1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Remark 3.1. There is a natural limitation on the possible size of N in (3.1). To

fix ideas, suppose that F is finite and is equipped with the uniform measure: if we

take an = .A11" 0 (n) for some 7ro E Fin (3.1), one obtains

L l>-7ro(n)l2 «t: Q~IFI,

n,,;,N

which puts a natural limit on N, since one expects (by Rankin/ Selberg theory) that

L l>-11" 0 (n)l2 »t: (NQ11" 0 )-t: N.

n,,;,N

3.1.1. Why large sieve?

The denomination large sieve inequality is a bit misleading since the inequality itself

has apparently little to do with sieving. The term goes back to the work of Linnik

[Linl], who showed how such inequalities could be very helpful in several prob-

lems related to the sieve. The purpose of this lecture is to provide several examples

of this sieving technique.

Given a (finite) family F = { 7f} and a subset £ c F, a primary objective of the

sieve is to improve on the trivial bound

(for the purpose of showing that the complement F \ £ is not empty; for instance).

In various cases it is possible to use a general inequality like (3.1): suppose one

can find N ;? 1 (large) and a sequence (a;)n,,;,_N, depending on the set£ we are

interested in, normalized by

and satisfying, for any 7f E £, the lower bound

I L a~.A11"(n)I :? N a

n,,;,N

for some a > 0. This inequality means that the linear form l:n,,;,N a;>-11"(n) takes

large values at the 7r's belonging to £^1 and so acts as a detector for£. Applying the

large sieve inequality (3 .1) to the sequence (a;)n,,;,N (with μF the uniform measure,

for instance), one obtains

hence

which is non-trivial at least if N is large enough.

(^1) and presumably not for 7r E :F\ £