1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LARGE SIEVE INEQUALITIES 225

3.1.2. Large sieve inequalities for characters

Simple examples of large sieve type inequalities are the ones for additive or mul-

tiplicative characters of modulus q; these follow easily from the Dirichlet orthogo-

nality relations:

(3.2)

(3.3) 2::-(

1

) I L anx(n)l^2 « (1 + N) L lanl^2.

x(q) <p q n~N q n~N

(n,q)=l

These inequalities exhibit quasiorthogonality on average over n ~ N « q. Deeper

and stronger are their extensions (due to Bombieri) when an extra averaging over

the modulus q is performed ([Bo]):

Theorem 3.1. For N, Q ~ 1 and any a E CN, we have

(3.4) L L I L ane(an)l2 « (Q^2 + N) L lan l^2 ;

q~Q a(q) n~N q n~N

(a,q)=l

here the summation is restricted to primitive characters and where the implied constant
is absolute.

Theorem 3.2. For N , Q ~ 1 and any a E C N, we have

(3.5) L L x <prq) I L anx(n)l2 « (Q

2

+ N) L lanl

2

;

q~Q x(q) n~N n~N

here the summation is restricted to primitive characters and the implied constant is
absolute.

This inequality means that on average over the moduli q ~ Q, quasiorthogonal-

ity holds when N « Q^2 • Note that the proof of the additive version uses spectral

analysis on (R, +) (i.e. Fourier analysis) and the duality principle presented be-
low; interestingly, the multiplicative version is obtained from the additive one: the
switch from additive to multiplicative characters is done by means of Gauss sums.
The most important consequence of this inequality (together with Siegel's The-
orem) is the:

Theorem (Bombieri/Vinogradov). For any A~ 1, there exists B(= 3A + 23) > 0
such that for any Q ~ x^112 / log^8 x, one has

1

L Max l'l/i(x ;q,a) - -( )'l/i(x; 1, 1)1 « A x/(logx)A

q~Q a(q),(a,q)=l <p q

where the implied constant depend on A only (but is not effective).

This inequality means that the primes less than x are very well distributed
in arithmetic progressions of modulus q, on average over moduli q up to Q =
x^112 / log^8 +^1 x ( see [BFil] for an elegant proof). For many applications, this is
nearly as strong as GRH (the only difference being that, under GRH, B can be taken
to equal 2). The Bombieri/Vinogradov theorem is a key ingredient (among others)
in the proof of Chen's celebrated theorem which states that every sufficiently large
even integer is the sum of a prime and an integer with at most 2 prime factors.