1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

By (3.14) and the prime number theorem, one has for any f E 3pxotic(q, x)

N1/12

L anAJ(n) » logN - O (logq)

n~N

(this amounts to counting the primes coprime to q and ~ N^1112 ), and

N1/12

L lanl2 « logN.

n~N

Thus

and hence the bound follows by taking N = q.

D

Remark 3.6. J. Ellenberg made the following observation [El]: for q is squarefree

and f E 81 ( q, x), and p any prime dividing q, the local representation of the inertia

subgroup at p is isomorphic to

(PJ )11v ':o:' 1 EB (Px)IIv·

In particular, PJ(Ip) is isomorphic to its image in PGL 2 (C) which is a cyclic sub-

group contained in A 4 , 84 or A 5 ; hence x^60 is unramified at any plq hence is trivial.

It follows that for q squarefree, the number of x(q) that are nebentypus of a form

of weight 1 and level q is bounded by Oe:(qe:) for any c: > 0. Hence if we denote by

3pxotic(q) the number of exotic forms of level q (regardless of their nebentypus),

one has

3.3.3. Zero density estimates

Given n E A~(Q), one can show by standard methods that the number of zeros of

L( n, s) within the critical strip and with height less by T ~ 1 is

(3 .15)

N(n,T) = l{p, L(7r,p) = 0, 0 ~1Rep, i<smpl ~ T}I,..., Tlog(Q1rT), T ~ +oo.

Of course, GRH predicts that all such zeros are on the critical line; however, for

many applications, it is sufficient to know that few zeros are close to 1Res = 1. To

measure this, we set for a ~ 1 /2,

R(a, T ) := {p E C , 1Rep ~a, i<smpl ~ T},

Z(7r;a,T) = {p, L(7r,p) = O, p E R(a,T)} and N(7r;a,T) := IZ(7r;a,T)I.

A zero density estimate is a bound for N(n; a, T) that improves on (3 .15) for a >
1 /2. One can mollify the problem further - again this is sufficient for many appli-
cations - and ask for a non-trivial bound for N(n, a , T) on average over a family
F.
There are many sorts of zero density estimates; in this section, we describe a
very general version that is obtained with the large sieve; for families of Dirichlet
characters, this approach started with Linnik and was developed in the works of
Barban, Bombieri, Montgomery and others [Bar, Bo, Mon]. For example, zero