1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. IARGE SIEVE INEQUALITIES 237

in [Mon]) by using an existing large sieve inequality for F; for instance, under the

assumptions of Theorem 3.5, one can show that

(3.18) L L [Lan A:~n) [^2 «d,c: (Q:FNt(N + [F[^2 Q}) L 1~;~

2

.

rrEF pEZ(rr) n~N n

Exercise. Prove (3.18) by using the duality principle and the proof of Theorem 3.5.

To deduce zero density estimates, we apply this bound to a zero detecting

polynomial and to the set of zeros of 7r, denoted Z(7r; a, T), contained in R(a, T).

Note that the well-spacedness assumption for Z(7r) is not a severe restriction: by

standard methods one can show that for any T E R, the number of zeros of L( 7r, s)

inside the critical strip and satisfying [~mp-T[ ( 1 is bounded by Od(log Qrr([T[ +

3)). In particular, Z(7r; a, T) can be subdivided into Od(log Qrr(ITI +3)) well-spaced

subsets. To keep control of the range on n, we need a dyadic decomposition of the

zero detecting polynomial D N(7r, s) given in (3.17): we divide the summation over

then variable into O(log N) dyadic subintervals of the form [M, 2M[ for M = 2v,

0 ( v « log N, obtaining

[DN(7r,s) [^2 « (logN)^2 Max[ '°' an Arr(n)[^2.

M ~ n s

n~NI

n~N

We now apply (3.18) to the polynomials "L,n~M an >-.';..~n) and obtain by (3.17),

L N(7r;a,T) «c:,d,fi (logN)^2 (1ogQrr([T[ +3))(QFN)c:(N + [F[^2 Q})N^1 -^20

« c ,d,fJ N2c:+2(1-o) ,

granted than N ;;?: [F[^2 Q}. We conclude by taking

N = Max(Q;f(^2 o-i), [F[^2 Q}).

Such estimates can be applied to fairly general families of L-functions.

0

An interesting family is the set of dihedral modular forms attached to class
group characters of an imaginary quadratic order of discriminant - D < 0. Using
the above precise density estimate and an analog of the zero-repulsion phenome-
non in this context, one can deduce a polynomial bound for the smallest Euler
prime of discriminant - D (a direct analog of Linnik's theorem above).
Theorem 3.11. There is an absolute constant A (effectively computable) such that
for any D ;;?: 1, the smallest prime (p D say) of the form

p=m^2 +Dn^2 , m,nEZ

is bounded by

PD« DA,

where the implied constant is absolute and effective.

Another type of application concerns the distribution of the values of automor-

phic L-functions at the edge of the critical strip (at s = 1 for instance).

Indeed, a zero density estimate (like the one above) gives, for almost all 7r E F

a rather large zero-free region. Having such a good zero-free region at hand, one