1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LARGE SIEVE INEQUALITIES 235

density estimates have been applied by Linnik and then by others to study the

following:

Question. Given 1 :::;; a < q two coprimes integers, give a bound for the smallest prime

>a, p(q, a ) say, congruent to a modulo q.

Clearly p(q,a) »c: q^1 - ". The GRH implies^3 p(q,a) «c: q^2 +", while the Hada-

mard/de la Vallee-Poussin zero-free region and Siegel's theorem give the much

weaker bound p(q, a) «c: exp(q" ) for any r:: > 0 (the implied constant being ineffec-

tive). A formidable achievement is the theorem of Linnik [Lin2], from 1944 , which

gave an unconditional bound, qualitatively as good as the GRH bound:

Theorem (Linnik). There exists an absolute constant A > 2 such that

p(q, a) « qA;

moreover, A and the constant implied are effectively computable.

Remark 3.7. Subsequently, several people have computed admissible values of A

(among others, Chen, Jutila and Graham) and currently the sharpest exponent is

due to Heath-Brown and isfairly close to 2: A = 5.5.

Basically, Linnik's proof combines three main ingredients: the Hadamard/de

la Vallee-Poussin zero free region, the quantitative form (due to Linnik) of the

Deuring/Heilbronn phenomenon, see Lecture 1) and the following zero density

estimate (also due to Linnik):

Theorem 3.9. For q and T;? 1, and o:;? 1/ 2 one has

(3.16) L N(x ; o:, T) « (qT)c(l-a)'

x (q)

where c and the constant implied in « are absolute and effectively computable.

Some comments are in order concerning this last bound: by (3.15), (3.16) is
non-trivial only when o: is close to 1 (i.e. > 1 - 1/c). Other zero density estimates
are non-trivial for any fixed o: > 1 /2: for instance, by using (3.3), Bombieri proved
([Bo] Theorem. 18)
L N(x; o:, T) « TAq^3 ~=~ logB q,
x(q)

for some absolute A , B. This is much sharper than the Linnik density theorem
when o: is small but weaker when o: is very close to 1 (i.e. 1 - o: = 0(1/ log qT)),
because of the logB q factor. This latter feature and the exceptional zero repulsion
phenomenon are critical to balance the influence of the exceptional zero.
Recently there have been various extensions of zero density estimates to gen-
eral families of automorphic forms; see for instance [KMl, KM3, Lu3]. For exam-
ple, one has the following analog of (3 .16) [KM3]:

Theorem 3.10. With the notations and assumptions of Theorem 3.5, one has, for
o: > 3 / 4 and T ;? 1,
L N (n;o:, T) « T BIQFlc(l- a),
7rE F

(^3) Note that the Bombieri/Vinogradov theorem implies the bound p (q, a ) «., q (^2) +" for almost all q ~ Q
as Q-+ +oo.