1549380232-Automorphic_Forms_and_Applications__Sarnak_

196 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

this method (this is also accessible to the Hadamard/ de la Vallee-Poussin method).

More importantly, the method of Eisenstein series applies to £-functions that can-

not be handled by the Hadamard/ de la Vallee-Poussin method: a notable example

is the symmetric 9-th power, sym^9 ?r, of a 7r E A~(Q) with trivial central character.

In [KiSh2], it is shown that L(sym^9 ?r, s) ::/= 0 in {~es~ 1}\[1, 71 /67]. Moreover, it

was shown recently by Sarnak (in the simplest case of Riemann's zeta), how this

method can be made effective to provide zero-free regions inside the critical strip

[SaS].

1.2.2. The Landau/Siegel zero

Before enlarging the standart region for £-functions, the most urgent task is to

rule out the exceptional zero in Theorem 1.4 for self-dual representations. This is

one of the deepest and most important problems of analytic number theory: the

possible existence of this single zero, close to 1, limits severely the strength of

several techniques. As we will see below, the hardest case is for d = 1 and for

7r = x a quadratic character. For that reason, we discuss some applications related

to Dirichlet character £-functions and how the exceptional zero limits them.

Zero-free regions are used to show cancellations in the sums of the >..,,.(p) over

the prime numbers. For Dirichlet characters, for instance, one can use these cancel-

lations to analyse the distribution of primes in arithmetic progression. The standart

zero-free region (1.15) yields:

Prime Number Theorem in Arithmetic Progressions. Given a, q two coprime

integers and x ~ 1, set

As x ---+ +oo, one has

(1.16)

with

1/J(x; q, a)= L logp.

pk~x

p=a(q)

x

1/J(x; q, a)= cp(q) + Err(x; q, a)

Err(x; q, a)= 0( cp~q) x^13 q- l) + O(x exp(-c~));

here (Jq denotes the largest exceptional zero of the real characters of modulus q and

c > 0 is an absolute constant.

Since (Jq < 1, it follows that for fixed q and x ---+ +oo,

(1.17) 1/J(x; q, a),.._, x/cp(q).

However, for more advanced applications, one needs an asymptotic uniform for
q varying in some interval depending on x. From the expression of Err(x; q, a)
above, it is clear that a (Jq close to 1 limits the size of the available range for q. A
consequence of Dirichlet Class Number formula^2 is that

(1.18) 1-(Jq » lj.Jq(logq)^2 ,

where the implied constant is absolute and explicit (see below). Hence (1.17) holds
uniformly for q = o(y'logx), which is a rather short range in applications; on the
other hand, if no exceptional zero exists, the asymptotic hold in the much larger

(^2) (1.18) can also be obtained by purely analytic means