1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. IARGE SIEVE INEQUALITIES 227

Thus (3. 7) shows quasiorthogonality of the Fourier coefficients on average over

n ~ N « qk (note that qk/12 is approximately the size of Bk(q, x)). For nonlacu-

nary sequences (an), the above inequality is much stronger than the one obtained
from the individual estimates (2.32), (2.33) of the previous lecture. The proof of
(3.7) is obtained through Petersson's formula, which transform the above quadratic
form into another one with Kloosterman sums (weighted by Bessel functions). The
proof follows by opening the Kloosterman sums and by reduction to the large sieve
inequality (3.2) for additive characters.

Proof. By Theorem. 2.1, the lefthand side of (3.7) equals

""""' L lan l +27ri 2. -k L """"' - 1 """"' L -amanSx(m,n;c)Jk-1( 4 7r -j7Tm ).

c c

n~N c=O(q) m,n~N

To apply the large sieve inequality (3.2) we need to separate the variables m and n
in the Bessel function: we do this by means of the Taylor expansion (which we use
for 0 ~ x/2 ~ 1),

We assume first that 27r N / q ~ 1; then by opening the Kloosterman sum, and apply-
ing Cauchy/Schwarz and (3.2), one has for c = O(q) and l? 0,

~ L am - an S x ( m, n,. c )(27r-j77m)^21 ~ ~ """"' L I'"' L an (27rn)^1 e (nx)12

c m,n~N c c x(c) n~N c c

(x,c)=l

« L lanl

2

·

n~N

On the other hand, by Weil's bound (2.31) one has

1 """"' _ 27r-j77m 21 NT

2

• L amanSx(m, n; c)( ) ~ (c) """"'^2
  c c c^112 L lan l ·
  m,n~N n~N

It follows that (since k? 2)

""""' L c 1 """"' L amanSx(m, - n; c)Jk-1( 47r -j7Tm c )

c=O(q) m,n~N

(27r)k-l 2 Nk-l. NT^2 (c) (27r)k-l Nlog^2 (N)

« (k - 1)! L lanl L ck-l mm(l, cl/2 ) « (k - 1)! q.

n~N c=O(q)

Hence (3.7) is proved for N ~ q/27r. The remaining case is proved by the following

trick which goes back to [11]: one choose p a prime such that pq « 27r N ~ pq;

the basis [ro(q):r~(pq)J1! 2 Bk(q, x) can be embedded into some orthonormal basis
Bk(pq, x) of Sk(pq, x). The remaining case follows from the inequality

L IL ann1/2P1(n)l2 ~ [fo(q): fo(pq)] L IL ann1/2PJ(n)l2·

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

0