powers (Hypothesis H over E and Q ) . The results provide an evidence that n should be (noncyclic) base change of e distinct cuspidal representations T I , . . . ,.ire of GL,(Qa), if it is invariant under the Galois action. A technique used in this article is a version of Selberg orthogonality for automorphic L-functions (Lemma 6.2 and Theorem 6.4), which is proved unconditionally, without assuming T and T I , . . . , .ire being self-contragredient.

<

+

K e y w o r d s : automorphic L-function, nontrivial zero, n-level correlation, base change 2000 M a t h e m a t i c s S u b j e c t Classification: l l F 7 0 , l l F 6 6 , 115141

Supported by China NNSF Grant Number 10125101. Project sponsored by t h e USA NSA under Grant Number hIDA904-03-1-0066. T h e United States Government is authorized t o reproduce and distribute reprints notwithstanding any copyright notation herein.

N u m b e r Theory: Tradition and Modernization, p p 11 9-1 52 W. Zhang and Y . Tanigawa, eds. 0 2 0 0 6 Springer Science + Business hfedia, Inc.

120

1.

J. Liu and Y.Ye

Introduction.

According t o Langlands' functoriality conjecture, the L-function L(s, n ) attached to an automorphic irreducible cuspidal representation T of GL,, over a number field E should equal a product of L(s, n j ) for certain cuspidal representations -irj of GLm3 over Q . Arthur and Clozel [ArtClo] proved that this is indeed the case when E is a cyclic Galois extension of Q of degree e and ;r is stable under Gal(E/Q). In fact in this case, n- is the base change of exactly !nonequivalent cuspidal representations ;rQ, ;rM @ qE/fJ, . . ., ITQ @ of GL,,(Q*), where 7jElfJ is the nontrivial character of Q:/QX attached to the field extension E / Q according t o the class field theory. Consequently (cf. [Bor] and [Lang])

~&>b

where the L-functions on the right side are distinct. When E is noncyclic over Q , factorization of L(s, n ) into a product of L-functions of GLmj over Q is unknown. Recently, the authors [LiuYe2] proved that for any L-function such a factorization, if exists, must be unique. In particular, we proved that the L-function L(s, nQ), attached t o an automorphic irreducible cuspidal representation TQ of GLm(Qa), cannot be factored further as a product of automorphic L-functions for GLm3(Qa). In other words, L ( s , ;rQ) is primitive in the sense of [Sell (see [Murl] and [Mur2]). On the other hand, Rudnick and Sarnak [RudSar] proved that the ) n-level correlation of normalized nontrivial zeros of this L(s, T ~ Jfollows a GUE (Gaussian Unitary Ensemble) model, for a class of test functions whose Fourier transforms have restricted support. Here the zeros are normalized according to their density. For distinct L-functions L ( s , q), with n j being cuspidal representations of GLm(QA),j = 1, . . . , e, the authors [LiuYel] proved that the n-level correlation of normalized nontrivial zeros of the product L(s, n l ) . . . L ( s , ne) follows a superposition distribution of individual GUEs from L(s, q )and products of GUEs of lower ranks, under Selberg's orthogonality conjecture and under a conjecture on the convergence of a sum over prime powers (Hypothesis H: see 52) for m 2 5 . Recently, Liu, Wang, and Ye [LiuWangYe] proved this Selberg orthogonality conjecture for automorphic L-functions. There4 fore, this n-level correlation is now known unconditionally for m and under the Hypothesis H for m 5 . Note that here the test functions also have a restricted support for their Fourier transforms. In this paper, we will use this GUE correlation to study the factorization of L(s, n-). We will show that the nontrivial zeros of L(s, n ) behave in the same way as the nontrivial zeros of L(s, n l ) . . . L(s, -ire).

>

<

Zeros of automorphic L-functions and noncyclic base chunge

121

Theorem 1.1. Let E be a Galois extension of Q of degree k', and n a n automorphic irreducible cuspidal representation of GL,(EA) with unitary central character. A s s u m e (i) Hypothesis H both over E and over Q w h e n m 2 5 and (ii) Conjecture 2.1 w h e n m 3. Suppose that n r n u for all 0 E Gal(E/Q). T h e n the n-level correlation of normalized nontrivial zeros of L ( s , n ) is equal t o the n-level correlation of normalized nontrivial zeros of a product o f t distinct automorphic L-functions attached t o cuspidal representations of GL,(QA).

>

Note that Theorem 1.1is an unconditional result for m = 1 and 2. For m = 3 and 4, it is also unconditional when (30, k') = 1 and (210, k') = 1, respectively. See $2 for details. Theorem 1.1 provides an evidence that L(s, n ) should factor into a product of L-functions k' distinct automorphic L-functions attached t o cuspidal representations of GLm(Qa), and suggests that the base change and factorization of L-functions such as (1.1) should hold for noncyclic extension fields as well. We remark that our results contain much less information than what was achieved in [ArtClo], as we cannot see individual representations T Q , . . . , naq @ as in (1.1). Since we are seeking less information, we can determine through zero distributions the base change structure of n beyond the scope of [ArtClo]. The reason behind this is indeed the universality of the n-level correlation of zeros discussed in [RudSar]: The n-level correlation of normalized nontrivial zeros of L ( s , naq) is independent of naq, as long as it is a cuspidal representation of GLm(Qa). Our computation of n-level correlation will be carried our for test functions f whose Fourier transforms a(() as in (2.8) below have rejJjl < 2/m. This restriction of support is good stricted support Cl<j
O?$h

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J. Liu and Y. Ye

alence relationships among local representations T, at places v lying above completely split primes. This information will therefore be described by orthogonality, and eventually lead t o the n-level correlation of normalized nontrivial zeros.

2.

Not at ion and assumption. Let E be a Galois extension over Q of degree t. Let EA =

fl; E, be its adele ring, where v runs over all places of E, and denotes a E,, where restricted product. For any prime p, we have E @Q Qp = v with vlp are places of E lying above p. Since E is Galois over Q,all E, with v J p are isomorphic. Denote by tp the degree, by ep = ordvz(p) the order of ramification, and by fp the modular degree of E, over Q for vlp. Then tp = epfp and qv = p f ~is the module of Ev. On the other hand, E @Q R is either = R,or = $,,,C. Let .rr be an automorphic irreducible cuspidal representation of GL,(EA) with unitary central character. The (finite part) L-function attached to T is given by the Euler product L ( s , T)= fl,,, L ( s , T,) for u > 1. The definition of the local factor L ( s , T,) is given by

n'

evlp

< <

where a,(j, v), 1 j m, are complex numbers given by the Langlands L(s, T,) the correspondence. For any prime p, denote by Lp(s, T)= product of local factors above p. Then

nvlp

Similarly, the product of Archimedean local factors is given by L,(s, = L(s, T,), where

n,,,

T)

if v is real, and

if v is complex. Here p,(j, v) are again complex numbers given by the ~ ) ,rc(s) = ( 2 7 r P S r ( s ) . Langlands correspondence, Fps (s) = T - ~ / ~ F ( S /and

Zeros of automorphic L-functions and noncyclic base change

123

We will need a bound for cr,(j, v ) :

This bound holds for any T,, either ramified or unramified. It was first observed by Serre [Ser] and appeared in published form in [LuoRudSar]. A complete proof is given in [RudSar] for the case of E = Q,using an argument of Landau [Land]. When T , is unramified, the generalized Ramanujan conjecture claims that l%(j,v)l = 1. (2.2) The best known bounds toward this Ramanujan conjecture over an arbitrary number field are

for m = 2 ([KimShah]),and

for general m ([LuoRudSar]),where vlp. We will not assume (2.2), but assume a bound 9, for it for any p which is unramified and does not split completely in E as stated in the following Conjecture 2.1. For a n y p which i s unramified and does n o t split completely in E , w e have, for a n y vlp,

where

Qp =

112 - 1/(2fp) - E for a small

E

> 0.

We remark that e, = 1 and hence fp = tpwhen p is unramified. Since p does not split completely in E, we know that f, 2 2. Consequently Conjecture 2.1 is known for m = 2, according to (2.3). Conjecture 2.1 is trivial for m = 1, Recall that f,lt. Thus conjecture 2.1 is known when all prime factors of! are > ( m 2 1)/2. For m = 3 this means that any pit is 1 7 , while for m = 4, Conjecture 2.1 is true when any pit is 2 11. We also need the Hypothesis H ([RudSar]) generalized t o E .

+

Hypothesis H. Let i7 be a n automorphic irreducible cuspidal representation of GLm(EA) with unitary central character. T h e n for a n y fixed k 2 2 log2 P 2 "!'i~,l < m. (2.5)

CTE V I P 1 P

l<j<m

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J . Liu and Y.Ye

We note that Hypothesis H is an easy consequence of the generalized Ramanujan conjecture (2.2). Since there are only finitely many p which are not unramified in E, the sum in (2.5) may be taken over all unramified p. As we have assumed Conjecture 2.1, we know that

C

p unramified, not split completely

2

vlp

l<j<m

o unramified. not'split completely

Consequently under Conjecture 2.1, Hypothesis H claims that for any fixed k 2

>

p splits completely

vlp ' l < j < m

As in the case of E = Q,Hypothesis H is t'rivial for m = 1. For m = 2, it can be proved using the bound in (2.3). In fact, (2.3) implies that laE(j, Vi) ( 5 pkfplg and hence

>

2. In Appendix, we will prove Hypothesis H for m = 3. For when k m = 4, Hypothesis H is a consequence of [Kiml], Proposition 6.2, as pointed out by [KimSar] and proved in [Kim2]. Let gj be a compactly supported smooth function on R. Then its Fourier transform (2.6) is entire and rapidly decreasing on R. We denote h = ( h l , . . . , h,) and define

Given

E

C;(IRn) (c meaning compactly supported) we define

where x = ( X I , . . . , x,), = (I, . . . , J,), x . ( = xi(1 the Dirac mass a t zero, and e(t) = e2i7it.

+ . . . + x,(,,

6(t) is

Zeros of automorphic L-functions and noncyclic base change

125

The n-level correlation of normalized nontrivial zeros of the L-function L ( s , n ) is given by

where the sum is taken over distinct indices i l , . . . , i n , of nontrivial zeros 1/2+iyi,, v = 1 , . . . , n, of L ( s ,n).Without assuming the Riemann log T provides Hypothesis, yi, are complex numbers. Here the factor the normalization for zeros pi,. In the following sections, we will first compute the same sum as in (2.9)but taken over all indices of nontrivial zeros. We will denote this latter sum by Pi, =

By an argument similar t'o that in [RudSar] and [LiuYel],we may deduce (2.9) from (2.10).

3.

The main theorems.

Theorem 3.1. Let E be a Galois extension of @ of degree i?, and T an automorphic irreducible cuspidal representation of GLm(Em)with unitary central character. Let a, 1 a i?, be the number of elements a E Gal(E/Q) with x E n o . Assume m 4 or Hypothesis H over E for m 5. Also assume Conjecture 2.1 when m 2 3 and that there is pi! such that p 5 (m2 1)/2. Then

>

< < <

+

where the sum on the left side is taken over all indices of zeros pj = 1 / 2 + i y j , j = 1, . . . , n, of L ( s , n ) . Here g j E C,00(R), Q E C ; ( R n ) , f are given in is supported in C l < j < nItj I < 2 / m , while hj, ~ ( h )and (2.6) through (2.8),-r~spectivel~j. When a = t, i.e., when 7r is invariant under the action of Gal(E/@), we apply the combinatorial sieving argument in [LiuYel], 59, directly

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J. Liu and Y.Ye

to (3.1) to deduce the following corollary. We need the notation. A set partition H of fi = (1,.. . , n) is a decomposition of & into disjoint subsets = [HI,.. . , H,,], where v = v ( H ) is the number of subsets in H. For a given set partition H, define

where K ( x ) = (sin n x ) / ( ~ x if) x

# 0, and K ( x ) = 1 if x = 0.

Corollary 3.2. W i t h notation and assumption as i n T h e o r e m 3.1, we assume that 7r i s invariant u n d e r the action of Gal(E/Q). T h e n the n-level correlation of normalized nontrivial zeros of L ( s , n ) is given by

where the s u m o n the left side is t a k e n over distinct indices of zeros Pij = 112 i y i j , j = 1 , .. . , n, of L(s, T ) .

+

The following theorem and its corollary were first proved in [LiuYel] on the assumption of Selberg's orthogonality conjecture, which was removed in [LiuWangYe]. Theorem 3.3. Let T I , . . . , T! be automorphic irreducible cuspidal representations of GL,(Qa) with unitary central character, such that .iri .irj for a n y i # j . A s s u m e m _< 4 o r Hypothesis H over Q for m 1 5 . T h e n

where the s u m o n the left side is taken over all indices of zeros pj = 1/2+ iyjlj = 1, . . . , n, of L(s, 7 r l ) . . . L ( s , n!). Here g j E C,OO(R),
Zeros of automorphic L-functions and noncyclic base change

127

is supported in Cl<j
>

Corollary 3.4. Assume m 5 4 or Hypothesis H over Q for m 5. With the same notation as in Theorem 3.3, the n-level correlation of normalized nontrivial zeros of L ( s , n l ) . . . L ( s ,ne) is given by

i l , ...,in distinct

where the sum on the left side is taken over distinct indices of zeros = 112 iyij, j = 1 , .. . , n , of L(s, n l ) . . . L(s, ire).

Pi,

+

Comparing the right side of (3.2) and (3.4), we conclude that the n-level correlation of normalized nontrivial zeros of L ( s , n ) equals the nlevel correlation of normalized nontrivial zeros of a product of t distinct L-functions attached to cuspidal representations T I , . . . , ne of GL, over Q , when ir is invariant under the Galois action. This is the proof of Theorem 1.1. In the following sections we will prove Theorem 3.1.

Again let E be a Galois extension over Q of degree t. Denote @ ( s ,ir) = L, ( s , n)L(s, ir), for 0 > 1. Then by a classical result of [GodJac], @ ( s ,n ) extends to an entire function with the exception of [(s), which has a simple pole at s = 1, @(s,n ) also has a functional equation @ ( s , ir) = E(S,ir)@(l - S, z ) ,

where the automorphic irreducible cuspidal representation i.r is contra~ . &, > 0 is the conductor of gredient to n , and E ( S , ir) = r ( n ) Q ~ Here n ([JacPSShall]), r ( n ) E C X ,Qi, = Q,, and T(T)T(%)= Q,. Let

a,(pk) = 0 if fp Ij k, and c,(n) = A(n)a, (n), where the von Mangoldt function A(n) is defined by A(n) = logp if n = pk and zero otherwise.

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J. Liu and Y. Ye

Then a5(pk) = 5, (pk). Then for a

> 1, we have

By the bound in (2.1) we have

for any n,, ramified or unramified. If bounds based on (2.3) and (2.4):

T,

is unramified, we have sharper

for m = 2, and

>

for m 3. We will need an explicit formula for the L-functions of smooth type as in [RudSar]. Let gJ be a compactly supported smooth function on R. Define h3(r) and n(h) as in (2.6) and (2.7). Let p = 112 + iy be a nontrivial zero of the L-function L(s, T ) . By applying the same arguments step by step as in [RudSar], we prove that

where the sum on the left side is taken over nontrivial zeros p = 112 + iy of L ( s , n), and S(T) equals 1 if the L-function is ((s)) and zero otherwise. Here ~ ( t =)log Q,

+C

rf 1 C (2 ( 2 + p,(j, v) + it)

vjoo lsjsrn

= log Q,

+C

c (2(A +

vloo l<j
r (-1 + ( +rc 2

re 2

) - it))

p n ( j ,U )

if E @Q

+ it)

R = @,@.

(4.5)

Zeros of automorphic L-functions and noncyclic base change

129

Denote L = m log T. The explicit formula (4.4) can be rewritten as

where

In the following, we will not consider the term with S(n) on the right side of (4.6), as it is non-zero only for the Riemann zeta function [(s).

Rankin-Selberg L-functions.

5.

For two irreducible automorphic cuspidal representations .ir and i.rf of GL,(EA) and GL,I (E*),respectively, their Rankin-Selberg L-function is given by

for a

> 1, where the local factor is

Denote

J. Liu and Y. Ye

Then for a

> 1,

where

and ~ , ~ a / ( = p 0~ if)

fp

+ k . In particular, when " .ir

TI,

Let STx?/ be the finite set of primes p such that there is vlp with either T, or T ; being ramified. Therefore for any p E ST,, = S, we have

On the other hand, absolute convergence of (5.1) for a with (5.5), implies that

> 1, together

Zeros of automorphic L-functions and noncyclic base change

131

By partial summation, we also have

where c, (n) = A(n)a, (n) and a, (n) is given in (4.1). Recall that when sr' E .j.r @ I det liTo for some TO E R, L ( s , n- x 7i-) has simple poles at s = 1 ir0 and irO.([JacPSShal2] and [MoeWal]). Otherwise L ( s , n x 2) is entire. Note that the Archimedean part of the Rankin-Selberg L-function is

+

We will need a trivial bound Re p,,+(j, k ; v) > -1. Denote by @ ( s ,n x 7i-') = L, ( s ,sr x 5')L(s, n- x 3') the complete RankinSelberg L-function. Then by a classical result of Shahidi ([Shahl], [Shahs], [Shah3], and [Shahill), @(s,sr) has a functional equation

where

6.

E(S,

n ) = T ( T x f1)Q,;,,

. Here

&,,,I

> 0 is the conductor

Orthogonality.

When n and sr' are cuspidal representations of GL, (Qa) and GL,, (Qa), T I , Liu, Wang, and Ye [LiuWangYe] proved the respectively, with sr following Selberg orthogonality

if a t least one of n and sr' is self-contragredient. In [LiuWangYe], (6.1) was proved as a consequence of a stronger prime number theorem with weights for a Rankin-Selberg L-function, and hence a zero-free region of the classical type was required (cf. [Morl], [Mor2], [GelLapSar], and [Sar]). This is the reason why we have to assume that a t least one of sr and n' is self-contragredient in (6.1). In this section, we will use the approach in [LiuYe2] to get a weighted version of (6.1) for cuspidal representations n and d over E, avoiding the use of the zero-free region and the self-contragredient assumption. Then we will apply an argument of Landau [Land] to remove the weight.

