1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 195

Lemma 1.2.1. Let D( s) be a Dirichlet series with non-negative coefficients, absolutely

convergent for ~es > 1, which is also an Euler product, say, of degree d. Suppose that

D(s) satisfies afunctionnal equation of the form (1.1),

q;{^2 D 00 (s)D(s) = w(D)q£-s)/^2 D 00 (l - s)D(l - s );

with a pole a s = 1 of order m > O; assume as well that D 00 ( s) D ( s) is of order 1 as
Is I _, +oo. Then there is a constant cd, m > 0 depending only on d, m such that D ( s)
has at most m zeros in the interval [l -cd,m / log( Q D)]. Here Q D denotes the analytic
conductor of D.

Suppose we are in the first case (if# 7f Q9 1.1^2 ito) and that for a constant c > 0,

L(7r, CJ+ ito) vanishes for CJ E [l - c/ log(Q7r(ltol + 2)), l]. Then D(s), as given by

(1.13), would have a pole of order 4 in this interval, while it has only a pole of

order 3 at s = 1, thus contradicting Lemma 1.2.1. When if= 7f Q91.1^2 ito we consider

D(s) given as (1.14) which has a pole of order 4 at s = 1. In that case it is possible
that L(7r, s + it 0 ) has a zero CJ E [1-c/log(Q7r(ltol + 2)), l]; however, Lemma 1.2.1
prevents it from having two (counted with multiplicity). Thus we have:

Theorem 1.4. There exists a constant cd > 0 (depending on d only) satisfying: for
any 7f E A~(Q) such that if is not equivalent to 7f ® l·lito for any t 0 ER, L(7r, s) has
no zeros in the region

(1.15)

Cd

~es;;:: 1 - ;

log(Q7r(l~msl + 2))

if 7f is self-dual ( 7f ~ if), then L ( 7f, s) has no zeros in the region (1.15), except for an
hypothetical simple real zero /311" E [1-cd/log( Q11" ), 1). If such a zero occurs, it is called
the exceptional (or Landau/Siegel) zero.

It is remarkable how little the matter of enlarging Hadamard/de la Vallee-
Poussin's region has progressed during the past century. Even for Riemann's zeta,
the most significant progress goes back to the SO's: the Vinogradov/Korobov zero-
free region
c
((s) # 0 for ~es;;:: 1 - log( It I+ 2)2/3 log log( I ti+ 3)1/3.

This zero-free region was established by means of far reaching exponential sums
techniques invented byVinogradov and perfected by his students; the exponent 2/3
has not moved since [Ko, Vi]. On the other hand, many efforts have been made to
find unconditional substitutes for GRH (i.e. density theorems: these state roughly
that, given a family of £-functions, very few of its elements have zeros close to
the edge of the critical strip ~es = 1). We will barely touch on these questions in
these lectures. Note also that, althought relatively elementary, the Hadamard/de
la Vallee-Poussin method was used by Deligne as a key initial step in his proof of
the Weil conjectures: i.e. GRH for £-functions over finite fields (see [De3] and also
[Morl]).
To conclude this section, we wish to mention that the method of Eisenstein se-
ries (the Langlands/Shahidi method) provides an alternative for showing the non-
vanishing of £-functions near the edge of the critical strip. In the mid-seventies
Jacquet/Shalika used this method to show that L( 7f, s) # 0 on ~es = 1 (before the
general theory of Rankin/Selberg £-functions was completed) [JSl]. In [Shal],
Shahihi proved the non-vanishing of L( 7f ® 1f^1 , s) on ~es = 1 for general ( 7f, 7r') by