1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDIVIDUAL £-FUNCTIONS 197

(althought still small) range q «A (log x )A for any A? 1. Finally, under GRH one

has Err(x; q , a) « x^112 log^2 (qx) and (1.16) is an asymptotic formula uniformly for

q « o(x^1 /^2 /log x).

Given a primitive quadratic Dirichlet character x( mod q), the existence of the

exceptional zero f3x was studied (among other people) by Heilbronn, Landau and

Siegel.

A first remark (basically due to Hecke and Landau) is that f3x being far from

s = 1, is essentially equivalent to L(x, 1) being large: to see this, consider

D (s) = ((s)L(x, s),

which has non-negative coefficients. If L(x, s) has an exceptional zero (i.e. satis-

fying 0 < 1 - f3x « 1/ log q), one considers for some x? 1 the complex integral

along the line ~es= 2:

~ J D(s + /3 )f(s)x^5 ds = ~ (l * x)(n) e-n/x

2ni x L..,, nf3x.

(2) n;;;,1

(where * denotes the Dirichlet convolution of arithmetic functions). Since

0 :::;; (1 x)(n) :::;; (1 l)(n) = r(n), the righthand side is bounded below by

? e-^1 /x » 1 and bounded above by« exp(O(logx/logq))log^2 x. On the other

hand, moving the line of integration to ~es = -/3x + 1 /2 < 0, we pass a simple

pole at s = 1 - f3x with residue f(l - f3x)x^1 -f3x.£(x, 1) and no pole at s = 0 since

D(f3x) = O; moreover, the resulting integral is bounded by O(qAx-^112 ) for some

absolute A(= 1) (see Section 1.3 for instance). Taking x = q^2 A+^1 , we infer that

(1.19) 1 « L(x, 1) « log2 q

1 - f3x '

where the implied constants are absolute and explicit.

Remark 1.6. If L(x, s) has no zero in [1 - c/ log q , 1], the same argument shows

that L(x, 1) » (log q)-^1 by taking f3x := 1 - c/ log q and noting that D(f3x) < 0.

In fact several results tend to show that if such a zero ever exists, it should be

unique. For example, one has:

Theorem (Landau/Page). There exists an (effective) constant c > 0 such that for
any Q? 1, the product

II II L(x, s)

q~Q x(q)

has at most one real zero within the interval [1 - c/ log Q, 1 ]; here the inner product
runs over the primitive real characters of modulus q.

Proof. In view of Theorem 1.4 we may restrict our attention to the quadratic x's.
Suppose that some quadratic x 1 of modulus :::;; Q has an exceptional zero in the

interval [1-c/ log Q, 1]; now consider any quadratic x "I-x 1 of modulus:::;; Q. Then

if c is sufficiently small (but fixed), it follows, from the Lemma 1.2.1 applied to

D(s) = ((s)L(x, s)L(x1, s)L(x 1x, s),

that L(x, s) "I- 0 in the interval [1 - c/ log Q, l]. D