J . Liu and Y.Ye

132

Lemma 6.1. Let n and T' be irreducible automorphic cuspidal representations of GL,(EA) and GL,/(EA) with unitary central characters, respectively. T h e n

Proof. The proof closely follows [LiuYe2],and hence we will only give a brief sketch here and point out the difference. Let X ( s ) = minnlo 1s- n / . Denote by C(m, m') the region in the complex plane with the following discs removed:

if v is real, and

if v is complex. Then for s E C(m, m') and all j, k , and v loo,

if v is real, and

'(s

+ ~ , x ' % ' ( j >k ; '1)

2

1 16mm/(

if v is complex. Let P(j, k; v) for j, k in the above range be the fractional part of Re p,,?~ (j,k; v) . In addition we let P(0,O;v) = 0 and p ( m + l , m l 1 ; v ) = 1. Then all P(j, k;v) E [O, 11, and hence there exist P(j1, k1; vl), P(j2, k2; v2) such that P(j2, k2; v2) - P(j1, kl; vl) 2 1/(3mmt!) and there is no p ( j , k; v) lying between P ( j l , kl; vl) and P(j2, k2; v2). It follows that the strip

+

is contained in C(m, m'). Consequently, for all n = 0, -1, - 2 , . strips

. . , the

Zeros of automorphic L-functions and noncyclic base change are subsets of C(m, m'). Differentiating (5.1), we get

(-)LfL

( s , rr x

2) =

C (logn)A(n)a,x%/(n) nS

n>l

for a > 1. B y the same method of proof as in [LiuYe2], 54, using the fact that the Rankin-Selberg L-function is of order one away from its possible poles ([Gelshah]),we have the following estimates: For /TI > 2 there exists T with T 5 T T 1 such that when -2 a52

< +

<

if s is in some strip S, as in (6.4) with n 5 -2, then

d t << ZIT. TO The last two integrals are bounded by << &?(x/t2) compute the first integral on the right side of (6.7), we choose a0 with -2 < 0-0 < -1 such that the line a = a0 is contained in the strip S-:,c C ( m , m f ) . Let T x and choose T with T T T 1 such that (6.5) holds. Denote this T simply by T. Now we consider the contour

< < +

Note that the two possible poles, some trivial zeros, and certain nontrivial zeros of L ( s 1,a x f f ) ,as well as s = 0, -1 are passed by the shift of the contour. The trivial zeros can be determined by the functional equation (5.7): s = -1 - p,,%/ ( j , k ; v). T h e trivial bound Re p,,,/ ( j ,k ; v) > -1 will give us O(1ogx) for the size of the residues from trivial zeros. Integrals along C1 and C3 are bounded by O{x ~ o ~ ~ ( Q , , ~ I T )using /T~)

+

134

J. Liu and Y.Ye

(6.5), while the integral on Cz is bounded by O ( l / x ) using (6.6). The residues at s = 0, -1 will contribute K ( 1 ) K(O)/x. The two possible poles are ir0 and ire - 1 which can only happen when m = m' and n' E n 8 ldetliTO for some TO E EX. If TO # 0, these are double poles and they will contribute O(1ogx) to (6.7). If TO = 0, i.e., if n 2 n', then s = 0 becomes a triple pole. Its residue contributes $ log2x O(1ogx) to (6.7). By the fact of L(s, n x 5')# 0 for a 1 ([Shahl]), the contribution from nontrivial zeros is also O(1ogx). This completes the proof of Lemma 6.1. 0

+

>

+

Lemma 6.2. Let n be an irreducible automorphic cuspidal representation of GLm(EA)with unitary central character. Then

Proof. Note that this is t,he case when n E n', and hence by (5.3) the series on the left side of (6.2) and (6.8) consists of non-negative terms. By the method of proof in [RudSar], p.282, we can remove the weight 1 - n l x from (6.2) when n E n'. 0 Lemma 6.3. Let n be an irreducible automorphic cuspidal representation of GLm(EA)with unitary central character. Then

~ ( l o g n ) A ( n ) a , , B ( n ) << x logz.

(6.9)

n<x

Proof. This is deduced from (6.8) by a standard argument of partial 0 summation. We remark that under the generalized Ramanujan conjecture, an asymptotic formula was proved in [LiuYe3]for cuspidal self-contragredient representation of GL, over Q: ~ ( l o g n ) A ( n ) a T X ~=( nx )l o g x

-x

+O ( x e x p ( - c 6 ) )

n<x

for a positive constant c. The upper bound in (6.9) is nevertheless unconditional and valid for cuspidal representations not necessarily selfcontragredient . Theorem 6.4. Let n and n' be irreducible automorphic cuspidal representations of GL,(Ea) and GLm,(EA)with unitary central characters, respectively, such that n' 4:. Then

Zeros of automorphic L-functions and noncyclic base change

135

Proof. The removal of the weight 1 - n l x from (6.3) is by an argument of Landau [Land] using (6.9). The proof is given in [LiuWangYe]. To make the present paper self-contained, we reproduce the proof in [LiuWangYe] below. Denote c(n) = (log n)A(n)aTX5l( n ) / n , and

Then

lx

~ ( t ) d=t C ( x - n)c(n) = x D ( s ) . n<x

We begin with an observation that

where v = &. By (6.3)) we have vD(x as n'

+ v) << v log x

(6.12)

n. The last term in (6.11) is

say. By (5.4) and Lemma 6.3 we have

<< vlogx,

(6.13)

and

<< v log x.

(6.14)

J . Liu and Y.Ye

Putting (6.12) through (6.14) into (6.11), we get

Now consider the difference

by the same argument as above. The desired result (6.10) now follows 0 from this and (6.15). Under the generalized Ramanujan conjecture, a much precise version of (6.8) and (6.10) was proved in [LiuYeS]for self-contragredient cuspidal representations over Q:

Ex; rr 8 aztfor any t ER,

if n1E rr 8 aim for some = c~

+O { e s p ( - c 6 ) )

if

d

TO €

where c, el, ..., c3 are positive constants.

7.

Orthogonality over primes.

In this section we will rewrite Lemmas 6.2 and 6.4 as orthogonality relation over primes using Conjecture 2.1 and Hypothesis H. First, according to (6.8) we have

Zeros of automorphic L-functions and noncyclic base change

Using the bound for cu,(j, u i ) in (2.1), we have

Therefore Hypothesis H implies

From (7.1) and (7.2) we conclude that

What we will show below is that the outer sum in (7.3) can be taken over all primes p _< z which split completely in E. Theorem 7.1. Let T be an automorphic irreducible cuspidal representation n of G L m ( E A ) with unitary central character. Assume Hypothesis H over E when m 2 5. Assume Conjecture 2.1 when m 2 3 and that there is pl!? such that p 5 ( m 2 1 ) / 2 . Then

+

Proof. We know that there are only finitely many p with its order of ramification ep > 1. Thus we can ignore these primes in (7.3) and consider only unra~nifiedprimes p with fp = tP.If such a prime p does not split completely in E, then fp 2 2. Under Conjecture 2.1, we then have

not- split completely

138

J . Liu and Y.Ye

Consequently (7.4) holds.

0

As in [RudSar], p.300, we can apply partial summation to (7.4) and prove an asymptotic formula for a weighted sum.

<

Lemma 7.2. Let $ ( u ) be a C1 function supported on iul ( 1 - 6 ) / m for some positive 6 . Let n be an irreducible automorphic cuspidal representations of GL,(EA) with unitary central character. Assume Hypothesis 3 and that H over E when m 2 5 . Assume Conjecture 2.1 when m there is pl! such that p (m2 1 ) / 2 . Then we have

<

>

+

P<X

splits completely

where the implied constant depends on m and 6 . Now let us turn to the case of n p n'. By Cauchy's inequality, we may remove terms on prime powers from the left side of ( 6 . 1 0 ) . Using the same proof as for Lemma 7 . 1 , we can further remove terms on those primes which do not split completely in E .

Lemma 7.3. Let n and .irf be irreducible automorphic cuspidal representations of GLm(EA)and G L , / ( E A ) with unitary central characters, T . Assume Hypothesis H over E when m 5 respectively, such that n' or mf 5 . Also assume Conjecture 2.1 when m 3 and that there is pit such that p 5 (m2 1 ) / 2 , or when mf 2 3 and that there is pit such that p 5 (m12 1 ) / 2 . Then

>

+

+

P S ~

>

>

I-

splits completely

<< log z.

(7.5)

Partial summation now can be applied to ( 7 . 5 ) and the following weighted sum can be obtained as in the proof of Proposition 4.5 in [LiuYel], p.436.

<

Lemma 7.4. Let $(v) be a C1 function supported on Iul (1 - S ) / m for some positive 6 . Let T and T' be irreducible automorphic cuspidal

Zeros of autom,orphic L-functions and noncyclic base change

139

representations of GL,(EA) and GL,I ( E A )w i t h unitary central characters, respectively, such that T' y T . T h e n u n d e r the s a m e a s s u m p t i o n as in L e m m a 7.3, we have

C P<X

T$(x)x( log2P

1 % ~

aT(jl,ui))(

vilp l<jl<m

splits completely

~r/(~i,vi)) l < j z <m

where the implied constant depends o n m and 6

8.

The n-level correlation.

Let E C1(IRn) be supported in the region 101 < 21772. Define f (z)as in (2.8). We will compute the left side-of (3.1):

<

where L = m l o g T and the sum over each yJ ( 1 j 5 n) is taken over all non-trivial zeros pj = 112 + i y j of L ( s , T ) . Now we use the Fourier transform and get

To compute the product, we set i, = 0 or k 1 for ,LL = 1 , .. . , n, and use i, to indicate which one of g p ~S,t , S i appears in the term:

C, ( f , h, T ) =Tn

c

-l
J

n g p ~i,=l ( ~ ~ ip=-1 t11 ~s;(c,) ~)n~;

ip=O

Now we use (4.7) and (4.8) to expand S:([,). = c,(n,). We have

Recall that A(n,,)a,(n,,)

140

J. Liu and Y. Ye

and -

m)A ~ , . .(.n~,T) ., 112

for

i i~ --1 -

i,#O

np

+ . . + En)dl.

x @(l)d(El

(8.2)

*

We will follow [RudSar] and [LiuYel] closely to estimate C, ( f ,h, T ) and point out differences between the situation at present and those discussed there. First, in the definition of R, ( r ) and hence of gT(x), These will there are sums of Fk/Fc or possibly more terms of cause no problem, since either over 62 or over R, Re p,(j, V ) > -112 by [RudSar]. By Stirling's formula, we thus have

rk/rw.

Therefore, Lemma 4.1 in [RudSar] is still true here. That is, (1 - e P X l 2 )+ epUxlog T:

if 1x1 << log log T ,

(log T

and

r

Consequently, the integral in (8.2) converges absolutely. We are now going to prove Proposition 4.1 of [RudSar] in our case. That is, we will show that

-

G I , . , . ,in

if

@(I)in

( T ) = Ctl,,..,in ( T )

(2.8) is supported in

+ o(T'-~'~)),

1111+ . . . + /[,I

< (2 - d)/rn, where

Eil,..i,,( T ) = lsn,<
i,#O, i,=l

n~

t,=-1

np

(8.4)

Zeros of automorph,ic L - f u n c t i o n s and noncyclic base change

and

= T n ~T.L<J:
n'-

9p~(~~<1)

To do this, we first point out that Lemmas 4.2 and 4.3 of [RudSar], i.e., Lemmas 7.1 and 7.2 of [LiuYel], hold in the present case.

Lemma 8.1. Ai

( n , T ) = 0 unless In,

<< T for

I,

n,,<<

# 0 and

~ 2 - 6

Lemma 8.2. A i l .i,(n, T ) = 0 unless In, : and I-Ii P--1 n,,= n,.

ni+

~ 2 - 6

<< T

for i,

# 0)

n,

<<

Lemma 4.4 of [RudSar] then holds in our case, since it is based on the bounds for g~ and Lemmas 8.1 and 8.2 above.

Lemma 8.3. If M N

<< T

~ - then ~ ,

Now we can write Cil..,i,( T ) = Q l i , ( T )

+ Cdiag + COB

where

(

x Ai

in ( n , T

)-

...in( n , T

))

142

J. Liu and Y . Ye

and

l
i,=l

n~

ip=-1

n~

by Lemma 8.3. We will prove later that

from which we can deduce

Cdiag << ~

'

~ Denote ~ 1

~

.

Then using (5.6) we can still prove Lemma 4.5 of [RudSar]:

Using the same arguments as in [RudSar], pp.293-294, we get

for any E > 0.Therefore, (8.4) is valid if we prove (8.7) Back to (8.6), we change variables to

{ ~+ ~ ~ ~ ,

if i, = 0,

YP =

i, log n,)

if i,

#0

Zeros of automorphic L-functions and noncyclic base change and get

i l log nl . , , -YnL ' 'TL

il log nl -

in log nn L

, . * * !- inlognn) L

+

0(~-1+6/3

))I

x,

where V is defined by y j = 0 , y , << 1 for i,, # 0 , and / y , << T 6 I 3 for i, = 0. By Stirling's approximation formula and ( 8 . 3 ) ,we get frorn (4.5)that

for r

> 1. Consequently 2n

hl ( r )

h,(r)R,(~r)'dr

=

k k

i

-L 2n

+ O(L"').

and hence

where k = n - r - s is the number of p with i, = 0 . This is Lemma 4.6 of [RudSar]. Back to ( 8 . 6 ) )we now have for r s > 0 that

+

Ail...'..

( n ,T ) =

Ten-'-' Lr+s-l

~ ( h ) i l log n l 2n L '""

-a(-

-

in log n.,

T

From (8.4) and (8.5) we thus obtain

x

a(-

i l log nl

! . . . , - in log nn L

) + O ( T ' - ~ / ~ ) (8.8)

144

J. Liu and Y. Ye

when i l l . . . , in are not all zero, and

If we assume Lemma 8.4 below, then (8.8) can be rewritten as

+

when r s > 0. This is indeed Lemma 4.7 of [RudSar]. Now let us turn to Lemma 3.8 of [RudSar]. From the bound for a,(pk) in (4.2) we get Ic,(pk)) 5 m u % P)Pk(ll2-PI with /3 = 1/(Qm2 1) > 0, which is essentially the same bound as in [RudSar] and [LiuYel]. Consequently for any K > 1/P we have (4.54) of [RudSar]:

+

From (5.5) we deduce that

for fixed k and 6 > 0. By an argument in [RudSar], we can prove that

is bounded for s < K fixed. Together with (8.11), this implies that Lemma 3.8 of [RudSar] is valid:

Zeros of automorphic L-functions and noncyclic base change

145

when r + s 2 3. Now we prove the following lemma which is Lemma 3.9 of [RudSar].

Lemma 8.4. Assume m = 2 or Hypothesis H.If 1 < r

< s , then

Proof. Using the argument in [RudSar] after (4.59)' the Lemma is reduced t o estimation of

by Cauchy's inequality. Using our results in (7.1) from the RankinSelberg L-functions, we conclude that

This proves the Lemma.

0

Lemma 8.4 implies (8.7) and hence (8.4). It also completed the proof of (8.10). Let S,, ...,, be the set of bijective maps from the set { p : i, = 1) onto the set { p : i, = -1) when r = s. Then (8.10) can be further

146

J. Liu and Y.Ye

reduced t o

c. . 21"'Zn

C C L2r p,
~ ( hT)L (T) =27r

for

n i,=l

lcn(")12

PP

where z, = -(log p,)/L and xu(,) = (log p,)/L if i, = 1, and z, = 0 if ill = 0,when r = s > 0.When r # s, Ci i,,(T) = O(T) by Lemma 8.4. Consequently we can go back to (8.1) and (8.9) to write

,...

+

where on the left side, for each j = 1,.. . , n,pj = 112 i y j is taken over all nontrivial zeros of L(s,n). Estimation of (8.12) is thus reduced to determining the asymptotic behavior of

<

c1

function supported in Ivl (1 - 6)/mfor some where 4(v) is a we get positive 6. By the definition of c,(n) in (4.1),

2

m log x

vlp l<j<m

Zeros of automorphic L-functions and noncyclic base change

147

Proof of Theorem 3.1.

9.

In this section we will assume that E is a Galois (not necessarily We are going to compute (8.12) and cyclic) extension of Q of degree l. deduce the n-level correlation of normalized nontrivial zeros of L ( s , T) from (8.12). The estimation (8.12) is based on the asymptotic behavior of (8.13). Note that only those primes p with fp = 1 contribute t o (8.13). Consequently, the outermost sum on the right side of (8.13) is actually taken over (i) primes p x which split completely in E, and (ii) primes p 5 x above which each local field E, is fully ramified (fp = 1). Note that there are only finitely many primes p which satisfy (ii) but not (i). We can thus rewrite (8.13) as

<

I ,c, , , ( + ,

log2

-

logp

PIX

splits completely

Now we distinguish two cases. First let us assume that for any a E Gal(E/Q), T and TO are equivalent. Under this condition, we have T,, E T , ~and

for any 29 and v2 lying above a completely splitting p. Consequently (9.1) becomes

=

e

e PIX

splits completely

C

llog2i p~ ( logp s)~ VIP

lIj<m

2

a n ( j l r v ) J +0(1)

J . Liu and Y . Ye

By Lemma 7.2, we get

This implies the asymptotic formula

Next let us consider the case that n is not stable under the whole Gal(E/Q). Let G, = { a E Gal(E/Q) / nu S 7r) be the subgroup of Gal(E/Q) fixing 7r. Denote by a the number of elements in G,, so that all and 1 5 a < l. Then according to [Rog], for any two places vl and v2 lying above a completely splitting p, 7r,, 2 n,, if and only if E,, is mapped onto E,, by some a E G,. Consequently (9.1) becomes

log2P a€Gal(E/Q) x%ra

P<X

splits completely

1 % ~

Zeros of automorphic L-functions and noncyclic base change

149

The estimation of the first term on the right of (9.4) is t,he same as in the first case, following (9.2) and (9.3):

On the other hand, the estimation of the second term on the right of (9.4) is done by Lemma 7.4; we get O(1og T). Consequently,

17

This proves Theorem 3.1.

Appendix. Proof of Hypothesis H for GL3 over E. Now we prove Hypothesis H over E for m = 3. Over Q,this was proved in [RudSar]. We follow their approach closely. Since the series on the right side of (5.1) converges absolutely for 0 > I , we know that for any S > 0

Taking k = 1, we get

On the other hand, as in [RudSar], p.283, we can have la,(2,v)l = 1 and /cu,(l, v)l = l/la,(3, v)l 1, because the central character of n is

>

J . Liu and Y . Ye

assumed to be unitary. Therefore

Consequently

The first sum on the right side of (A.2) is 5 Y ~ , ( l o g p ) 2 / p k< cc when k 1 2. Applying the bound in (4.3) to / an(j,n)12k-2, we see that the last sum in (A.2) is

z;=,

by ( A . l ) . Therefore Hypothesis H is true for m = 3.

Acknowledgments The second author wishes to acknowledge support from the Obermann Center for Advanced Studies, the University of Iowa.

References [ArtClo]

Por] [GelLapSar] [Gelshah] [GodJac]

J . Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies, no. 120, Princeton Univ. Press, Princeton, 1989. A. Borel, Automorphic L-functions, Proc. Symp. Pure Math. 33 (1979), part 2, 27-61. S.S. Gelbart, E.M. Lapid, and P. Sarnak, A new method for lower bounds of L-functions, C.R. Acad. Sci. Paris, Ser. I 339 (2004), 91-94. S. Gelbart and F.Shahidi, Boundedness of automorphic L-functions in vertical strips, J . Amer. Math. Soc. 14 (2001), 79-107. R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math. 260, Springer-Verlag, Berlin, 1972.

