1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYI'IC PROPERTIES OF INDMDUAL £-FUNCTIONS 193

by subtracting the N - 1 sum from the N sum. Taking N =pk (with k --? +oo),

one deduces, for all p and all i E {1, ... , d^2 }, the bound

1 2

ia7r 0 7r,i (p) I ~ P -d^1 +^1 ;

in particular, for p unramified, one has la7r,i(P)l^2 ~ p^1 - d^2 ~1. A more carefull analy-

sis of the ramified factors implies that the latter bound is valid at the remaining

non-archimedean places. D

1.2. Zero-free regions for £-functions

The most important problem of the analytic theory of £-functions is the Generalized

Riemann Hypothesis, giving the optimal zero-free region of an automorphic £-

function:

Generalized Riemann Hypothesis (GRH). Given 7f E A~(Q), the product

L 00 (7r, s)L(7r, s) does not vanishforfRes -=I-1/2.

We have already seen that zero-free regions for L(K 0 ii", s) are usefull for pro-

viding bounds for the local parameters of Jr. Note that GRH is also expected for

L( 7f 0 ii", s), since the latter is expected to be automorphic; for instance, one can

show that GRH for L(K 0 ii", s) implies the bounds of (1.8) with() = 1/ 8 for Jr.

However, the main application of zero-free regions is to provide good (in fact opti-

mal) control of the sums Lp:;;;x ..>..7r(p), when p ranges over the prime numbers: for

instance, under GRH (for L(K, s)), one has

L ..>..7r(p) = c57r=trivX + Od(x^1 /^2 log

2

(Q7rx)), as x--? +oo.

p:;;;x

In this section we review the most basic approximations to GRH (zero-free regions)

and their most classical applications. We refer to [Ser2] for some other arithmetic

applications.

1.2.1. The Hadamard/de la Vallee-Poussin method

Even in the case of Riemann's zeta function, very little is known about this conjec-
ture (but we have theoretical and extensive numerical evidence to support it). At
the end of the 19th century, J. Hadamard [HJ and Ch. de la Vallee-Poussin [VP]

proved (independently) that ((s) -=I- 0 for ~es = 1. The non-vanishing of (at the

edge of the critical strip turns out to be equivalent to the Prime Number Theorem
(PNT):

Prime Number Theorem. As x --? +oo, one has

L

x x
1=--+o--log x ( log x ) ·
p:;;;x
A little later, de la Vallee-Poussin [VP2] extended the non-vanishing of ( to an
explicit region inside the critical strip:

Theorem 1.3.

(1.11)

There exists a c > 0 such that ((s ) does not vanish for

c

~es ~^1 - log(ltl + 2) ·