Minimal theta functions

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by HUGH L. MONTGOMERY! (Received 3 May, 1986). 1. Statement of results. Let /(u) =f(ult ... Science Foundation Grant NSF DMS 85-02804. Glasgow Math.
MINIMAL THETA FUNCTIONS by HUGH L. MONTGOMERY! (Received 3 May, 1986) 1. Statement of results. Let /(u) =f(ult u2) = au\ + buiU2 + cu\ be a positive definite binary quadratic form with real coefficients and discriminant b2-Aac = -l. Among such forms, let h(u) = ~/x(wi + u1u2 + uf). The Epstein zeta function of / is V3 denned to be

£/(*)= 2

f{m,n)-'.

(m,«)#0

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s>0,

£>(*) ^ £„(*).

(1)

We prove a corresponding result for theta functions. For real a > 0, let

This function satisfies the functional equation 6,(1/a) = ae,(a).

(2)

(This may be proved by using the formula (4) below, and then twice applying the identity

(8)0 THEOREM 1. For any

a>0,

If there is an a>0 for which df(a) = dh(a), then f and h are equivalent forms and

ef - eh. Let Us) = tf(s)T(s)(2jzr. Then

= 7TT - ; + f S

J.

S

(X

J\

for all complex numbers s other than 0 and 1. From Theorem 1 it follows that %f(s) — %h(s) for all real s, with equality only when f ~ h. Hence we obtain the earlier result (1) as a corollary. On differentiating in (3), we see moreover that %fk)(s) > %ffk\s) for all real s, and that ^2k+l\s) > %fk+1\s) for all 5 > 1/2. (For s < 1/2 the inequality is reversed in the case of derivatives of odd order.) t Research supported in part by National Science Foundation Grant NSF DMS 85-02804.

Glasgow Math. J. 30 (1988) 75-85.

76

HUGH L. MONTGOMERY We may factorize / ; / = a(u! + zuz){ux + zu2),

where z = x + iy, and without loss of generality y > 0. Since b2 — 4ac = - 1 , we deduce that a = l/2y, so that f-

— (

-

\ - — (

\2

i

2

2y 2y Since 6f and £y are determined by the complex number z in the upper half-plane, we dispense with the notation df, t,f, and henceforth write 6(a;x, y), £(s;x, y) instead. In particular, we see that

6(a; x, y) = 2 e"^"2 2 e—^"-1-")^.

(4)

Two forms are equivalent, fi~f2, if there is a C = [c,7] e 5L(2, Z) such that/ 2 («) =/i(CM). In this case

The form / is reduced if— a\, this is >-le~m. (1 - e-2jl)~2 < 1-004 for n > 1, f > 1, so that

( 1

+

e~2m}

"^y

}



On the other hand (1 - ia'/y. Hence the sum above is at most y 00

00

Z, n e

-e

n=2

2J n=2

n e

Suppose that a>y. In this case the right hand side of (14) is e-Wna'y> while the last sum above is /i2e"*("2"2)/4 = 0-868649. . . < 1. n=2

This gives the result in this case. Suppose that a0 oy holds when z = x + iy lies in the fundamental domain 3). To this end, we first prove the following subsidiary result.

82

HUGH L. MONTGOMERY LEMMA

5. Ifa>0,

0 < x < | , andx2 + y2>l,

then

2

2 3 d 6(a;x, y) > 0. —5 6(a; x, y) H dy y dy Numerical experiments suggest that this inequality holds if a>0 and _y>0.71. However, it fails when a = 9/4, x = 1/2, y = 0-70. For any Y > 1/2 there is a C(Y) such that the inequality holds when a^ C(Y), y > Y. On the other hand, at the saddle point (x, y) = (1/2, 1/2), we have 36_

3*0

' dy2

dy

for all a > 0. By differentiating in (13) it is easy to see that ' dy

2

' dy2

4

as y/a—Kx>, a > 1. Thus it is obvious that the stated inequality holds when ff>l and y/a is large. Proof. In view of (2), we may assume that ar> 1. By direct calculation in (4) we see that the quantity in question is - _ (m-™)2\2£-2x«f y

1

2jt(X

^

y

n2e-*»«f=

m,n

2 - 2 , 1

2

say. In Si, the terms (m, n) = (±1, 0) contribute an amount where K denotes the first factor on the above right, and Px the second. The terms (m, n) = (0, ±1) contribute to Si an amount 2(JTO-) 2 (1 - x*iyiye-'«