Zeros of automorphic L-functions and noncyclic base change

151

[JacPSShall] H. Jacquet, 1.1. Piatetski-Shapiro, and J . Shalika, Conducteur des re'pre'sentations du groupe line'aire, Math. Ann. 256 (1981), 199-214. [JacPSShal2] H. Jacquet, 1.1. Piatetski-Shapiro, and J . Shalika, Rankin-Selberg convolutions, Amer. J . Math. 1 0 5 (1983), 367-464. [Kiml] [Kim21 [KimSar] [KimShah] [Land] [Langl

H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth ofGL2, J . Amer. Math. Soc. 1 6 (2003), 139-183. H. Kim, A note on Fourier coefJicients of cusp forms on GL,, t o appear in Forum Math. H. Kim and P. Sarnak, Appendix: Refined estimates towards the Ramanujan and Selberg conjectures, Appendix to [Kiml]. H. Kim and F . Shahidi, Cuspzdality of symmetric powers with applications, Duke Math. J . 1 1 2 (2002), 177-197. E. Landau, ~ b e rdie Anzahl der Gitterpunkte in gewissen Bereichen, Gott. Nachr. (1915), 209-243. R.P. Langlands, Base Change for GL2, Annals of Math. Studies, no. 96, Princeton Univ. Press, Princeton, 1980.

[LiuWangYe] Jianya Liu, Yonghui Wang, and Yangbo Ye, A proof of Selberg's orthogonality for automorphzc L-functzons, t o appera in Manuscripta Math. [LiuYel] [LiuYe2]

[LiuYeS]

Jianya Liu and Yangbo Ye, Superposition of zeros of distinct L functions, Forum Math. 1 4 (2002), 419-455. Jianya Liu and Yangbo Ye, Weighted Selberg orthogonality and uniqueness of factorization of automorphic L-functions, Forum Math. 1 7 (2OO5), 493-512. Jianya Liu and Yangbo Ye, Selberg's orthogonality conjecture for automorphic L-functions, Amer. J . Math. 1 2 7 (2005), 837-849.

[LuoRudSar] W . Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for G L ( n ) ,Proc. Symp. Pure n l a t h . 6 6 (1999), part 2, 301310. [MoeWal]

C. Moeglin and J.-L. Waldspurger, Le spectre re'siduel d e G L ( n ) ,Ann. Sci. ~ c o l eNorm. Sup. (4) 2 2 (1989), 605-674.

[Morl]

C . J . hloreno, Explicit formulas in the theory of automorphzc forms, Lecture Notes Math. vol. 6 2 6 , Springer, Berlin, 1977, 73-216.

[Mor2]

C.J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J . Math. 1 0 7 (1985), 163-206.

[Mur11

hl. R a m Murty, Selberg conjectures and Artin L-functions, Bull. Amer. Math. Soc. 3 1 (1994), 1-14.

[Mur2]

M . Ram Murty, Selberg's conjectures and Artin L-functions 11, Current trends in mathematics and physics, Narosa, New Delhi, 1995, 154-168.

[Rogl

J . D . Rogawski, Functoriality and the Artin conjecture, Proc. Symp. Pure Math. 6 1 (1997), 331-353.

[RudSar]

Z. Rudnick and P. Sarnak, Zeros of principal L-functzons and random matrix theory, Duke Math. J . 8 1 (1996), 269-322.

Par1

P. Sarnak, Nonvanishing of L-functions on 3 ( s ) = 1, Contributions t o Automorphic Forms, Geometry & Number Theory, ed. by Hida,

J . Liu and Y . Y e

wl Per] [Shahl] [Shah21 [Shah31 [Shah41

Ramakrishnan and Shahidi, Johns Hopkins Univ. Press, Baltimore, 2004, 719-732. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Collected Papers, vol. 2, Springer, Berlin 1991, 47-65.

J.-P. Serre, Letter to J.-M. Deshouillers (1981). F. Shahidi, O n certain L-functions, Amer. J . Math. 103 (1981), 297355. F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for G L ( n ) ,Amer. J . Math. 106 (1984), 67-111.

F. Shahidi, Local coejjicients as Artin factors for real groups, Duke Math. J . 52 (1985), 973-1007. F.Shahidi, A proof of Langlands' conjecture o n Plancherel measures; Complementary series for p-adic groups, Ann. Math. 132 (1990), 273330.

ANALYTIC PROPERTIES OF MULTIPLE ZETA-FUNCTIONS IN SEVERAL VARIABLES Kohji Matsumoto Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan [email protected]

Abstract

We report several recent results on analytic properties of multiple zetafunctions, mainly in several variables, such as the analytic continuation, the asymptotic behaviour, the location of singularities, and the recursive structure. Some results presented in this paper have never been published before.

Keywords: multiple zeta-function, analytic continuation, Mellin-Barnes integral 2000 Mathematics Subject Classification: 1l M 4 l

1.

Euler-Zagier sums

Let r be a positive integer. We begin with the discussion on the Euler-Zagier r-fold sum

where S I , . . . , sr are complex variables. This multiple series is convergent absolutely in the region

The case r = 2 of (1.1) was already investigated by L. Euler in the eighteenth century, and the general r-fold case has recently been studied by Zagier [48]and others. In particular, their research on special values 153 Number Theory: Tradition and Modernization, p p . 153-173 W. Zhang and Y . Tanigawa, eds. 02006 Springer Science Business Media, Inc.

+

154

K. Matsumoto

of CEZ,, a t positive integers shows the great importance of this function in various fields of mathematics and mathematical physics. The meromorphic continuation of (1.1) to C T has been achieved by various methods; see Arakawa and Kaneko [5], Zhao 1491, Akiyama, Egami and Tanigawa [I],and t'he author [25]. It can also be regarded as a special case of Essouabri's general result [9];see Section 3. We shall describe the method of [I] which is based on the EulerMaclaurin summation formula. Let n l be a positive integer, 7j be a real number, Rs > 1, ai 2 0, Bj(x) the j-th Bernoulli polynomial, and B~(x) = Bj (x - [XI). Akiyama and Ishikawa [2] proved a modified version of the Euler-Maclaurin formula, including a parameter 7,which implies

+

for any positive integer J, where (s):, = r(s j ) / r ( s ) (see Lemma 1 of [2]). In [I], formula (1.2) (with a = 7 = 0) is applied to the sum with respect t o m, on the second member of (1.1),and an expression of CEZ,,(SI,. . . , s,) as a sum involving C ~ z , , - l ( s i , .. . , S r - - 2 ) ~ ~ - lS, j) (-1 5 j J) is obtained. Hence the analytic continuation can be shown by induction on r , because the integral on the right-hand side of (1.2) converges in a wider region of s when J becomes larger. On the other hand, the basic tool of the author's method is the MellinBarnes integral formula

+ +

<

where s,X E C , Rs > 0, /argXI < n-, A # 0, and C E R, -Rs < c < 0. The path of integration is the vertical line from c - ioo to c ioo. The key point is to apply (1.3) (with X = m , / ( m l + . . . + m T p l ) ) to the factor

+

on the last member of (1. I ) , and express CEZ,, ( s l , . . . , s,) as an integral s, z) as a factor. whose integrand includes <Ez,,-l(sl, . . . , s,-2, s,-1 Then the analytic continuation can be shown by shifting the path of integration suitably. We can also see the location of possible singularities by both of the above methods. Akiyama, Egaini and Tanigawa [I] considered this matter more carefully, and proved

+ +

Analytic properties of multiple zeta-functions i n several variables

155

Theorem 1. (Akiyama, Egami and Tanigawa [I]) Singularities of C E Z , r ( ~ l , . . . , s,) are located only on

and

where No denotes the set of non-negative integers. All of the above sets are indeed singularity sets. In the same paper [I],they also studied the values of CEZ,r ( s l , . . . , s,) at non-positive integers. This direction of research has been continued by Akiyama and Tanigawa [3], and Kamano [16]. It is an important problem to generalize the analytic theory of EulerZagier sums to a more general situation. Akiyama and Ishikawa [2] studied the series

and

<

<

<

crk < 1 (1 5 k r) and XI,(1 5 k r ) are Dirichlet where 0 characters of the same conductor. It is clear that (1.5) can be expressed as a linear combination of several series of the form (1.4). Akiyama and Ishikawa [2] applied (1.2) to the right-hand side of (1.4) to obtain an expression involving

C E Z , ~ - I ( S. ~. ., , sr-2,

ST-I

+ sr + j ; ai, . . . , a,-1)

(-1

<j < J).

This expression gives the analytic continuation of (1.4). Akiyama and Ishikawa also discussed the location of singularities of (1.4) and (1.5). Ishikawa [13] further studied the location of poles of (1.5) in the special case s l = . . . = s, = s , and applied the result to the evaluation of certain multiple character sums (Ishikawa [14]).

156

K. Matsumoto

The author [26], [27] considered a further generalization of (1.4))that is the series

ml=0

m,=O

x . . . x (a,

where

ak,

wk (1

+ mlwl + . . . + m , ~ , ) - ~ ' ,

< k < r ) are complex parameters. Let

-T

(1.6)

< 0 < .ir and

If we assume that wk E H ( 0 ) (1 5 k 5 r ) , then the series (1.6) is convergent absolutely when Rsk (1 k 5 r) are sufficiently large. Under the same assumption, the author proved the meromorphic continuation of (1.6) to CT by using the Xlellin-Barnes formula (1.3), discussed the asymptotic behaviour with respect t o w, and the order estimate with respect t'o Ss,. The aim of introducing the above generalized form (1.6) is to treat the Barnes multiple zeta-function

<

CB,,(s;a;wl,. . . , w,) =

x x ...

ml=0

(a, + m l w l + . . + m , w , ) - '

(1.7)

m,=O

as a special case s l = . . . = s,-1 = 0 and s, = s . The author first ~ ; w2) in [22] by a considered the asymptotic behaviour of C B , 2 ( ~ ; w1, different method (contour integration), and then by using (1.3) in [25]. These studies have applications to Hecke's zeta and L-functions attached t o real quadratic fields; see Corrigendum and addendum of [22],and [23]. Multiple Dirichlet series of the Euler-Zagier type with general coefficients, of the form

have been introduced and studied by hfatsumoto and Tanigawa [30], under the assumption that the series CEZlak(m)m-s (1 5 k I: r ) have nice properties. The above (1.8) includes the multiple L-series of Arakawa and Kaneko [6]. In [30],the analytic continuation and a certain order estimate of (1.8) have been obtained by using the Mellin-Barnes integral (1.3). It seems that the method of using the Mellin-Barnes integral is suitable to consider upper bound estimates of multiple zeta-functions. The case of the Euler-Zagier sum CEZSr ( s l , . . . , s,) was studied by Ishikawa

Analytic properties of multiple zeta-functions i n several variables

157

and Matsumoto [15]; especially, non-trivial estimates in the cases r = 2 and r = 3 have been obtained. However it is still not clear how is the real order of magnitude of C E Z , T ( ~ l , . . . , s r ) . Recently, Matsumoto and Tsumura [31] introduced further generalized series

>

<

1, a k , wk E R, 0 < a k - C Y ~ - 1 wk, in connection with a where u study of certain generalized multiple polylogarithms. (As for multiple polylogarithms, see, for example, [7].)

2.

Multiple series defined by linear forms

Let AN, = (anj)l < n < N , l < J < r be an ( N ,?-)-matrix, where a,j are nonnegative real numbers. Assume that all rows and all columns of An,, include a t least one non-zero element. Let

The Euler-Zagier sum (1.1)is a special case of (2.1). Shintani [35], [36] considered the situation when all a,j are positive (with characters and additional constant terms). Shintani actually treated the case s l = . . . = s ~but ) Hida [I11 introduced multi-variable Shintani zeta-functions. Other typical examples of (2.1) are the Mordell-Tornheim multiple series

(2.2) and the Apostol-Vu multiple series

Both of the above series (2.2) and (2.3) were introduced in the author's paper [28], though the history of some special cases goes back to Tornheim [38], Mordell [34], and Apostol and Vu [4]. The following theorem has been proved by the author in [24] for the case r = 2, and in [28] for general r .

158

K . Matsumoto

Theorem 2. T h e series ( 2 . 2 ) and ( 2 . 3 ) can be continued meromorphically t o cr+l. T h e possible singularities of ( 2 . 2 ) are located only o n the subset of c r f ldefined by one of the following equations:

Also, the possible singularities of ( 2 . 3 ) are located only o n the subset of CT+l defined by one of the following equations:

T h e proof of Theorem 2 in [24] [28]is again based on ( 1 . 3 ) . As in the case of the Euler-Zagier sum, using ( 1 . 3 ) we obtain

C ~ I T , ~ (. .~, sIr,; .s r + l )=

1

J 2x2

-

(c)

r(s,+l

+ 4J3-x)

%+l)

where C ( . ) is the Riemann zeta-function. In the case of the Apostol-Vu series, we have

where

Note that p,,, = CAV,r and

T h e relations ( 2 . 4 ) and ( 2 . 5 ) imply the recursive sequences

Analytic properties of multiple zeta-functions i n several variables

159

along which the proof of Theorem 2 goes inductively. The discussion in Section 1 implies another recursive sequence

Thus we find a recursive structure in the family of multiple zeta-functions. This viewpoint is discussed in the last section of [28]. In [47], Maoxiang Wu introduced the X-analogues of (2.2) and (2.3). Let X I , . . . , X, be Dirichlet characters of the same modulus q (> 2), and define cc -

00

xl (m1) . . x,(m,) ,

C . . . m,=l C m;' . . m F r ( m l + ml=l

These series are convergent absolutely for X s k 0. Wu proved the following two theorems.

+m,)s~+l'

> 1 (1 < k

(2.7)

< r ) , R S , + ~>

Theorem 3. (Wu [47]) T h e series (2.7) can be continued meromorphically t o C T + l . If n o n e of the characters X I , .. . , X, are principal, t h e n LMT,, is entire. If there are Ic principal characters x j , , . . . , xj, a m o n g t h e m , t h e n possible singularities are located only o n the subsets of cT+l defined by one of the following equations:

where 1

< h < Ic, 1 < i(1) < . . . < i ( h ) 5 k , e E No.

Theorem 4. (Wu [47]) T h e series (2.8) can be continued meromorphically t o c T + l , and possible singularities are located only o n the subsets of CT+ldefined by o n e of the following equations:

where 1

< h < r , l E No.

160

K. Matsumoto

Since Wu [47] is unpublished, we briefly outline his proof of these two theorems here. The proof of Theorem 3 is just a direct generalization of the argument developed in [28]. We omit the details, only noting that the basic formula, corresponding t o (2.4), is

where L ( . ,x,) is the Dirichlet L-function attached to To prove Theorem 4, we define

x,.

Then corresponding to (2.5), we have

Hence the induction argument goes along the sequence

but

whose basic analytic properties has already been discussed by Akiyama and Ishikawa 121. As explained in Section 1, LEZ,T(sl,. . . , S r ; X I , . . . , xT) can be expressed in terms of C E ~ , ~ ( S. .~. , S r ; a', , . . , a,), and the latter can be expressed as a sum involving

Using these expressions, Wu [47] proved (by induction) that both of the ) L E Z , ~ ( S.I.,. , S T ;X I , . . . , X r ) are functions C E Z , ~ ( S ~. ,. . , S T ; ai,. . . , a T and

Analytic properties of multiple zeta-functions in several variables

161

of polynomial order with respect to J S s l1, . . . , ISs,I. Hence, using (2.11), we can show that Qj,, ( I 5 j 5 r) is also of polynomial order. Therefore it is possible t o shift the path of integration on the right-hand side of (2.11) freely. The remaining part of the proof is the same as in [28]. Special values of CMT,r, CAV,, and their relatives have been studied by several mathematicians, including Tornheim, Mordell and Apostol and Vu themselves. Special values in the domain of absolute convergence have been further studied by Huard, Williams and Zhang [12], Subbarao and Sitaramachandrarao [37], and Tsumura in his recent series of papers [39], [40], [41], [42], [43] and [44]. In those papers, various relations among special values a t integer arguments have been obtained. From an analytic point of view, however, it is important to reveal whether those relatzons are valzd only at znteger poznts, or valzd also at other values. Tsumura [45], [46] discovered that some relations a t integer points, proved in his previous articles [39], [40], [42], are actually valid continuously a t other values. These relations of Tsumura may be regarded as functional relations among multiple zeta-functions. Another functional relation has been found by the author [29], which implies, as a special case, a certain relation between C E z , 2 ( ~ l , s 2 ) and CEZ,2(1 - s 2 , l - s l ) . More generally, in [29] the author defined the double Hurwitz-Lerch zeta-function

<

1, w > 0, and proved a certain relation where 0 < a 5 1, 0 5 between C2(sl,s2; a, P, w) and C 2 ( l - s2,1 - s l ; 1 - P, 1 - a , w). Note that the case w = 1 of this function was already introduced by Katsurada [17] in his study on the mean square of Lerch zeta-functions. It is also possible to regard Proposition 1 of [29] as a double analogue of the functinal equation for Hurwitz-Lerch zeta-functions.

3.

Multiple series defined by polynomials

In [28],it has been shown that any multiple series of the form (2.1) can be continued meromorphically to the whole space c",by the method of Mellin-Barnes integrals. It is in fact possible to prove a much more general result by the same method. Let

162

K. Matsumoto

be polynomials, where a k ( n )E C , p j ( k , n ) E N o , and for any fixed j , a t least one of p j ( k , n ) ( 1 n N , 1 5 k K ( n ) )is positive. We assume that R a k ( n )> 0 for all k and n . Hence

< <

<

is smaller than 7r/2. Define

<

where m = ( m l ,. . . , m,) and s , = a, + i t , E C ( 1 5 n N ) . It is clear that there exists a positive constant a, = a,(Pl, . . . , P N ) such that the series (3.1) is absolutely convergent when an > a, for 1 5 n N . By a multiple strip we mean a set of the form

<

<

where a,l, a,z ( 1 5 n N ) are any fixed real numbers with anl < a,2. By F ( . ) we denote a quantity, not necessarily the same a t each occurrence, which is of polynomial order with respect to the indicated variables.

