1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

where Q:F = MaxQ"; here the exponents Band c and the constant implied in «

rrE:F
depend only on d, e and on the ratio ~! &J".

Proof. Since our objective is to describe a basic application of the large sieve, we
merely sketch a proof of the cruder bound
L N(7r; a, T) «c: TBQ"';:-+c(l-a)'
rrE:F
for any c: > O; the replacement of c: by 0 uses much more refined techniques (namely
mollification methods).
The first thing needed is a detector of zeros: a basic method to construct such a
detector is to form a Dirichlet polynomial

L

Arr(n)

D N(7r, s ) := a 71 --

n s

l,;:;n,;:;N

that takes large values on the zeros of L( 7r, s ) contained in R( a, T) and such that the

coefficients (an)n,;:;N (which have to be independent of 7r) are not too large. There

are several ways to construct efficient zero detecting polynomials (via mollifiers);

since we do not seek the best possible results we will choose a very crude one. Pick

p E R(a, T) a zero of L(7r, s) and consider the contour integral

1 J N log^2 N ~ Arr(n) nlog2 N

-. L(7r, s + p)f(s)(- 2 -)^8 ds = e--N-+ 0 --e-- N-;

27ri log N nP

(3) 2,;:;n

since L(7r, p) = 0, one may shift the above contour to ~es= 1/2- ~ep < 0 without

hitting any poles. By the convexity bound, one finds

e_log~N +~Arr(n)e_,, 101 i

2

0 nP N « c:,d,IJ (Q rr )"'(Q rr )l/4Nl/2-a=o c:,d,IJ (l)

n~2

granted that N is sufficiently large (i.e. N ;;:::: (Qrr)^1 1<^2 a -l) is sufficient, for in-

stance). By (1.23) one shows, by trivial estimation, that the tail ( n ;;:::: N) becomes

negligible when N is large and one deduces for such N's and for any zero in R(a, T)

that

(3.17)

thus D N ( 7r, s) is our zero detecting linear form.

For the next step, one uses the large sieve inequality, or, more precisely, a vari-

ant of it. Given Z c R(a, T), we say that Z is well-spaced if <;Sm(p - p') ;;:::: 1 when-

ever p =f=. p' E Z. To each 7r E F, we associate some (possibly empty) well-spaced

finite set Z(7r) c R(a, T), and given some Dirichlet polynomial L an>-rr(n)n-^8 ,

n,;:;N

we seek bounds for the averaged mean square of the values of this polynomial at

p E Z (7r):

2: 2: 12: ~: 1

2

• rrE:F pEZ(n) n,;:;N

By considering the pairs (7r, p), 7r E F , p E Z(7r) as a family :F' one can give large
sieve type inequalities for such sums, either by applying the duality principle or (as