1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Theorem 3.5. Let F be a finite subset of A3(Q) and Q;:: = sup,,.EF Q,,.. We assume
that any 7r E F satisfies (1. 8) for some fixed e < 1/4, and that two distinct 7r, 7r^1 E F
are never twist of each other by a character of the form I det lit for some t E R; then
for all N ~ 1 and all (an) E cN, one has

LIL an.A,,.(n)l

2

«d,e (Q;::NnN + IFl

2

Q}) L lanl

2

.

n

Remark 3.5. While such a bound is not as strong as quasiorthogonality in the sense

of (3.1), it is still non-trivial (when N is sufficiently large) by comparison with the

trivial bound (Q;::N)elFIN:Z:n lanl^2.

Proof. By the duality principle, it is sufficient to prove that for (b,,. ),,.E;: one has

To do this we multiply the left hand side by g(n) where g is a smooth non-negative

function, compactly supported in ]O, N + 1], which takes the value 1 on [1, NJ.

Opening the square, one has to bound the sum

(3.11)

The fact that all 7r, 7r^1 E F satisfy (1.8) for some e < 1 /4 enables us to show:

(1) the following factorization holds: for R es > 3,

L >.,,. (n).A,,.1 (n)n-s = H(7r, ir', s )L(7r 0 ir', s)

n

for H ( 7r, 7r^1 , s) some Euler product, which, in the the domain ~es ~ 1 /2,

is absolutely convergent and uniformly bounded by C(c, d)Q'; 0 ir1, for any

c > 0.

(2) the convexity bound

L( 7r 0 7r I , s ) <<€ ,d Ql/4+e 1r®irl

for ~es= 1/2, and at s = 1, the bound

ress=1L(7r 0 7r

1

, s) «e,d Q'; 0 ,,.1,

the latter residue being non-zero iff 7r = 7r^1 • A shift of the s-contour to the line

~es= 1 /2 in (3.11) yields

2

~i J g(s) L .A,,.(n).A,,.^1 (n)n-^5 ds = g(l)ress=iL(7r 0 7r^1 , s) + Oe,d,e(N^112 Q;j~t,e).

(3) n

Hence, from the bound Q,,. 0 ir1 «d ( Q,,.Q,,.1 )d ~ Q'?j we obtain that

L b,,.b,,. 1 Lg(n)>.,,.(n).A,,.1(n) «e,d,e QJ:(N + N^112 1FJQ'.i!^2 ) L Jb,,.J^2.

rr,rr' n 7r

0