< <

Theorem 5. W h e n 0, < n / 2 ( 1 n N ) , the multiple zeta-function (3.1) can be continued meromorphically t o the whole space C N . T h e possible singularities of it are located only o n hyperplanes of the form

where c l , . . . , c~ E No and u ( c l , .. . , c N ) i s a n integer determined by c l , . . . , C N . Moreover, the estimate

holds uniformly in a n y multiple strip (3.2)) except i n neighbourhoods of possible polar sets (3.3). T h e case N = 1 of (3.1) was first studied by Mellin [32],[33].T h e Mellin-Barnes integral (1.3) already appeared in those papers. After Mellin, many people including K. Mahler, P. Cassou-Noguks, and P. Sargos continued his research. T h e multi-variable form (3.1) was first discussed by Lichtin [18],[19],[20],[ 2 1 ] ,and he proved the continuation of (3.1) when polynomials are hypoelliptic. Then Essouabri [9], [lo]

Analytic properties of multiple zeta-functions i n several variables

163

introduced the condition HoS, under which he proved the continuation. Here we do not give the exact definition of HoS, but only state that it is satisfied if all coefficients of polynomials have positive real parts. Moreover, though only the case N = 1 is discussed in [lo], Essouabri mentioned in his thesis [9] that his result can be generalized to the multiin which a twisted version of (3.1) variable case. See also de Crisenoy [8], (for general N ) was studied. of (3.1) in the above theoTherefore, the meromorphic cont~inuat~ion rem is included, as a special case, in Essouabri's theorem. Nevertheless we give a proof of the above theorem here, because of several reasons. First, our method is quite different from Essouabri's and rather simple. Secondly, formula (3.5) below, which is the key of our proof, implies the recursive structure similar to those discussed in the preceding section. Thirdly, our method is suitable t o obtain various explicit information, such as location of poles and order estimates, inductively. And finally, our method can be generalized t o the case with general coefficients (similar t o (1.8) and (1.9)).

Remark 1. When we write the (possible) polar sets of CT in the form (3.3), we can choose el, . . . , c~ whose common greatest divisor is as small as possible. We call such tuples (el, . . . , cN) primitive. Then, in the proof of Theorem 5 it will be shown that, for any fixed C,, there are only finitely many primitive tuples (el, . . . , cN) such that the (possible) polar sets of CT are of the form (3.3). Remark 2. For any fixed e l , . . . , c ~ there , exists a positive integer v ( c l , . . . , cN), by which the order of the singularity (3.3) is bounded uniformly for any l. Now we start the proof. We prove Theorem 5 with Remarks 1 and 2 by induction on N n=l

The argument is a generalization of the proof of Theorem 3 in [28]. First consider the ca,se K ( P l , .. . , PN) = 1. Then K(n,) = 1 for 1 _< n N , so all the P,'s are monomials and

<

164

K.Matsumoto

Hence all the assertions of Theorem 5, Remarks 1 and 2 clearly hold. a,, and at first Now consider the case K ( P 1 , . . . , PN) 2. Let a: assume that ( s l , . . . , slv) is in the region

>

B* = { ( s ~. .,. , S N )

>

I CT, > 2a:(1 1 n 5 N ) ) .

Since a t least one K ( n ) 2 2, changing the parameters if necessary, we may assume that K ( N ) 1 2. Then

where h I k ( N ) = m1~ l

00

x

( k N. ). , m,P " ~ ' ~ ) Hence, .

applying (1.3), we obtain

CO

...

Pl(m)-"'

PN-l( m )-SN-1 P* N ( m 1- s N - z p F (rn)zdZ,

and we can choose r as

Then the multiple series on the right-hand side of (3.5) is absolutely convergent and is the zeta-function

Since

Analytic properties of multiple zeta-functions i n several variables

165

by the induction assumption we see that (3.7) can be continued meroand possible singularities are of morphically to the whole space cN+l, the form

, E N o and u(cl, . . . , c ~ + l E) Z. If cN = C N + ~ then , where cl, . . . , c ~ + l! this is c1sl . . . C N S N = u ( c ~. ,. . , c N ) - e (e E N ~ ) , (3.8)

+ +

which is irrelevant to z. If

and if c~

-

CN -

c ~ += l do > 0, then

c ~ += 1 - e ~< 0, then

We write the first term on the right-hand side of (3.9) (resp. (3.10)) as D ( s l , . . . , S N ; C) (resp. E ( s l , . . . , S N ; c)) for brevity, where c = (cl, . . . , cN). Denote the set of all primitive tuples c = ( e l , . . . , cN) appearing in (3.8) (resp. (3.9), (3.10)) by To (resp. To,T E ) .These sets are finite because of Remark 1. The above (3.9) and (3.10) can be poles, with respect t o z, of the integrand on the right-hand side of (3.5). The other poles of the integrand are

and z=e

(emo).

We can assume that a: is so large that all the poles (3.9) and (3.11) are on the left of the line Rz = y,while all the poles (3.10) and (3.12) are on the right of Rz = y. Now, let (sy, . . . , be any point in the space C N , and we show that the right-hand side of (3.5) can be continued meromorphically to 0 . . , sj,J First, remove the singularities of the form (3.8) from the integrand. These singularities are cancelled by the factor

sk)

(4,.

(by Remark 2 as a part of the induction assumption). Let L be a sufficiently large positive integer such that, if a, 2 R s i (1 5 n 5 N ) ,

166

K. Matsumoto

does not hold for any c = (cl, . . . , cN) E To. Define

and rewrite (3.5) as

where

xCT(si,.. . , SN-1,

SN

+

Z , -2;

Pi, . . . , PN-1,P ,:

P ? ) ~ z .(3.14)

Then the integrand on the right-hand side of (3.14) does not have singularities of the form (3.8) in the region a, 32s: (1 n N). Since @ ( s l , .. . , sN)-' is meromorphic in the whole space, in order to complete the proof of the continuation, our remaining task is to show the continuation of J ( s l , . . . , s N ) . Let M be a positive integer, and s; = s: +Dl (1 n 5 N ) . We may choose M so large that (ST, . . . , s k ) E B*. Let Z1 be the set of all imaginary parts of the poles (3.9) and (3.11), and Z:! be the set of all imaginary parts of the poles (3.10) and (3.12), for (sl,. . . , s N ) = (ST, . . . , s;). Case 1. In the case Z1 n Z2 = 0, we join D(sT, . . . , s;; C ) and D(s7, . . . , s k ; C ) by the segment S ( D ;c) which is parallel t o the real axis. Similarly join E(s7, . . . , s k ; C) and E(s:, . . . , s;; C) by the segment S(E;c ) , and join -s; and - s k by the segment S ( N ) . Since Zl nZ2 = 0, we can deform the path XZ = y t o obtain a new path C from y - icc to y icc,such that all the segments S ( D ; c ) and S ( N ) are on the left of C, while all the segments S(E;c ) and the poles (3.12) are on the right of C (see Fig. 1). Then we have

>

< <

<

+

x C r ( ~ 1 ,. .. , SN-I,

SN

+

Z , -2;

Pi,.. . , PN-1,P i , P ? ) ~ Z (3.15)

in a sufficiently small neighbourhood of (ST, . . . , s k ) . Next, on the right-hand side of (3.15), we move ( s l , . . . ,sAr) from (ST,. . . , s & ) to (s?, . . . , s&) with keeping the values of imaginary parts of each s,. Since C, in the integrand satisfies an estimate of the form (3.4) by the induction assumption, this procedure is possible; and, in the course of this procedure, the path C does not cross any poles of the integrand. Hence the

Analytic properties of multiple zeta-functions in several variables E($,

. . . , s:)

167

E ( s 7 , . . . , skr)

X X X X X

X X X X X X X

D ( s ; ,. . . , s k )

:, . . . .

Fig. 1 expression (3.15) gives the holomorphic continuation of J ( s l , . . . , s N ) to a neighbourhood of (sy,. . . , s i ) . Case 2. Next consider the case Zl n Z 2 # 8.Then the imaginary part of some member of ( D ( s 7 , .. . , sTy; c ) , -sTy I c E T D }coincides with the imaginary part of some member of { E ( s T ,. . . , s h ; c ) , 0 / c E T E ) . We consider the case

for some cl and c2, because other cases can be treated similarly. The associated poles are D(s7, . . . , s h ; c l ) - dolel and E(sT, . . . , s&;c l ) ei112 ( e l , t 2 E No). When ( S T , . . , s&) is moved to ( s y , .. . ,sOh,),these poles are moved to ~ ( s y. .,. , s & ; c l ) - d i l e l and ~ ( s y ., .., s:; c l ) e0 l e 2 , respectively. In the case

+ +

168

K. Matsumoto

for any t1and t 2 ,we modify the argument in Case 1 as follows. Let q be a small positive number, and consider the oriented polygonal path S f ( D ;c l ) joining the points D ( s T , . . . , s L ; c l ) , D(sT +iq, . . . , sL+iq; c l ) , c l ) in that order. ~ ( s y i q , . . . , s& i q ;c l ) , and then ~ ( s y. .,. , Similarly define the path S f( E ;c z ) which joins E(sT, . . . , sL; c 2 ) ,E(sT i q , . . . , s&+iq; c 2 ) ,E(s:+iq, . . . , s%+iq; c 2 ) ,and then E ( s 7 , . . . , s:; c 2 ) . Then S f ( D ;c l ) lies on the lower side of the line

+

+

SON;

+

while S f ( E ;c z ) lies on the upper side of C. Because of (3.17),we can define the path C f , which is almost the same as C, but near the line C we draw C f such that it separates

U ( s ' ( D ;c l ) - d o 1 t l )

el E N O

and

U

( S 1 ( Ec: 2 ) ez €NO

+ eo1e2)

(see Fig.2). Then the expression (3.I s ) , with replacing C by C', is valid in a sufficiently small neighbourhood of ( S T , . . , s & ) . When ( s l , .. . , s N ) moves along the polygonal path joining ( S T , . . . , s k ) , (sT+iq, . . . , s&+iq), i q ) , and then (sy,. . . , in that order, the path C f (sy i q , . . . , encounters no pole, hence we obtain the holomorphic continuation. Case 3. The remaining case is that

+

SON +

SON)

el

holds for some and t2.Then this might hold for some other pairs of & ) . In this case we consider the path C f f which is almost the same as C, but near the line ,C we only require that S ( D ;c l ) is on the left c2) of C f f ,and that the points E ( S T , . . . , s h ; c 2 ) e i 1 t 2 , E (sy, . . . , eo '!2 are not on C f f for any 4'2. When we deform the path Rx = y on the right-hand side of (3.14) to C", we might encounter several poles of the form (3.10). Then we move (31,.. . , s N ) from ( S T , . . . , s i r ) to (s?, . . . , s ; ) ; again the path might encounter several poles of the same type. Hence, in a sufficiently small neighbourhood U of (sy, . . . , s;), the integral J ( s l , . . . , s N ) has the expression

(el,

+

SON;

+

where R ( s l , . . . , s N ) is the sum of residues of the above poles. Hence ) a (finite) sum of residues of the form we see that R ( s l , . . . , s , ~ is

Analytic properties of multiple zeta-functions in several variables

I ' ( s N ) - l @ ( s l ,. . . , S N ) R(e2),where

+

with x(12) = E ( s l , . . . , s N ; c ~ ) e o i e a , if the order of the pole is h . This implies that all possible singularities of R ( s l , . . . , s N ) are polar sets. Therefore expression (3.19) gives the meromorphic continuation of J ( s l , . . . , sni) to U . Now the meromorphic continuation of C,(sl, . . . , S N ; P I , . . . , P N ) has been proved. Next we show that all the possible polar sets of Cr ( s i , . . . , slv;P i , . . . , P N ) are of the form (3.3). This is clear for the

170

K. M a t s u m o t o

polar sets of
<<

/

CO

exp

( ~ ( I ~ -N ItN I

+ 4 - l ~ l )f oN(ltN + YI + 131))~

( t ry)dy> ,

-00

(3.21) which is 0(ee"ltNIF(tN)) by Lemma 4 of [26]. Hence we obtain the desired assertion in Case 1, and the treatment of Case 2 is similar. In Case 3, we have to estimate R ( s l , . . . , s N ) . Since

where K: is a small circle around the point z ( t a ) ,we see that R ( s l , . . . , sN) satisfies an estimate of the form (3.4) with respect to t l , . . . , tN-1. As , relevant exponential factor is the same as the exponential for t ~ the factor in (3.21), and hence we can obtain the desired estimate as above. The proof of Theorem 5 is now complete.

Acknowledgments The author expresses his gratitude to Dr. Yasushi Komori for helping him to draw the figures, and to Professor Hirofumi Tsumura for pointing out several misprints in the original version of the manuscript.

References [ I ] S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta functions and their values at non-positive integers, Acta Arith. 98 (2001), 107116. [2] S. Akiyama and H. Ishikawa, O n analytic continuation of multiple L - f u n c t i o n s and related zeta-functions, in "Analytic Number Theory", C. Jia and K . Matsumoto (eds.), Developments in Math. Vol. 6 , Kluwer, 2002, pp. 1-16. [3] S. Akiyama and Y . Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J . 5 (2001), 327-351.

A n a l y t i c p r o p e r t i e s o f m u l t i p l e z e t a - f u n c t i o n s in s e v e r a l v a r i a b l e s

171

[4] T . M.Apostol and T . H. Vu, Dirichlet series related t o the R i e m a n n zeta function, J . Number Theory 19 (1984), 85-102. [5] T . Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J . 153 (1999), 189-209. [6] T . Arakawa and M. Kaneko, O n multiple L-values, J . Math. Soc. Japan, 56 (2004), 967-991. [7] D. Bowman and D . M.Bradley, Multiple polylogarithms: a brzef survey, in "qseries with Applications to Combinatorics, Number Theory and Physics", B. C. Berndt and K. Ono (eds.), Contem. Math. vol 291, Amer. Math. Soc., 2001, pp. 71-92. [8] M . de Crisenoy, Values at T - t u p l e s of negative integers of twisted multivariable zeta series associated t o polynomials of several variables, Compositio Math., to appear. [9] D. Essouabri, Singularitis des se'ries de Dirichlet associe'es d, des polyn6mes de plusieurs variables et applications a la the'orie analytique des nombres, ThCse, Univ. Henri PoincarC - Nancy I , 1995. [lo] D. Essouabri, Singularite's des se'ries d e Dirichlet associies a des polyn6mes de plusieurs variables et applications e n the'orie analytique des nombres, Ann. Inst. Fourier 47 (1997), 429-483. [ll] H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Math. Soc. Student Text 2 6 , Cambridge Univ. Press, 1993.

1121 J . G . Huard, K . S. Williams and N. Y. Zhang, O n T o r n h e i m ' s double series, Acta Arith. 75 (1996), 105-117. 1131 H. Ishikawa, O n analytic properties of a multiple L - f u n c t i o n , in "Analytic Extension Formulas and their Applications", S. Saitoh et al. (eds.), Soc. Anal. Appl. Comput. Vol. 9, Kluwer, 2001, pp. 105-122. [14] H. Ishikawa, A multiple character s u m and a multiple L - f u n c t i o n , Arch. Math. 79 (2002), 439-448. [15] H. Ishikawa and K. Matsumoto, O n the estimation of the order o f Euler-Zagier multiple zeta-functions, Illinois J . Math. 47 (2OO3), 1151-1166. [16] K . Kamano, Multiple zeta values at non-positive integers and a generalizatzon of Lerch's formula, Tokyo J . Math. to appear. [17] M . Katsurada, A n applacation of Mellin-Barnes' type integrals t o the m e a n square of Lerch zeta-functions, Collect. Math. 48 (1997), 137-153. [18] B. Lichtin, Poles of Dzrzchlet serzes and D-modules, in "Thkorie des Nombres/Number Theory1',Proc. Intern. Number Theory Conf. (Lava1 1987), J.-M. De Koninck and C. Levesque (eds.), Walter de Gruyter, 1989, pp. 579-594. [19] B. Lichtin, T h e asymptotics of a lattice point problem associated t o a finite n u m b e r of polynomials I, Duke math. J . 63 (1991), 139-192; 11, ibid. 77 (1995), 699-751. [20] B. Lichtin, Volumes and lattice points - proof of a conjecture of L . Ehrenpreis, in "Singularities, Lille 1991n, London Math. Soc. Lect. Note Vol. 201, J.-P. Brasselet (ed.), Cambridge Univ. Press, 1994, pp. 211-250. 1211 B. Lichtin, A s y m p t o t i c s determined by pairs of additzve polynomials, Compos. Math. 107 (1997), 233-267.

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[22] K . hIatsumoto, Asymptotic series for double zeta, double gamma, and Hecke Lfunctzons, Math. Proc. Cambridge Phil. Soc. 123 (1998), 385-405; Corrzgendum and addendum, ibid, 132 (2002), 377-384. 1231 K. Matsumoto, Asymptotzc expanszons of double gamma-functzons and related remarks, in "Analytic Number Theory", C. Jia and K. Matsumoto (eds.), Developments in Math. Vol. 6, Kluwer, 2002, pp. 243-268. [24] K . Matsumoto, On the analytic continuation of various multiple zeta-functions, in "Number Theory for the Millennium 11, Proc. of the Millennia1 Conference on Number Theory" hI.A. Bennett et al. (eds.), A K Peters, 2002, pp. 417-440. [25] K . Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J . 172 (2003), 59-102. [26] K . 1CIatsumot0, The analytic continuation and the asymptotic behaviour of certain nzultiple zeta-functzons I, J . Number Theory 101 (2003), 223-243. [27] K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions II, in "Analytic and Probabilistic Methods in Number Theory", Proc. 3rd Intern. Conf. in Honour of J . Kubilius, (Palanga, Lithuania, 2001), A. Dubickas et al. (eds.), TEV, Vilnius, 2002, pp. 188-194. [28] K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, in "Proceedings of the Session in Analytic Number Theory and Diophantine Equations", D. R. Heath-Brown and B. Z. Moroz (eds.), Bonner Math. Schriften Nr. 360, Univ. Bonn, 2003, n. 25, 17pp. [29] K . Matsumoto, Functional equations for double zeta-functions, Math. Proc. Cambridge Philos. Soc. 136 (2004), 1-7. [30] K . Matsumoto and Y. Tanigawa, The analytic continuation and the order estimate of multiple Dirichlet series, J . Thkorie des Nombres de Bordeaux 15 (2003), 267-274. [31] K . Matsumoto and H. Tsumura, Generalized multiple Dirichlet series and generalized multiple polylogarithms, preprint. [32] H,Mellin, Eine Formel fur den Logarithmus transcendenter Funktionen won endlichem Geschlecht, Acta Soc. Sci. Fenn. 29, no. 4 (1900), 49pp. [33] H. Mellin, Die Dirichlet'schen Reihen, die zahlentheoretischen Funktzonen und die unendlichen Produkte won endlichem Geschlecht, Acta Math. 28 (1904), 37-64. [34] L. J. Mordell, On the evaluation of some multiple series, J . London Math. Soc. 33 (1958), 368-371. [35] T . Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-negative integers, J . Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 393-417. 1361 T . Shintani, On a Kronecker limit formula for real quadratic fields, ibid. 24 (1977), 167-199. [37] M. V.Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J . Math. 119 (1985), 245-255. [38] L. Tornheim, Harmonic double series, Amer. J . Math. 72 (1950), 303-314. [39] H. Tsumura, On some combinatorial relations for Tornheim's double series, Acta Arith. 105 (2002), 239-252.

