1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Remark 3.2. By using the spectral analysis on modular forms (based on the Peters-
son/Kuznetsov formula and its further developments by Deshouillers and Iwaniec
[DI]), and the dispersion method of Linnik, Fouvry/ Iwaniec obtained for the first
time results of Bombieri/Vinogradov type on the distribution of arithmetic
sequences (including the primes) in arithmetic progressions over special (i.e. well
factorable), but very large, moduli (i.e. with q ~ Q with Q = xf3 and f3 > 1 /2) [Foll,
Fol2]; this is beyond the possibilities of GRH! The ideas and methods of these pa-
pers were further polished and magnified by Bombieri/Friedlander/Iwaniec and
Fouvry in [BFil, BFI2, BFl3, Fol, Fo2]: among the various Bombieri/Vinogradov
type theorems beating GRH, one can quote the following two results from [BFI2,
BFil]. For any fixed a, for some A> O and for any Q ~ x^112 ,
1
L 1 1/J(x;q,a)--( )1/;(x; 1, 1)1 «a x/(logx)A.
q~Q <p q
(a,q)=l
At the expense of replacing the absolute value above by a more flexible function, it
is possible to pass far beyond the critical exponent 1/ 2 (and GRH): for any c > 0
and any A~ 1, one has
1
L >..( q) ( 1/;(x; q, a) - <p( q) 1/;(x; 1, 1)) «a,A,c: x/ (log x )A,
q~Q
(a,q)=l
for any Q ~ x^417 -"; here >..(q) denotes any bounded arithmetical function satisfying
an extra technical assumption (i .e. is well factorability), which is not limiting in
applications.
One cannot end this section without mentioning the large sieve inequality for
real Dirichlet characters of Heath-Brown [HB].
Theorem 3.3. For any complex numbers an we have
b b n 2 b
(3.6) L IL an(-)1 « (QN)"(Q + N) L lanl^2
q
q~Q n~N n~N
b
with any c > 0, the implied constant depending only on c. Here ~ indicates restric-
tion to positive odd squarefree integers.
Although this inequality looks very similar to the previous ones (observe that

the number of primitive real characters of modulus ~ Q is » Q), its proof is much

more involved; this is a powerful estimate, and we refer to [HB, So, IM] for some

applications.

3.1.3. Large sieve inequalities for modular forms

In the context of modular forms, the natural analog of the character values x(n)

are the Fourier coefficients pf ( n); not surprisingly, large sieve inequalities exist and

are consequences of the Petersson/ Kuznetzov formulae of the previous section. In

the case of holomorphic forms, one has the following [DFI3]:

Proposition 3.1. For ( an)n~N a sequence of complex numbers and k ~ 2, one has

I'(k - 1) ~ I~^1 ;^2 12 ( N log N ) ~^2

(3.7) ( 4 n)k- l L L ann PJ(n) = 1+0( qk ) L lanl.

fEBk(q,x) n~N n~N