Analytic properties of multiple zeta-functions in several variables

173

[40] H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc. 131 (2003), 3633-3641. [41] H . Tsumura, Evaluation formulas for Tornheim's type of alternative double series, Math. Comput. 73 (2003), 251-258. [42] H. Tsumura, Multiple harmonic series related to multiple Euler numbers, J. Number Theory 106 (2004), 155-168. [43] H . Tsumura, On Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 133 (2OO5), 2387-2393. [44] H . Tsumura, On a class of combinatorial relations for the Mordell-Tornheim triple series, preprint. [45] H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, preprint. [46] H . Tsumura, Certain functional relations for the double harmonic series related to the double Euler numbers, J . Austral. Math. Soc. to appear. [47] Maoxiang Wu, On analytic continuation of Mordell-Tornheim and Apostol-Vu L-functions (in Japanese), Master Thesis, Nagoya University, 2003. [48] D. Zagier, Values of zeta functions and their applications, in "First European Congress of Mathematics, Vol.11 Invited Lectures (Part 2)", A. Joseph et al. (eds.), Progress in Math. Vol. 120, Birkhauser, 1994, pp. 497-512. [49] Jianqiang Zhao, Analytic continuation of multzple zeta-functions, Proc. Amer. Math. Soc. 128 (2000), 1275-1283.

CUBIC FIELDS AND MORDELL CURVES Katsuya Miyake Department of Mathematical Science, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo,Japan, 169-8555 [email protected]

Abstract

The purpose of this article is to describe a relationship between several cubic polynomials and elliptic curves, and show a clearer view on it than that in the former half of our previous work [Mi-20031. For a monic irreducible cubic polynomial P ( u ) in u over Q , the curve E = E ( P ( u ) ) defined by the equation w3 = P ( u ) is an elliptic curve whose j-invariant is equal to 0. We describe the set E[Q] of all rational points of E over Q by use of a root ( of P ( u ) as

+

Then we show that the short form of E is a Mordell curve, y2 = x3 k , with a certain rational number k determined by the coefficients of P ( u ) . It is also pointed out that E ( P ( u ) )is essentially dependent on the polynomial P ( u ) rather than the cubic field Q ( ( ) even though E[Q] is completely described by the subset W ( E )of the cubic field. K e y w o r d s : Elliptic Curves, Cubic Fields, Mordell Curves. 2000 M a t h e m a t i c s S u b j e c t C1assification:Primary l l G 0 5 , llR16

1.

Introduction

In this paper we study those elliptic curves which are defined over the rational number field Q and whose rational points over Q are described by certain subsets of the associated cubic fields. The curve E = E ( P ( u ) ) we are interested in is defined by the equation

where P ( u ) is an irreducible cubic polynomial in u over Q.One of the points at infinity is rational over Q (although other two are not). Hence T h e author was partly supported by t h e Grant-in-Aid for Scientific Research (C) (2) No. 14540037, J a p a n Society for t h e Promotion of Science, while he prepared this work.

Number Theory: Tradition and Modernization, p p . 175-183 W . Zhang and Y.Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

176

K.Mzyake

E is an elliptic curve defined over Q. Let E[Q]denote the set of all rational points of E over Q. One of our main interests should be the Mordell-Weil rank of E ( P ( u ) ) which is an isogeny-invariant. We will not, however, go into its detailed study in this article. Take a root J of P ( u ) in the complex number field @I, and let K = Q(J) be the cubic field generated by E . P u t

Then there is a bijective map from W ( J ) onto E[Q]which maps 1 E W(J) to the point a t infinity in E[Q]. Note that the subset W(J) of K X is not well determined by the cubic field K itself but depends on ( or, we may say, on the polynomial P ( u ) ; indeed, if we take another root 7 of P ( u ) , the isomorphism between the cubic fields K and Q(7) bijectively maps W ( J ) t o W(7). In this way we can completely describe the set of all rational points of E over Q by the subset W ( J ) of the cubic field K. It is clear that W(J) = W(sJ t) for s, t E Q, s # 0. Thus in Section 3 we reduce the curve E ( P ( u ) ) to two typical elliptic curves Fo(a) and Eo(b) which are defined as

+

and parametrized respectively by a, b E Q X . The first F o ( a ) is isomorphic t o the well known pure cubic twist X3 y3 a Z 3 = 0 of the cubic Fermat curve. Its short form is a nilordell curve M5(a) defined as

+ +

(cf. e.g. Cassels [Ca-501). In Section 6 we will show our main result as Theorem 1; it states that a short form of Eo(b)is

and that this has a rational point of infinite order except two special values of the parameter. (We use the indices of curves here in such a way as they are t o be consistent with the ones in the previous paper [Mi-20031.) As is well known, every cubic field is generated by a root of R(b; u) := u3 bu b for a suitable choice of b E Q (cf. Section 2). If we restrict ourselves to cyclic cubic fields, then

+ +

177

Cubic Fields and Illordell Curves

is generic for them over Q (cf, e.g. Jensen, Ledet and Yui [JLY-20021). Here we only note the fact that every cyclic cubic field is generated by a root of Q ( s ;u ) for a suitable s E Q , and the converse also holds as far as Q ( s ;u ) is irreducible over Q . A short form of the elliptic curve

(see Section 5, Proposition 7 ) . We also determine the subfamily of Eo(b) corresponding to E 4 ( s )in Section 5 , Proposition 6.

2.

Miscellaneous Preliminaries

+

+ +

Let P ( u ) = eu3 au2 bu c be an irreducible cubic polynomial in u over Q , and E = E ( P ( u ) )is a curve defined by the equation

Proposition 1. One of the points of E at infinity is rational over Q if and only zf the leading coeficient e of P ( u ) is a cube in Q (other than 0). In this case, other two points of E at infinity are not rational over Q but over Q ( a ) . This is almost obvious, and the proof is omitted. As we stated in the introduction, we suppose in this paper that one of the points of E ( P ( u ) )a t infinity is rational over Q. Then by Proposition 1 we have e = r3,r E Q x . Replacing the variable w in the defining equation of E by rw,we may hereafter assume e = 1. Let J be a root of P ( u ) in C,and I< = Q ( J )be the cubic field generated by J . Furthermore, let' W ( J )be the subset of K X defined in Section 1. Proposition 2. There exists a bijective map from W ( J ) onto E[Q] which maps 1 E W ( J )to the point at infinity in E [ Q ] .

Proof. Let J , J' and

J'l

be the three roots of P ( u ) in

C.Then we have

+

Hence for k E Q , we have P ( k ) = N K I Q ( k- J ) . For a = q< r E W ( J ) with q # 0, put u = -r/q and w = - l / q . Then we have w3 = P ( u ) because N K I Q ( a = ) 1; that is, we have a point ( u ,w ) = (-rlq, - l / q ) of E [ Q ] .For a = r E Q,N K I Q ( a )= r3 = 1 if and only if a = 1. Define a

178

-

K. Mzyake

map p : W ( [ ) E[Q]by p(a!) := (-rlq, - 1 / q ) for cr = q<+r E Q with q # 0 and p ( 1 ) := the point a t infinity in E [ Q ] .It is clear that p is well defined and injective. There is only one point a t infinity in E[Q]. Let ( u ,w ) = ( k ,I ) , k , I E Q be a point of E [ Q ] .Then we have L3 = P ( k ) and hence 1 # 0 because P ( u ) is irreducible over Q. Therefore ( k , I ) = p(a) with a! = ( - l / I ) [ k/L E W ( < ) .The proof is completed. 0

+

Lemma 1. W (<) = W ( s t

+ t ) for

s, t E Q,s # 0

The proof, being almost obvious, is omitted.

3.

The curves Fo ( a ) and Eo( b )

+

+ +

Let us see that our curve E ( P ( u ) )for P ( u ) = u3 au2 bu c is isomorphically reduced to either Fo(a)or Eo(b)defined in Section 1 without changing the set W ( < ) . First put x = u a/3. Then we have P ( u ) = s3 b'x c' with

+

+

+

Since P ( u ) is irreducible over Q,c' is not equal to 0. If b' = 0, then we have the curve Fo(a): w3 = u3 a ;

+

here we replaced x and c' by the letters u and a, respectively, for simplicity. s. Then we see Suppose now b' # 0.P u t y =

5

We have, therefore, the curve

5

here we replaced . w, y and b' by the letters w, u and b, respectively, for simplicity. We easily see by Lemma 1 in the preceding section that the original W ( J )coincides with either W(*) for Fo(a) or W ( 7 )for Eo(b) where 7 is a root of R(b;u ) . Thus we obtained the following proposition. Proposition 3. The elliptic curve E ( P ( u ) ) is isomorphic over Q to either Fo(a) or Eo(b)for some a or b in Q with the same subset W ( < ) of the cubic field Q(<).

179

Cubic Fields and Mordell Curves

4.

The curve E l ( c ) In our previous work [Mi-20031, we studied a family of elliptic curves

where P ( c ; u) is an irreducible cubic polynomial in Q[u]. This polynomial is related to R(b; u) := u3 bu b as follows: as before let be the fixed is a root of P ( c ; u) with c = b-l. This relation root of R(b; u). Then may give us a morphism between the two algebraic curves Eo(b) and El(c). If we try to pick up such one, however, it may not be defined over Q as we do not have a natural correspondence between W ( < ) and w ( < - ~Hence ) . we should adopt the reduction process in the preceding section to obtain a homomorphism of elliptic curves defined over Q .

+ +

+

Proposition 4. Suppose 27c 2 # 0. Then E l ( c ) is isomorphic to Eo(b)for b = -27/(27c 2)2 over Q.

+

Proof. In this case, indeed, the defining equation

of El (c) is equivalent to

0

as is easily checked.

Remark 1. In case of 27c + 2 = 0, the cubic polynomial P ( c ; u) = u3 u2 c is reducible over Q.

+ +

It should be noted that c may not always be obtained from b in Q 2)2. Hence the family of elliptic through the relation b = -27/(27c curves E l ( c ) , c E Q, covers a strict subfamily of Eo(b),b E Q. We restate a few results on El(c) given in [Mi-2003, Prop. 11 for the convenience of readers.

+

Proposition 5. (i) The short form of E l ( c ) for c E Q - (0, -4127) is Ml(c) which is defined by the equation

(ii) M l (c) has a rational point

180

K. Mzyake

This is not a torsion point for c # -2127, and hence the Mordell- Weil rank is greater than or equal to 1 if c # -2127. (iii) The exceptional curve MI(-2/27) : y2 = x3 + ( 4 / 3 ) 3has MordellWeil rank = 0 . Its torsion group is of order 2 and generated b y the point (-413, 0 ) .

5.

The curve E 4 ( s ) associated t o cyclic cubic fields As we stated in Section 1, the polynomial

is generic for cyclic cubic fields over Q (cf. e.g. Jensen, Ledet and Yui [JLY-20021).This implies that every cyclic cubic field is generated by a root of Q ( s ;u ) for a suitable s E Q,and the converse also holds as far as Q ( s ;u ) is irreducible over Q . In this section we study the elliptic curve

E 4 ( s ) : w3 = Q ( s ; u )= u3 - ( s - 3)u2- s u - 1. First we give a subfamily of Eo(b)which is covered by E q ( s )

Proposition 6. Suppose s # 312. Then E q ( s ) is isomorphic to Eo(b) for b = -27(s2 - 3s 9 ) / ( 2 s- 3)2 over Q .

+

Proof. In this case, the defining equation

of E 4 ( s )is equivalent to

0

as is easily checked.

Remark 2. In case of s = 312, the cubic polynomial Q ( s ;u ) = u3 ( s - 3)u2- su - 1 is reducible over Q. Proposition 7. The short form of E 4 ( s ) is

M ~ ( S: )y 2 = ~3

+ 24(s2

-

3~

+ 912.

Let 8 be a root of Q ( s ;u ) and K = Q ( 0 ) . Then -8

-

1 belongs to

W ( 8 )= { a = a8+ b I N K / Q ( a )= 1,a,b E Q ) .

Cubic Fields and Mordell Curves

181

The rational point ( w ,u) = (1,-1) of E 4 ( s ) corresponding to -Q mapped to the rational point (x, y) = (0, 4(s2 - 3s

-

1 is

+ 9))

of i&(s). This is a torsion point of order 3. The proof may be obtained in a standard manner and is omitted (cf. Section 6 below).

6.

The short form of Eo(b)

In this final section, we show our main result on Eo(b)which corresponds to Proposition 5 on E l ( c ) . The author owes t o M. Imaoka for a portion of the calculation.

Theorem 1. (i) The short form of Eo(b)for b E Q - (0, -2714) M 7 ( B ) ,B = 4b/9 E Q - (0, -31, which is defined by the equation

is

(ii) M7(B) has a rational point

(zo,yo)

=

(B

+ 4, 3 B + 8).

This is not a torsion point unless either B = -4 or B = -813, and hence its Mordell- Weil rank is greater than or equal to 1 zf B # -4, -813.

+

(iii) The ezceptional curves M7(-4) : y2 = x3 16 and &I7(-813) : y2 = x3 - (4/3)3 have Mordell- Weil rank = 0. Their torsion groups are of order 3 and of 2, respectively, and generated by the point (xo,yo) (= (0, -4) and = (413, O), respectively).

Proof. (i) P u t W = w

- u.

Then we obtain

+ +

from the equation w 3 = u3 bu b. TOmake a perfect square out of the left-hand side, multiply both sides by 12W , and get

The constant term of the right-hand side is a square. Hence by multiplying both sides by b2/W4, we obtain

where we put

Then we utilize the method of Mordell [Mo-69, Ch.10, Th.21. If we put a = b 9, p = 9(b 6), and y = -3(b 9)2 then we have 3 a 2 y = 0 and -cry - P2 a3 = b2(4b 27). Therefore if we put

+

+

+

+

we easily obtain

+

+

+

y2 = x3 - (4b)2(4b 27). If we divide both sides of the equation by

then we have

Hence we obtain (i) of the theorem by replacing x/9 and y/27 by x and y, respectively. (ii) The former half is confirmed by direct calculation. We prove the latter half. P u t P = (xo,yo) for simplicity and suppose that it is of finite order. We know by Fueter [Fu-301 (also see [GPZ-98, Prop.11) that the torsion part of the rational points of a Mordell curve is cyclic and of order 1, 2, 3 or 6; furthermore, a rational point on it is of order 2 or 3 if and only if its y-coordinate is equal to 0 or its x-coordinate is equal to 0, respectively. We see immediately that yo = 0 if and only if B = -813; in this case, we have M(-813) : y 2 = x3 - (4/3)3 and the point P is of order 2. It is also clear that xo = 0 if and only if B = -4; in this case, we have M(-4) : y2 = x3 16 and the order of P is 3. Let us now check if P might be of order 6. Suppose that xo # 0 and yo # 0. The x-coordinate x ( 2 P ) of 2 P = P P on M 7 ( B ) is given by

+

+

cf. e.g. Silverman and Tate [ST-921. We know xo if and only if

# 0.

Hence x ( 2 P ) = 0

C,ubic Fields and Mordell Curves

and hence if and only if

Since the algebraic integer 7 B is contained in Q,it is a rational integer which divides the constant term -64 . 7 2 . Such a divisor is, however, unable t o make the cubic polynomial equal to 0 as is easily checked. Therefore we conclude that the point P is not of order 6. This completes the proof of (ii). (iii) It is sufficient t o show that the Mordell-Weil ranks of the curves M(-813) : y2 = x3 - (4/3)3 and M(-4) : y2 = x3 16 are equal to 0. The latter curve is on Table 1 of Cassels [Ca-501; as for the former, it is isomorphic to y2 = x3 - 33 . 26 which is also on the table. 0 The theorem is now completely proved.

+

References [Ca-501

J . W. S. Cassels, The Rational Solutions of the Diophantine Equation, = x3 - D, Acta Math. 82 (1950), 243-273. R. Fueter, Ueber kubische diophantische Gleichungen, Comm, Math. Helv. 2 (1930), 69-89. J . Gebel, A. Petho and f i . G. Zimmer, On Mordell's Equation, Comp. Math. 110 (1998), 335-367. y2

[Fu-301 [GPZ-981 [JLY-20021

C. U. Jensen, .4. Ledet and N. Yui, Generic Polynomials, Constructzve Aspects of the Inverse Galois Problem, Cambridge U.P., Cambridge, 2002.

[Mi-20031

K.Miyake, Some Families of Mordell Curves associated to Cubic Fields, Jour. Comp. and Applied Math. Sciences 160 (2003), 217-231.

[Mo-141 [r\/Io-691

L. J . hiordell, The Diophantine Equation y 2 - k = x3, Proc. London Math. Soc. 13 (1914), 60-80. L. J . Mordell, Diophantine Equations, Acad. Press, London and New York, 1969.

[Si-861

J . H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.

[ST-921

J . H. Silverman and John Tate, Rational Points on Elliptic Curves, Springer-Verlag, New York, 1992.

TOWARDS THE RECIPROCITY OF QUARTIC THETA-WEYL SUMS, AND BEYOND Yoshinobu Nakai Department of Mathematics, Faculty of Education and H u m a n Sciences, University of Yamanashi, Kofu, Yamanashi, 400-8510, Japan [email protected]

Abstract

In the van der Corput method, one of the three principal methods of exponential sums, one treats the vdC reciprocal function f * ( y ) = f ( x y ) - yxy ( f l ( x U )= y). In the case where f (x) is a polynomial of degree K , i.e. the Weyl sum, one encounters a situation similar to the elliptic transformation and the quadratic and cubic polynomial cases were successfully treated by the author. The main idea is to introduce the decomposition x = nK-' + m and think of n as the global variable of K the function f (n) = v c u x ~ - ' F ( ( ~ ) Then ). the I;-th vdC reciprocal function f'(y) given by (2.9) is essentially of the similar form to f (in terms of n1, y = n1K - l + m l ) and the length of interval of summation remains the same order, establishing the k-th reciprocation (under elliptic transformations). The behavior of f " under parabolic transformation is postponed for later researches. Instead, the inductive representation of f * is also given, with the concrete examples of the quartic case.

Keywords: the van der Corput method, the reciprocal function of van der Corput type with respect to the global variable n , theta-Weyl sums, Weyl sums, finite theta series, elliptic transformations 2000 Mathematics Subject Classification: 1lL15

Throughout in what follows we use the following Notation W - the set of all positive integers Z - the set of all integers R - the set of all real numbers e(J) = exp(2.j~-J) - the complex exponential function (( E R) N - a large variable, appearing as the limit of summation N t - a variable associated to N, 0 5 N t << N , used as the length of the interval of summation 185 Number Theory: Tradition and Modernization, pp. 185-204 W. Zhang and Y . Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

k - a non-negative integer used in the context of the k-th iterate of "theta reciprocation" - a (k 1) dimensional vector ( t ) = (tol tl,. . . , & ) E K - a fixed integer indicating the degree of the polynomial-like function f ( n ) a - the leading coefficient of f ( n ) , used as the variable of integration (in "circle method") For a real-valued function f with the positive second derivative, n y is defined by

+

f l ( n y ) = Y,

Y

( f l ( N ) ,f l ( N

+ N1)],

(ny E ( N , N

+ N1]),

f * = f *(Y)= f ( n y ) - yn, - the van der Corput reciprocal function in y (the vdC reciprocal function for short) For a differentiable function H = H ( ( E ) ) = H ( E o ,E l , . . . , E k ) and n > 0, let

G

=

G((E)) = 6

K-1

(F),

C ( H ) = (5,( H ( ( z ) ) - H ( 0 , 0, . . . , 0 ) ) , 1+X=

(K- l)a Y

x,,

with xy = nF-' + m

+

where m comes from the decomposition x = nK-' m with 0 5 m (n I ) ~ - ' nK-'. Q ( t ) = QK(t) is the unique function determined by (2.6):

+

(QK(t) - t)K-2 (QK(t)

+ ( K - 2)t) = 1

and

9 ~ ( 0 =) 1.

<

Towards the reciprocity of quartic theta- Weyl sums, and beyond

1.

187

Introduction

One of the fundamental tools in analytic number theory invented in the 20th century is the method of exponential sums (or sometimes referred t o as trigonometrical sums). There are three mutually related treatments of these sums known due to H. Weyl, I. M. Vinogradov, and J. G. van der Corput, respectively (cf. e.g. [2] or [10]). We are concerned with the third type of treatment, the van der Corput method, whose main ingredient is the following lemma.

+ +

Lemma 1. Let f (n) be a real-valued function o n [N, N N'] with the positive second derivative. For y E (f ' ( N ) - 0, f ' ( N N ' ) 0)) we define n, and f * as in Notation: f '(n,) = y and f * = f * (y) = f ( n y ) - yn,. T h e n we have

+

where y r u n s through integer values i n the mapped interval ( f l ( N ) f l ( N f N') 711 with 0 < 'Tj <

A.

+

'Tj;

Lemma 1 is often applied, with f a polynonlial (in which case the exponential sum is often called the "Weyl sum"), to an integral with respect to the leading coefficient of f up to a constant factor or sometimes applied repeatedly k times, i.e, the k-th iterate. I f f ( n ) = ~ a n K f o r K E N ( K > l ) ,a > O a n d N < n < N N' (<< N ) , then

+

and

< < (K - l)a(N+ N')~-'.

( K - l ) c u ~ ~ - y'

(1.2)

Hence for K = 2. we have and

f * ( y ) = - - a1 2

-1

y2 ,

so that both sums appearing in Lemma 1 are finite theta series and Lemma 1 may be thought of as claiming the theta reciprocation (i.e.

the elliptic transformation under the action of

( il))

sending n into

-a-'. This situation may be termed as the "theta-Weyl" sum, meaning that the Weyl sum admits a treatment as a (finite) theta series. Thus for K = 2 we have a (quadratic) theta-Weyl sum which we define by

with a slight change of notation, where ,6 > 0, t E R (cf. 161; see also [7], Chap. 2 and [I]). If P > 1, then decomposing P into ,6 = a PI-' with a = [p] (the integral part of P) and P I > 1, we have a relation between 0(P, t ;N , N N1) and 0 ( ~ ' - ' , tl; N , N N1) where t' admits a numbertheoretic expression. This corresponds to the parabolic transformation

+

+

+

(

)

;a , which we shall keep off in this of ,8 under the action of paper. In view of (1.2) we distinguish two cases /3'-'N1 << 1 or /3'-'N1 >> 1, and in the former case we appeal to another lemma of van der Corput while in the latter case, Lemma 1 in conjunction with (1.3) g'lves a 0-reciprocity-like formula between O(pl-', t'; N , N') and O(P1,t"; pl-' N, pl-I (N N')). We may then take it for granted that the quadratic theta-Weyl sum is satisfactorily treated, since all variables andparameters contained are interpreted in a good number-theoretic manner, in terms of the regular continued fraction expansion of p. The estimation of the error term being important in applying Lemma 1, we shall confine ourselves to the elliptic transformation of a , i,e, the K vdC function f * with the admissible error term O(x= N - ~ - ' ) and the y-interval (f l ( N ) , f l ( N + N1)]. However, (1.2) shows that the length of the y-interval is x N K P 2 N 1 , which is longer than N' if K 2 3, invalidating the nai've theta-series like treatment. It took the author quite a long time to overcome this difficulty and hit on the following natural setup: Viewing n as the global variable ranging over ( N , N + N 1 ] we write x = nK-'+m, where nK-' 5 x = n K - l + m < ( n I ) ~ - ' ,or 0 5 m < (n - nK-' << nKW2and we view m as a local variable. Then as n ranges over ( N , N N'] we have x x NK-I and as (1.2) implies (cf. also Lemma 2 below), y x N ~ - ' ,SO that they are well-balanced. \

/

+

+

+

+

189

Towards the reciprocity of quartic theta- Weyl sums, and beyond

With this in mind and with the purpose of incorporating the k-th reciprocation, we consider

where x = nK-' + m as above and F ( ( E ) ) = F ( E o , .. . , Ek) is a function in (k 1) variables, having ( K 1)-th derivatives for iEol, . . . , iEk1 << N-', and F ( ( 0 ) ) = 1. In what follows we let to,.. . , ti, be real parameters satisfying << N ~ ( h- = 0~, . . . , k), so that according to Notation, Z h = lies in the range lEhl << N-l.

+

+

%

2.

The van der Corput reciprocal function f *

We are to find the vdC reciprocal function f * to f given by (1.4). First we shall find ny in Lemma 1.

Lemma 2. Viewing n as a real variable (z= nK-I

a

+ m), we have

K-2

-f (n) = ( K - l ) a x ( l - Q)K-lG((E)), an where, as in Notation, Q = x-'m, Eh = x - 'th,(0

We may choose an ny E [N, N

(2.1)

< h 5 k), and

+ N'] such that

where we put nK-l

x ~ =y

+ m.

Proof. (2.1) follows by formal differentiation with respect t o n and K-2 -K-2 factoring out (K - 1 ) a x n x I < - 1 , thereby noting that nK-2

J:

-E K-1

= (l -

Q)E,

Secondly, as n and m range over the interval [N, N+N1] and [0, ~ ( n ~ - ~ ) ] , respectively, x range over ( N K 1 , ( N N ' ) ~ - ' )whence so does f ( n ) ,

+

&

190

Y.-N. Nakaz

+

because G = 1 O(N-l). Hence, for any given y x NK-I , we may find an n y E [N, N N'] satisfying (2.2). (2.1) and (2.2) together imply 0 (2.3), completing the proof.

+

Now appealing to Notation, we have the data (K-')"zy 1+ X , $ = = Hence (2.3) raised t o the the ( K - 1)-th power with these data reads

&,5 &.

a

Note that by (2.4))X << N-' and << N-l. Now as in Notation we introduce the unique function @(t) = QK(t) determined by the functional equation (\liK(t) - t)K-2(*K(t)

+ ( K - 2)t) = 1,

*K(0) = 1

(2.6)

and put

where T << N-l. We may now state

Lemma 3. Notation being as above, we have

+

a+

Proof. Since 1 X T satisfies (2.6) with t = T, (2.8) follows by 0 uniqueness of the solution. We may now state an expression for the vdC reciprocal function.

Lemma 4. The vdC reczprocal function f * of (1.4) is given in the first instance by

Proof. In the definition of f *,

Towards the reciprocity of quartic theta- Weyl sums, and beyond

we rewrit'e the second term as

I<

=

and factor out

YK - l ( ( K- 1 ) a ) - A ( 1

-T((K -1)a)-nyK-1

+ X - a)&

K

1

to obtain

after slight transformation. Using ( 2 . 5 ) we see that the second term

+

K(K-2)

8)"';

in the third factor become - & ( I Xcompleting the proof. 0 Although they are not in perfect conformity because of different processes, it is instructive and worth while reviewing our previous result for the cubic case [8]. First recall the notations in this special case.

The vdC reciprocal function f * of

where y = n:

+ m l , 0 5 ml 5 2n1, &N

+ N ' ) , and

5 nl 5 &(N

for U,V E R,U,V << N - l . This ( Q in Lemma 4) works as the Ek+l in the ( k 1)-st recipY rocation. We close this section by stating some properties of the function Q K ( t ) .

+

( K - 2)t Q, ( t ) ( K - 3)t ' K-2 K-3 Q,(t) = I + t2 - ( K - 2 ) . -t3 2 3 2K-5 K - 4 +(K-2).---t4 2 4 3K-7 K - 5 - ( K - 2 ) ( K - 3 ) . ----- .-t5 3 5 4 K - 9K - 6 t6 + ( K - 2) . 2K-7 . 3K-8 . 2 3 4 6 Q'K(t)=

(2.10)

+

. . -

+ ....

(2.11)

It seems that the coefficient of t K is 0. First a few functions are

and

3.

Inductive represent at ion for the vdC reciprocal function

To prove Proposition 1 below which provides an inductive representation for f *, we first establish Lemma 5 below giving the same for the variable xi,= n i P 1 m under the notation:

+

Towards the reciprocity of quartic theta- Weyl sums, and beyond

-

Sj

in general Aj-1 and

Proof, On writing

-

-

GK-l

-

1 X rn = 1 - rn, we have, by Taylor's formula,

C x

=:

a G K - ~ )((E)) (% =q1 -X sh2(- -G K - ~ )((E) (1 - 6-1 X 1S-x ashlaEh2 l+x

==h.

h

-X

are defined by

'

"

with suitable 0 < 6 < 1. Multiplying this by 1 and using the notation

we deduce that

)

)

+ X, subtracting 1 + X ,

-

Substituting the data X = O ( N P 1 ) , S h = O(NP1), and c(GK-~(E)) = O ( N P 2 ) ,we conclude that

which proves (3.2) in the case j = 0. Now suppose (3.2) is true for j 0. Then by writing 1 X = E 2 j +3 1 E ? + ~and 1 - -1 , and arguing l + Z , + ~ 2 ~ + 3 l+X3 ( I + Z ~ ) ( I + Z ~ + E ~ ~ + ~ ) E2j +3 as above (with X replaced by ), we obtain (1+Xj)(1+X,+E2j+3)

>

+Xi+

+

This reduces further, on account of the induction hypothesis, to

+

1 in place of j , Substituting (3.4) in (2.7)) we conclude (3.2) with j completing the proof. 0 Note that to apply Lemma with admissible error term, we must choose j so that 2 j 3 2 K 1, or j :(K - 1); hence for K = 3,4, we must 1. take j We are now in a position to state the main result of the paper.

>

+

+

>

Theorem 1. The vdC reciprocal function f * of f given b y (1.4) admits the inductive representation with admissible error term:

Towards the reciprocity of quartic theta- Weyl sums, and beyond

195

> max(1, y) and for the (k + 1)-st reciprocation y is to be decomposed as y = nlK-I + m l (as in (2.4))) ( ( K - 1 ) a ) A N < n1 5

where j

+

"-'

+

((K - l)a)&(N N') and n l 5 nlK-I m l < ( n l 0-constants not depending on N , m , Eh (h = 0,1, . . , k ) .

+ I)"-',

the

The proof is elementary but long, and we shall not state it here. Instead we shall prove Proposition 1, which gives an exact formula for f * ( y ) as well as an induction representation with a weaker error term: Proposition 1. The vdC recportial function f * admits the exact representation

Moreouer, (3.5) holds with 0 ( ~ - ( ~ 7 + ~ ) if) ,j 2 max (I,

v).

Proof. (3.6) follows from Lemmas 4 and 5, and the second assertion, (3.5) with weaker error term, follows from (3.6) by order estimates of E2j+3and E2(j-l)+3 in Lemma 5: (2.6) and the fact that F - GK begins with the second order term, which in turn is a consequence of

This completes the proof. In applying Theorem 1, the dependence of the implied constant on a , k , j, F, K, etc, must be made explicit, and we shall do it elsewhere.

We close this section by stating some formulas on whose grounds the intermediate order term cancel each other in the proof of Theorem 1.

Lemma 6 . We have for j

2 -1

and

-

Proof. (3.8) follows from Lemma 5 for j and (2.10). To deduce (3.10), we first derive

2 0.

(3.9) follows from (2.6)

The left-hand side of (3.10) is equal to

which turns out to be equal to the right-hand side of (3.10) by (3.11).

4.

Refinement of the expression and the quartic Theta - Weyl sums By the same reasoning as used in the proof of Lemma 5, we may prove

Towards the reciprocity of quartic theta- W e y l s u m s , and beyond

Lemma 7. For j

Lemma 8. For K

> 1,

> 4 and j > 1 we have

Using these and more elaborate argument, we may prove the following refinement of Theorem 1. Proposition 2.

For j 2 1 we have

198

Y.-N. Nakai

Remark 1. (i) What we reported in the China-Japan seminar in Xi'an is

If we apply Lemma 8, this becomes (4.1) with j = 1. (ii) If we need the error term o ( N - ( ~ J + ~ or) o ) ( N - ( ~ J + ~then )), j 2 1 or j 2 is sufficient, respectively. (iii) We expect to obtain the expression of f * with the error term o ( N - ( ~ + ~for ))K 5 in which case the "main term" is of the form suggested as in the following Theorem 2. Hereafter, we shall dwell on the proof of the following theorem which is the case j = 1, K = 4 of Proposition 2 with the "main term" expressed in more symmetrical form.

>

>

Theorem 2. T h e udC reciprocal f u n c t i o n of

+ a)

and w = +(-I where F = F((E)) and p = a(l primitive 6 - t h and 3-rd root of 1, respectively.

+ G)are

Towards the reciprocity of quartic theta- Weyl sums, and beyond

Proof. Since for j

-

=

1, the term containing Grc

K-1

199

disappears from (4.1)

on account of X-l = 0, it is enough to express the main term

in the form given above. We appeal to the Taylor expansion

+

which we wish t o express as c:=~ ahcl,th O(t6). Using the orthogonality of c:=~ p'j for 1 E Z, we further decompose

where

5

A~ =

C ahpphil

5

Bi (t) =

h=O

C cn(plt)'. k=O

We determine ck as the Taylor coefficient of (1 + t);, i,e.

whence

+

Bj(t) = (1 $ t ) i

+ 0(t6)

(4.6)

We determine ah's by equating (4.3) and (4.4), therewith substituting the values of c k : a0 = l , a l = -1,a2 = -2,as = -1,a4 = 1 + g9 , a 5 to be specified presently. Substituting these values, we get, after some transformation,

Choose as = 2 and classify the values of j mod 6 t o obtain

Noting 1 - p

-

2p2 = 3p, we see that mod 6),

and similarly for j E 5 mod 6 . Substituting (4.6) and (4.7) in M4(t) = deduce that

A c : = ~ A ~ B ~ (+~O(t5), ) we

c:=~+

d ( l wit2): with an Finally, replacing the term &t4 by -$ error O(t5), we conclude that M4(t) is of the form given in Theorem 2. 0 Remark 2. (i) In order to treat general quartic polynomials, it is enough to consider the function of the form

where cl and cz are constants, L ( ( J ) )is a linear function in (I)and F2 is a similar function as F. This will be studied elsewhere. (ii) As we remarked after the proof of Proposition 1, we need to make the dependence of error terms on parameters, especially on a , explicit for practical applications of Theorem 2 (like Theorem 1). To be entitled to call our process theta-Weyl sum, we need to study the parabolic transformation ([8, $61) of the "k"-ic continued fraction expansion ([8,$21). These tasks are to be conducted elsewhere.

Appendix: Three methods for treating exponential sums

x

e2""(") or Weyl sums X
The Weyl-Hardy-LittlewoOd method

e2"2"f(n)

X
The van der Corput method

The Vinogradov method

Estimates from above using the mean value thorem Inequalities

Schwarz inequality; triangular inequality

1

< f ( ~ ) g ( x ) d x= .fix) x g(z)dz

Hijlder inequality (both for sums and integrals); double sum methods

in integral calculus (f(x) is positive and monotone decreasing)

Euler's summation formula (Poisson formula in finite form), esp., tools

differencing (polarization in case of polynomials)

c 00

< z >= -

v=1

sin 27ruz ------, 7ru

the saw-tooth Fourier series

the most effectively applied objects

Wcyl sums

sums with F such that F" is of constant sign, or F' is monotone

Newton formula for the relation between power sums and fundamental symmetric functions; the number of integer-solutions of a system of linear equations as L1norm

Weyl sums with the core polynomial having real coefficients; also effective for exponential sums

contd. on next page

results obtained first

alteration of summation intervals at each step of application

local type

inside I(- - . ) 1, the interval remains invariant, with summation over consecutive integers;outside I (. . . ) 1, the length of interval increasing geometrically, where I (. . . )I means the inner sum of

E

,

K

-

1)

(K

local type

mean value type

changes into F f ( t h e previous interval) , summation is extended over consecutive integers

inside I(. . . )I, the length of intervals changes from X into x ' - " ~ outside, the length of intervals increases of order xl'" . (. . . ) , where (. . . ) means a similar sum as in W-H-L; division into equal subintervals (Vinogradov) or selection as arithmetic progressions (Linnik-Karatsuba), both putting subintervals together using parallel translation of subintervals

the order of F" in the form

upper bounds for

+

H l)crzKpHy, yH)l and outside means the sum over yl, . . . , YH in the case f = azK

estimate of Diophantine approximation used (all linear type)

upper bounds for

IFI1(x)I x X on [X,X

+ N]

(in paractice, use is made of I F ( ~ + ' ) ( Z )=: ~ X K + ~using differencing)

~ F ( ~ + ' ) ( z=c ) l X K f l on [X,X

+

NI

contd. on nexf. page

T

k

Towards the reciprocity of quartic theta- Weyl sums, and beyond

203

References [I] H. Fiedler, W . Jurkat and 0. Korner, Asymptotic expansions of finite theta series, Acta Arith. 32 (1977), 129-146. [2] S. W . Graham and G . Kolesnik, Van der Corput's method of exponentzal sums, London Mathematical Society Lecture Note Series 126, Cambridge Univ. Press, London, 1991. [3] J.-I. Igusa, Lectures on forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 59, T a t a Inst., Bombay, 1978. [4] T . Kubota, On an analogy to the Poisson summation formula for generalised Fourier transformation, J . Reine Angew. Math. 268/269 (1974)) 180-189. [5] W . Maier, Transformation der kubischen Thetafunktionen, Math. Ann. 111 (1935), 183-196. [6] Y.-N. Nakai, On a 8-Weyl sum, Nagoya Math. J . 52 (1973), 163-172. Errata, ibid. 60 (1976), 217. [7] Y.-N. Nakai, On Diophantine znequalitzes of real zndefinite quadratic forms of additive type in four variables, Advanced Studies in Pure hlathematics 13, (l988), Investigations in Number Theory, 25- 170, Kinokuniya Compa.LTD., Tokyo, Japan. (This series is now published by Math. Soc. Japan). [8] Y.-N. Nakai, A penultimate step toward cubic theta- Weyl sums, Number Theoretic Methods, Future trends, ed. by S. Kanemitsu and C.-H. Jia, (2002), 311338, Kluwer Acad. Publishers. [9] W . Raab, Kubischen und biquadratische Thetafunktionen I und 11, Sizungsber. ~ s t e r r e i c h Akad. . Wiss. Mat-Natur. K1. 188 (1979), 47-77 and 231-246. [lo] E. C. Titchmarsh, The theory of the Riemann zeta-function, Oxford Univ. Press,

1951, 2nd ed. 1986 (edited by D.R. Heath-Brown).

EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 210093, People's Republic of China [email protected]

Abstract

In this paper we establish some explicit congruences for Euler polynomials modulo a general positive integer. As a consequence, if a , m E Z and 2 f m then mk+l

a -l aE X ) z for every i= 0 . 1 . 2 , . . ., 2 2 which may be regarded as a refinement of a multiplication formula.

-E"(m

+

Keywords: Euler polynomial, Congruence, q-adic number 2000 Mathematics Subject Classification: Primary l l B 6 8 ; Secondary l l A 0 7 , llS05.

1.

Introduction

Congruences for Bernoulli numbers have been a very intriguing objective of research since the time of L. Euler, and they recently got revived in connection with p-adic interpolation of L-functions. Congruences for Euler numbers, being cognates of Bernoulli numbers, have also received much attention from the same point of view of p-adic interpolation. In [S4]the author determined Euler numbers modulo powers of two, while Euler numbers modulo any odd integer are essentially trivial. As a natural further step, we are led to consider congruences among Bernoulli and Euler polynomials, the latter of which will be our main concern in this paper. We prove the integrity of coefficients of f k ( x ;a, m) (defined by (1.7)))which are related to the summands in the multiplica-

Supported by t h e National Science Fund for Distinguished Young Scholars (No 10425103) and t h e Key Program of NSF (No. 10331020) in China.

205 Number Theory: Tradition and Modernization, pp. 205-218 W . Zhang and Y . Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

206

Z.- W. Sun

tion formula (1.6) for Euler polynomials, and establish number-theoretic generalizations thereof (Theorems 1.2 and 2.1). Hereafter, the labelled formulae with star indicate those known ones which have their counterparts for Bernoulli or Euler polynomials. Hopefully, these will serve also as a basic table for these polynomials (for more information, the reader is referred to [AS], [El, [Sl]). In referring to them, we omit the star symbol. Euler numbers Eo,El,E2,. . . are defined by

It is well known that they are integers and odd-numbered ones El, ES,E5, ' are all zero. For each n E No = {0,1,2, . . .), the Euler polynomial En(x) of degree n is given by e

n-k

k=O Note that

En = 2 n E n ( l / 2 ) . Here are basic properties of Euler polynomials:

and

From now on we always assume that q is a f i x e d i n t e g e r g r e a t e r t h a n one, and let Zq denote the ring of q-adic integers (see [MI). For a , p E Z q , by a = p (mod q) wemean that a + = qy for some y E Zq. A rational number in Zq is usually called a q-integer. In this paper we aim at establishing some explicit congruences for Euler polynomials modulo a general positive integer. We adopt some standard notations. For example, for a real number a , jaJ stands for the greatest integer not exceeding a, and {a) = a - La] the fractional part of a, (a, b) the greatest common divisor of a , b E Z, and A ( P ( x ) ) the difference P ( x 1) - P ( x ) of a polynomial P ( x ) . For

+

207

Explicit congruences for Euler polynomials

a prime p , n E No and a E Z , we write pnlla if pn / a and pn+l 1 a. For a , b E Z \ { 0 ) , by a ~2 b we mean that both 2nlla and 2nIj b for a common n E No. For convenience we also use the logical notations A ( a n d ) , V ( o r ) , # ( i f a n d o n l y zfl, and the special notation:

=

1 if A holds, 0 otherwise.

By ( 1 . 1 ) and ( 1 . 2 ) it is easy to get

1 2 E o ( z ) = 1 , E l ( x ) = x - - and E a ( x )= x 2

-2,

and verify that the polynomial

has integral coefficients for k = 0 , 1 , 2 . This phenomenon is not contingent but universal as asserted by the following general theorem.

+

Theorem 1.1. F o r each k E No a n d a , m E Z w i t h 2 m, w e h a v e

This result is remarkable in that (1.6) can be expressed as the vanishing arithmetic mean

Theorem 1.1 follows from the following more general result whose proof will be given in Section 2 .

Theorem 1.2. L e t k E N o , d , m E N a n d d I m. L e t c be a real n u m b e r , a n d let P ( x ) d e n o t e t h e polynomial

208

Z.- W . Sun

T h e n P ( x ) E Zq[x]. Furthermore, if q is odd, t h e n q- 1

P(x)2

(-l)j (x

+j m ) k

(mod q);

(1.8)

j=O 2$(d-l)j+ly]

if q i s even, t h e n

+

1;

-[41 d + 1 ] ( A ( x k ) + [4/ q ] A ( x k - l ) ) ( m o d q ) i f 2 2 k m ,

N o w we derive various consequences o f T h e o r e m 1.2. Corollary 1 . 1 . Let k E No and m E N,and let x be a q-integer. If q i s odd, t h e n

If q i s even, t h e n

[ ![4

1 m+1

A (k = 1 V 2

1k

V 2 l / q ) ]( m o d q )

otherwise,

209

Explicit congruences for Euler polynomials

Proof. Just apply Theorem 1.2 with c = x and d = m , and note that 0 = s x (mod q) if k > 0 (cf. [S3, Lemma 2.11).

izk

Corollary 1.2. Let a E Z, k E No, m E Z+, 2 1 q and ( m , q) = 1. Then

+ ![4

1 m + l] ( a ( x k ) + [2 ( k

A 4 1 q ] ~ ( x " l ) ) (mod q).

Proof. Clearly 2 { m and ~ ( ( x + ak ,) - n ( x k ) E 2Z[x]. Applying Theorem 1.2 with c = a , d = m, and x replaced by x a , we obtain the desired 0 congruence.

+

Proof of Theorem 1.1. Suppose first that m

> 0.

Then by Corollary 1.2,

If p is an odd prime, then E k ( x ) / 2 E Zp[x], and also by (1.5)

Thus every coefficient of the polynomial f k (x) = f k (x; a , m) is a p-integer for any prime p, which amounts to f k ( x ) E Zjx]. For the negative modulus case (-m > 0), using

and 2 1 m, we may express mk+l 2

--Ek

(

x

fk

(x; a , -m) as

+ (a + m ) )

+

Ek (4.

Thus the positive modulus case applies and the proof is complete. In the spirit of Sun [S3],Theorem 1.2 can also be used to deduce some general congruences of Kummer's type for Euler polynomials. However, in order not to make this paper too long, we will not go into details.

210

2.

Z. - W. Sun

Proof of Theorem 1.2

We introduce the Bernoulli polynomials B n ( x ) (n E No) by the generating power series

Their values B,(O) a t x = 0 are rational numbers, called Bernoulli numbers and denoted by B,; it is well known that Bzk+l = 0 for k = 1 , 2 , 3 , .. .. Raabe's multiplication formula (counterpart of (1.6)) reads m-1 x r (2.2). mn-' B, (--nL) = ~ , ( x ) for any m E I"].

C

+

r=O

Other properties include (2.3)* and (2.4)* t'he last one links the Euler and the Bernoulli polynomials

Lemma 2.1. Let k be a positive integer and y be a real number. Then

Proof. Observe that

Hence, if 2 1 [y], the right hand side of (2.5) is

which coincides with the left hand side of (2.5) in view of (2.4). Now, if 2 f Ly] , then the right hand side of (2.5) is

+

(T)

(q)

We may express Bk x+{Y)+' as 21pkBk(x {y}) - B~ by (2.2). x+{Y> Then what remains is - 2 ( 2 k ~ k - Bk(z {y))) which equals the 0 left hand side of (2.5). This completes the proof.

(T)

+

2 11

Explicit congruences for Euler polynomials

Lemma 2.2. Let k E N o and m E ;Z \ (0). Then ( k Z j x ] . Furthermore,

+ l ) m k~ ~ ( x / mE )

Proof. First note that by (2.4),

for any L E No. Hence, if 1 is even then

while for odd I , we see that in the expression

the third and fourth terms are integers by the von Staudt-Clausen thepp. 233-2361) and Fermat's little theorem, respectively, so orem (cf. [IR, that (1 l ) E l ( 0 ) is an odd integer. as (1 1) we find that Using (1.5) and writing ( k 1)

+

lies in Z [ x ]and that

Now, since

+ (F)

+

(fz:),

2.- W . Sun

the third term lying in 2Z[x],we conclude that what we should subtract ( g ) is from (k 1 ) m ' " ~ k

+

=

zk

+ (x

-

km)

(x

+ m)k - xk , m

as asserted, and the proof is complete.

Lemma 2.3 ([S3, Theorem 4.1)). Let k E No, d, m, n E N, d 1 n, m. 1 qn, and 2 { d o r 2 f q or 2 1 $. Put d = (d,qn/m) and m = (m,qn/d). Then, for any real number y , the polynomial

,is in Z 4 [ 2 ]and is congruent to

modulo q . Now we establish a result more general than Theorem 1.2 ((2.9) and (2.10) below are generalizations of (1.8) and (1.9), respectively).

Theorem 2.1. Let k E No, d, m, n E N, d I n, m I qn, and 2 ij d or 2 { q or 2 1 Put d = (d, qnlm) and 6 = ( m ,qnld). Then for any real number y we have

z.

Explicit congruences for Euler polynomials

Moreover, zf q n l m is odd then

+ ![d 2

yn A d

$ 0 , l (mod 4)]kxk-' (mod y); (2.9)

if q n l m is even then

where

-

+

i [ d w 2 n A d $ O , 1 (mod 4)](A(xk) 14 1 y ] ~ ( x " ~ ) ) 2/2 1 km,

4 [ d w 2 n]([d $ 0 , l (mod 4)]

+ [2 1 n

A 2/(yn/m)])xk-1

zf 2 + k ( m - 1).

+

Proof. We observe that the (k 1)-degree terms in (2.7) cancel each other in view of d l m = 21%. Hence L ( x , y) is of degree a t most k. Writing

we see that 2"(1, fortiori that

and

Y)

E Zq[x], and similarly 2"($,

3) E Zg[x], and a

214

Z.- W. Sun

By (2.7) we can express the left hand side of (2.11) in such a way that we may apply Lemma 2.1 to deduce that

On the other hand, the right hand side of (2.11) is congruent to

modulo q, where

+ 12 f m

A 21n A 21q]~(x'-'))

(2.13)

and

Hence R(z,y) - 2k'1~

= C+r(x)

(mod q ) .

(2.15)

By the counterpart of (2.6), the second term on the right hand side of (2.14) becomes

in which we shall divide the sum into two parts via midpoint. Then

Explicit congruences for Euler polynomials

Whence, writing

for j = 0,1, . . ., we obtain

Recombination of terms yields

where

216

Z.- W . Sun

Thus, in view of the equality [2 1/ j] - [2 1 deduce that

-

j] = [2 / :](-l)j-',

we

Writing

and applying (1.4) and (1.5) successively, we obtain

whence separating the term with 1 = 0,

=0

>

2, so that if

x ( [ l = 11 + [l = 2 A 2 f n A 21Iq]) (mod q),

(2.20)

Now, by [S3, Lemma 2.11, ql-'11 0

(mod q) for 1

< 15 k, then

=

2

[d -2 n A d $ 0 . 1 (mod 4)]

and in particular, only two terms with 1 = 1 , 2 appear on the right hand side of (2.19) modulo q. Hence the sum on the right hand side of (2.19)

Explicit congruences for Euler polynomials

is congruent t o

modulo q. Thus, by Lemma 2.2, C1 is congruent to

modulo q. By (2.20) with 1 = 1 , 2 , the second term of the above expression is congruent to -(-l)qnlmT(x) modulo q, where

+

(x m ) k - x k 4 ~ ( x =) [d n A d $ 0 , l (mod 4) A 2 1 k] 2 m + s [ d - 2 n A d g 0 , l (mod 4) A 2 f k j 2 x ( x mlk-l - (x m)xk-I X m (x m)k-l - xk-l A 21kn A 4 1 d + l ] . (2.21) +![2)q m Thus qn d-1 C1 r -[2 -]mkEk (2.22) F(x) (mod q). 2 m Now, from (2.11)-(2.13), (2.18) and (2.22), it follows that

+

+

+

(c) +

2.- W . S u n

+

With the help of the binomial theorem, we can easily verify that r ( x ) F(X) = Rk(x) (mod q ) . If q n l m is odd, then either 2 q or 2 1 m, whence

4 R k ( x ) = - [d N 2 n A d $ 0 , l (mod 4)]kxkP1 (mod q). 2 and the desired results follow.

0

Proof of Theorem 1.2. Just apply Theorem 2.1 with n = m and y cmld.

=

References hf. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. A. Erddyi, W.Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions I, McGraw-Hill, New York, Toronto and London, 1953.

K . Ireland and hI. Rosen, A Classical Introduction to Modern Number Theory (Graduate texts in mathematics; 84),2nd ed., Springer, New York, 1990. K. Mahler, Introduction t o p-adic Numbers and their Functions, Cambridge Univ. Press, Cambridge, 1973. Z. W. Sun, Introduction to Bernoulli and Euler polynomials, a talk given a t Taiwan, 2002,http://pweb.nju.edu.cn/zwsun/BerE.pdf. Z. W. Sun, Combinatorial identities i n dual sequences, European J . Combin.

24(2OO3), 709-718. Z. W. Sun, General congruences for Bernoulli polynomials, Discrete Math. 262(2OO3), 253-276. Z. W. Sun, O n Euler numbers modulo powers of two, J . Number Theory, 2005, in press.

SQUARE-FREE INTEGERS AS SUMS OF TWO SQUARES Wenguang Zhai School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong, 250014, P.R. China [email protected]

Abstract

Let r(n) denote the number of representations of the integer n as a sum of two squares, p(n) the Mobius function and P(x) the error term p(n)lr(,n).In this of the Gauss circle problem. Let Q(x) := short note we shall prove that if the estimate P(x) = O(zs)holds, then Q(x y) - Q(x) = A y ~ ( y x - ~ 'x"~), ~ where A is a constant. In particular this asymptotic formula is true for 8 = 131/416. Our result improves Kratzel's previous result.

+

+

+

Keywords: Gauss circle problem, square-free number 2000 Mathematics Subject Classification: 1lN37

1.

Introduction

>

Let r(n) denote the number of representations of the integer n 1 as a sum of two squares and p(n) the Mobius function. The celebrated Gauss circle problem is to determine the smallest exponent a for which the estimate for the error term P(x):= C,<,r(n) - nx = O(xa+&) holds for every E > 0. It was Gauss who proved that P(x)= 0 ( x 1 P ) . The exponent 112 was improved by many authors. The latest result is due to Huxley [2], who proved that

It is conjectured that a = 114. For a historical survey on the circle problem, see E. Kratzel [6]. T h i s work is supported by National Natural Science Foundation of China (Grant No. 10301018).

Number Theory: Tradition and Modernization, pp. 219-227 W. Zhang and Y . Tanigawa, eds. 02006 Springer Science Business Media, Inc.

+

220

W. Zhai

Since lp(n)l is the characteristic function of the set of square-free integers, the function jp(n)lr(n) is the number of representation of a square-free integer as a sum of two squares. Let 1 f ( 4 = 1-44l r b ) . We have (for a

> 1)

where PI denotes the set of all primes which are congruent to 1 modulo 4. Then for the summatory function Q(x) of Ip(n) lr(n), K.-H. Fischer [I] proved that

where

A = Res F ( s ) . s=l

The exponent 112 in (1.3) cannot be reduced with the present knowledge of the zero-free region for ((s). E. Kratzel [5] studied the short interval case, and proved that if

and D 3 ( 2 ) :=

d3(n) = x(cl log2 x

+ c2 logx + c3) + 0 ( x 6 )

(6

< 112)

n_<x

(1.5) with some constants el, c2, c3, then the asymptotic formula

holds for With the present best known estimates 6 = 4 3 / 9 6 + ~ ,0 = 1 3 1 / 4 1 6 + ~ (see Kolesnik [4] and Huxley [2], respectively), (1.6) is true for y xo.4501...+~ . since 6 113, 8 114, the limit of E. Kratzel's method is Note that 43/98 = 0.4387. . . and 1311416 = 0.3149. . . . y In this short note we shall use the convolution method to prove the following theorem.

>

>

>

>

221

Square-free integers as sums of two squares

Theorm. Suppose (1.4) is true for some 114 < 8 < 113, then we have

Corollary. The asymptotic formula (1.6) is true for z'31/416+2E -Y< <

x. Notations. Throughout this note, E always denotes a fixed sufficiently small positive constant. For any fixed integers 1 a1 5 a2 5 . . . 5 ak,

<

d ( a l , a?,

. . , ak; n ) :=

1, d k ( n ):= d ( l , l , . . , I ; n ) . R=nal

..,

;k

[ ( s )is the Riemann zeta-function, and L ( s , X ) is the Dirichlet L-function associated to the non-principal characters x mod 4.

Proof of Theorem

2.

We recall from (1.2) that

where f (n)= i l p ( n ) l r ( n ) . The following lemma plays a crucial role in the proof. Lemma 1. Suppose Rs

> 1. Then we have

where [ ( s ) is the Riemann zeta-function, L ( s ,X ) is the Dirichlet Lfunction, respectively as above and M ( s ) is a certain Dirichlet series which is absolutely convergent for Rs > 115. Proof. By the well-known Euler product representations, we have

and

222

W. Zhai

where P3 signifies the set of all primes which are congruent to 3 mod 4. Hence

Now for Jul < 112, we note that

with E ( u ) = 1

x

+ O(u5). Hence we may rewrite (2.1) as

n

(1 - p-37-2 M~( s ) ,

PEP1

npEPl

where M l ( s ) := E(pPS): which has a Dirichlet series expansion, absolutely convergent for Rs > 115. Proof of (2.2) amounts t o substituting (2.5) (its powers) and replace the infinite products by (2.3) and (2.4). In fact we have

and

Square-free integers as s u m s of two squares

Hence we get (2.2) with

which has a Dirichlet series expansion, absolutely convergent for R s > 115. 0

By Lemma 1 , we have for R s

> 1,

F ( s ) = Fl ( s )F2 ( S ) F3 ( 3 ),

(2.10)

where 00

Fl ( s ) =

Cn

fl(n)-

[ ( s ) L ( s ,X ) ~ - 1 ( 4 s ) ~ - 2 ( X4 s)7~ ( s )

(2.11)

n=1

and

Then we have

Lemma 2. If (1.4) holds, then we have

where

A1 = Res F l ( s ) s= 1

Proof. We introduce the notation which we will use in this proof only.

and

W. Zhai

Then

h ( 4=

C fdm)0(1),

P(n) =

C

?(m)a(l)

and

By Perron's formula we see that

Hence

by partial summation, (1.5), and (2.15). Now we may appeal to Ivii [3] Theorem 14.1 to conclude (2.14). Or we may directly apply the hyperbola method as follows.

225

Square-free integers as sums of two squares

Since the generating function for f l (s) is F l ( s ) having a simple pole at s = 1, the main term i n c x must coincide in view of Tauberian argument 0 or Perron's formula: p1 c = A1 = Res,,~ F l ( s ) . Finally we prepare a lemma for estimating the error term.

Lemma 3. Let k and

> 2 be a fixed

integer, 1


< x be large real numbers

xxE

Then we have A(x, y; k , E )

<< yx-"

+ x1I4.

(2.16)

Proof. Without loss of generality, we may suppose k = 2. It is clear

where d ( l , 2 ; n)

D ( 1 , 2 ;u):= n
and

From Richert [7] we have

which we apply with the error estimate 0 ( x 1 I 4 ) . For

Lemma 3 follows from the above formulas.

Now we prove our Theorem. By (2.10) p(2.13))

Co,we have

W. Zhai

and

fi(n)

<< n E 2 ,i = 1 , 2 , 3 .

(2.17)

Hence we have

where

By Lemma 2 we have

where as in the proof of Lemma 2 , the coefficient of the main term

must coincide with A = Res,=l F ( s ) . Also

Square-free integers as sums of two squares

227

on writing nlni = n. Since d ( l , 3 ;nz) << rn"' we have by Lemma 3

Similarly, we have Now our Theorem follows from (2.18)-(2.21).

Acknowledgments The author deeply thanks Professor Kanemitsu, who kindly read the manuscript of the paper and proposed many helpful suggestions.

References [l] K.-H. Fischer, ~ b e rdie Anzahl der Gitterpunkte auf Krezsen mit quadratfreien Radien-quadraten, Arch. Math. 33 (1979), 150-154.

[2] M.N. Huxley, Exponential sums and Lattice points 111, Proc. London Math. SOC.87(3) (2003), 591-609.

[3] A. IviC, The Riemann zeta-function, John Wiley & Sons, New York, 1985. [4] G . Kolesnik, On the estimation of multiple exponential sums, in Recent Progress in Analytic Number Theory, Symposium Durham 1979 (VoLl), Academic, London, 231-246. 151 E. Kratzel, Squarefree numbers as sums of two squares, Arch. Math. 39 (1982), 28-31. [6] E. Kratzel, Lattzce poznts, Deutsch. Verlag Wiss. Berlin, 1988. [7] H.-E. Richert, ~ b e dze r Anzahl Abelscher Gruppen gegebener Ordnung I , Math. Z. 56 (1952),21-32; 11. ibid.58 (1953), 71-84.

SOME APPLICATIONS OF L- FUNCTIONS TO THE MEAN VALUE OF THE DEDEKIND SUMS AND COCHRANE SUMS * ZHANG Wenpeng Department of Mathematics, Northwest Unzversity X i 'an, Shaanxi, P. R. China [email protected]

Abstract

In this paper, the applications of L- functions to the mean value of Dedekind sums and Cochrane sums are described, and a few asymptotic formulae are presented.

Keywords: L- functions, Dedekind sums, Cochrane sums 2000 Mathematics Subject Classification: 1111120; llF20

1.

Mean value properties of L-functions

>

In the following we let q denote a fixed modulus, q 2 and x a Dirichlet character mod q. The Dirichlet L-function L ( s , X ) associated to x is defined by OC,

n=l

absolutely convergent for 8s > 1 if x = xO,the principal character mod q, and convergent for 8 s 0 if x # xO. They play a very important role in number theory, e.g. in the proof of Dirichlet's prime number theorem, in Dirichlet's class number formula, in the solution of D. H. Lehmer's problem on the parity of residues ( [12]) and so on. We summarize the mean value results on L-functions (for odd characters, x(-1) = -1) in $1 and the corresponding mean value results on Dedekind and Cochrane sums in 52 and 53, respectively, as the consequences of the results in 51, in conjunction with close relationships

>

*This work is supported by t h e N.S.F. (10271093, 60472068) of P.R.China

229 Number Theory: Tradition and A.;'odernization, pp 229-237 W. Zhang and Y. Tanigawa, eds. 0 2 0 0 6 Springer Science Business Media, Inc.

+

230

W. Zhang

between L-functions and the afore-mentioned sums, to be stated in the respective section. We shall focus on achievements of our own school. Let e(y) = e2"'y and let G(x, n ) denote the Gauss sum

for n

> 1 and x mod q. For n = 1 we write

Define the saw-tooth function ((x)) by

((XI>=

x-

[XI

-

1 2

-,

if x is not an integer, if x is an integer,

[XI

with denoting the integer part of x. For any primitive character x modulo q, H. Walum [lo] established the identity

-

where X(n) = ~ ( n means ) the complex conjugate of ~ ( n For ) q = p (a prime), he deduced the beautiful exact formula

For general q, S. Louboutin [8] and the author [13] obtained the following generalization of (1.2): Proposition 1.1.

where $(q) is the Euler function.

The mean value of the Dedekind sums and Cochrane sums

231

There are many mean value theorems on the Dirichlet L-functions. We state some of them. First,

XE

Proposition 1.2 ([4]). Let u and v be integers with (u, v) = d 2 2 , and X: be the principal characters modulo u and v respectively. T h e n we have for Rs 2 1

x mod d x(-I)=-1

and

x mod d x(-1)=1

The special case s = 1 is given in [14] Secondly, Proposition 1.3 ( [ 5 ] ) . Let q = uv,where (u, v) = 1, u be a squarefull n u m b e r o r u = 1, v be a square-free number. T h e n for a n y positive integers n and m we have

and

232

W. Zhang

where

x

x*

m e a n s the s u m m a t i o n over even o r odd primitive char-

mod m x(-l)=&l

acters mod m respectively. Proposition 1.4 ([18]). Let q the asymptotic formulae

2 3 be a n odd modulus. T h e n we have

x mod q

2.

Mean value of Dedekind sums For any integer h , the Dedekind sum is defined by

whose reciprocity law amounts to the transformation formula for the Dedekind eta function [9]. In the same paper [lo] referred to above, H. Walum showed that for a prime p,

which suggests the existence of some connections between Dedekind sums and L-functions. J. B. Conrey, E. Fransen, R. Klein and C. Scott [3]studied the higher power mean value of the Dedekind sums and proved the following by elementary methods:

The mean value of the Dedekind sums and Cochrane sums

where

233

XI denotes the summation over h relatively prime to q, ( h , q ) = 1, h

and f,(k)

is defined by the Dirichlet series

( ( s ) denoting the Riemann zeta-function. There are improvement over ( 2 . 2 ) , due to Chaohua Jia [2]in the case m > 1 and due to the author [14]in the case m = 1, the latter reading

with pa 1) q indicating that p a J q but pa+l J q . This depends on the following identity [13] expressing the relation between the L-functions and the Dedekind sums, alluded t o in 51:

x(-l)=l

Moreover, let

which is related to the Dedekind sum through s z ( h ,q ) = 2 s ( h ,q / 2 ) - s ( h , q )

for even q. By this identity, we were able to obtain an analogue of ( 2 . 3 ) : Proposition 2.1 ([16]).Let q = 2PIkl with ,O have

2 1 and 2 V M .

T h e n we

234

W. Zhang

Huaning Liu [6, 71 obtained asymptotic formulae for the 2m-th power mean value of sums analogous to Dedekind sums. Furthermore, we have obtained the following results suggesting the relationships between Dedekind sums and the Hurwitz zeta function f) and those between the sum s l ( h , q ) and the Ramanaujan sum Rq( h ), respectively:

~(1,

Proposition 2.2 ([15]). For q 2 3 and a n y fixed positive integer m, we have the asymptotic formula

(

+ 0 q2m+l/2 exp

Proposition 2.3 ([19]). For q totic formula

3.

(2)) . In In q

> 3 a n odd modulus, we have the a s y m p -

Mean value of Cochrane sums

For an arbitrary integer h, we define the Cochrane sum in analogy to (2.1):

the sum being taken over all a , relatively prime to q, and inverse of a mod q.

a denoting the

The mean value of the Dedekind sums and Cochrane sums

235

We have a counterpart of (2.4) 1211 expressing the relation between the Cochrane sum and L-functions:

x(-I)=-1

(3.2) Formula ( 3 . 2 ) enables us to obtain an upper bound for, and a mean value of, the Cochrane sum: Proposition 3.1 ([21]). For any integer h with ( h ,q ) = 1, we have the estimate

IC(h1 q ) I

<< @(q)

ln2 9,

where d ( q ) = CdJq 1 is the divisor function. Proposition 3.2 ([21]). For q = p an odd prime,

Now, corresponding to the k-dimensional analogue of ( 2 . 1 ) due t o L. Carlitz [ l ]we , may introduce the k-dimensional Cochrane sums

the case k = 1 being ( 3 . 1 ) . Xu Zhefeng and the author [ I l l obtained the k-dimensional analogue of ( 3 . 2 ) )which in turn is one of ( 2 . 4 ) )and succeeded in proving counterpart of Proposition 3.1 and 3.2:

236

W. Zhang

Proposition 3.3 ([ll]). For a n y integer h , ( h , q ) = 1 and a n y fixed positive integer k with ( q , k ( k 1))= 1, we have

+

where w ( q ) denotes the n u m b e r of all distinct prime divisors of q . Proposition 3.4 ([Ill). For q = pa a prime power, we have

nplip

denotes the product over all where E i s a n y fixed positive number, primes pl with pl # p, Ck = m ! / n ! ( m - n)!. The definition ( 3 . 1 ) gives an impression that there may be some relation between the Cochrane sum and the classical Kloosterman sum K ( h , n; q ) defined by

This impression is strengthened by the following mean value result of Cochrane sums with Kloosterman sum weights:

Proposition 3.5 ([17]). W e have

for a n y

E

> 0.

T h e proof rests on the identity ( 3 . 4 ) and some involved discussion of calculation. The author and Huaning Liu [20] improved the error estimate in Proposition 3.4 to O(ql+') by established the identity

The mean value of the Dedekind sums and Cochrane sums

237

References [I] L. Carlitz, A note on generalized Dedekind sums, Duke Math. J . 21 (1954), 399-404. [2] Chaohua Jia, On the Mean Values of Dedekind sums, J . Number Theory 87 (2001), 173-188. [3] J . B. Conrey, E. Fransen, R. Klein and C. Scott, Mean values of Dedekind sums, J . Number Theory 56 (1996), 214-226. [4] Hongyan Liu and Wenpeng Zhang, On the mean square value of a generalized Cochrane sum, Soochow J . Math. 30 (2004), 165-175. [5] Huaning Liu and Wenpeng Zhang, On the hybrid mean value of Gauss sums and generalized Bernoulli numbers, Proc. Japan Acad. Ser. A- Math. Sci. 80 (2004), 113-115.

[6] Huaning Liu and Wenpeng Zhang, On certain Hardy sums and their 2m-th power mean, Osaka J . Math. 41 (2004), 745-758. [7] Huaning Liu and Wenpeng Zhang, On the 2m-th power mean of a sum analogous to Dedekind sums, Acta Math. Hungarica 106 (2005), 67-81. [8] S. Louboutin, Quelques formules exactes pour des moyennes de fonctions L d e Dirichlet, Canad. Math. Bull. 36 (1993), 190-196. [9] H. Rademacher and E. Grosswald, Dedekind sum, MAA 1972. [ l o ] H. Walum, An exact formula for an average of L-series, Illinois J . Math. 26

(1982), 1-3. [ l l ] Zhefeng Xu and Wenpeng Zhang, On the order of the high-dimensional Cochrane sum and its mean value, J . Number Theory (in press).

[12] Wenpeng Zhang, On a problem of D.H.Lehmer and its generalization, Compositio Math. 86 (1991), 307-316. 1131 Wenpeng Zhang, On the mean values of Dedekind sums, J , de Theorie des Nombres de Bordeaux 8 (1996), 429-442. [14] Wenpeng Zhang, A note on the mean square value of the Dedekind Sums, Acta Math. Hungarica 86 (2000), 275-289. [15] Wenpeng Zhang, On the hybrid mean value of Dedekind sums and Hurwitz zetafunction, Acta Arith. 92 (2000), 141-152. [16] Wenpeng Zhang, A sums analogous to Dedekind sums and its mean value formula, J . Number Theory 89 (2001), 1-13. 1171 Wenpeng Zhang, On a Cochrane sum and zts hybrid mean value formula (11), J . Math. Anal. and Appl. 276 (2003), 446-457. 1181 Wenpeng Zhang, A problem of D. H. Lehmer and its mean square value formula, Japan. J . Math. 29 (2003), 109-116. [19] Wenpeng Zhang, On a hvbrid mean value of certain Hardy sums and Ramanujan sum, Osaka J . Math. 40 (2003), 365-373. [20] Wenpeng Zhang and Huaning Liu, A Note on the Cochrane Sum and its Hybrid Mean Value Formula, J. Math. Anal. and Appl. 288 (2003)) 646-659. [21] Wenpeng Zhang and Yuan Yi, On the upper bound estimate of Cochrane sums, Soochow J . Math. 28 (2002), 297-304.

Index

(7,€)-cone, 30 L-function automorphic -, 120 Dirichlet -, 229 factorization of , 121 Hecke -, 156 nontrivial zeros of -, 125, 128 primitive -, 120 Rankin-Selberg -, 121, 129, 131 trivial zeros of -, 133 n-level correlation, 120 p-solvability condition, 31 q-adic integer, 206 q-series identities of Ramanujan, 104 adele ring, 122 admissible n-tuple ( N 1 , . . . , N,), 32 analytic continuation, 154 Apkry, R., 100 Apostol-Vu multiple series, 157 Artin hypotheses, 30 Artin's conjecture for primitive roots, 13 base change, 120, 121 basis, 27 Bernoulli - number, 210 polynomial, 154, 210 beta-expansion, 1 Bombieri-Vinogradov type mean-value theorem, 41 -

canonical number system, 1 characteristic of vectors, 29 class field theory, 120 class number, 102 Cochrane sum, 234 k-dimensional -, 235 combinatoriai sieving, 125 completely split prirnes, 122 congruences for Euler polynomials, 206 cubic field, 175, 176 cyclic -, 176, 177, 180 cusp form, 23

D. H. Lehmer's problem, 229 decimal expansion, 68, 73 Dedekind - eta-function, 108 - sum, 232 Deligne's theorem, 25 density, 69 - function, 70, 73 Dirac mass, 124 Dirichlet character, 155, 159 distribution density, 75 elliptic curve, 175 supersingular -, 98 elliptic transformation, 188 equi-distribution property, 12 Essouabri's theorem, 163 estimate of the trigonometric integral, 34 Euler number, 206 - polynomial, 206 Euler product, 122 Euler, L . , 153, 205 Euler-hlaclaurin summation formula, 154 Euler-Zagier r-fold sum, 153 explicit formula, 128, 129 exponential sum, 44, 187 - in arithmetic progression, 44 -

Fermat curve, 176 finite theta series, 187 functional equation, 88, 104, 127 functional relation, 161 Gauss circle problem, 219 Gaussian sum classical -, 48 generalization of -, 48 ~ e n e r i l i z e dRiemann Hypothesis (GRH) , 12 generic, 177, 180 Goldbach-Vinogradov Theorem, 39 GUE model, 120

Index Handelman, D., 5 Hasse invariant, 102 Hasse, H., 13 Hauptmodul, 99 Hecke - congruence subgroup, 23 - eigenform, 25 - operator, 24 Hecke algebra, 8 1 Hilbert-Kamke problem, 28 Hooky, C., 13 Hua's problem, 34 Hypothesis H, 120, 121, 123, 149 Klein, F., 98 Kloosterman sum, 236 Kratzel, E., 219 Lagarias, J. C. and Odlyzko, A. hl.,16 Langlands correspondence, 122 Langlands' functoriality conjecture, 120 local L-factor, 122 local density, 84 mean value - of t h e Cochrane sums, 235 - of the Dedekind sums, 232 mean value theorem, 32 Mellin-Barnes integral formula, 154 meromorphic continuation, 154 Mertens' theorem, 121 method of Mordell, 182 modular - degree, 122 - relation, 104 Mordell curve, 176, 182 hlordell-Tornheim multiple series, 157 Mordell-Weil rank, 180, 181 multiple - L-series, 156 - character sum, 155 - Dirichlet series, 156 - polylogarithms, 157 - strip, 162 multiplicative order, 11 multivariate - Tarry problem, 35 - Vinogradov problem, 35 natural density, 11 newform, 23 noncyclic extension field, 121 normal distribution, 70, 74 Odoni, R . W., 13 old form, 23 order condition, 28

order of ramification, 122 parabolic transformation, 188 Perron-Frobenius theorem, 5 Pisot number, 2 point at infinity, 177 prehomogeneous vector space, 93 primitive, 163 quasimodular, 97 Rademacher, H., 40 Ramanujan conjecture, 123 Ramanujan's formula, 111 Ramanujan, S., 98 real quadratic field, 156 real solvability condition, 31 recursive structure, 159, 163 relative invariant, 82 representation contragredient -, 127 cuspidal -, 120 self-contragredient -, 131, 134 residual index, 12 Riemann Hypothesis, 125 Rogers-Ramanujan function, 98 Selberg orthogonality, 120, 121, 126, 131 set partition, 126 short form, 179-181 short interval, 220 Siegel, C . L., 5 singular series, 42, 63 singularities, 154 spherical function, 82 square-free integer, 220 superposition distribution, 120 supersingular, 101 the partial fraction expansion for the coth, 113 theory of multiple trigonometric sum with primes, 36 theta reciprocation, 187 theta-Weyl sum, 188 cubic-, 191 quadratic -, 188 Three Primes Theorem, 39 torsion part, 182 trigonometrical sum, 187 truncated Chebyshev function, 57 type (F), 88 uniform distribution, 74 universality of the n-level correlation, 121 van der Corput method, 187, 201 -

Index - reciprocal function, 186 van der Corput, J . G., 187 Vinogradov - method, 201 - problem, 30 - system of equations, 30 Vinogradov, I. M., 39, 187

Waring problem, 28 Weyl sum, 187 Weyl, H., 187

Weyl-Hardy-Littlewood method, 201 Zagier, D., 153 zero-free region, 131 zeta-function Barnes multiple -, 156 double Hurwitz-Lerch -, 161 Hecke -, 156 Hurwitz-Lerch -, 104, 161 L e r c h , 161 Shintani -, 157