1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

abuse of notation, we still denote by X· We take the x to be even, of prime moduli
q not dividing q'lr. Since q Y q'lr and x.7f j!. 7f, the L(x.7f 0 ir, s) are entire. Moreover,
we have the following facts (see [LRS]):

(1.35)

(1.36)

(1.37)

(1.38)

Qx_.7r®ir = q7r®irQ d^2 /2 ;

w(x.7f 0 ir) = w(7r 0 ir )x(q7r®ir )( ~)d

2

,

where Gx. denotes the Gauss sum. Hence by (1.35), the bound {3 0 ~ 2Bd follows
from:

Proposition 1.2. For any f3 > 2Bd, one has

I: I:

q~Q x.(q)

- Q2

L(x.7f 0 7r,/3) »7r,r3 logQ'

xlx.o,even

where q ranges over primes. In particular, for any f3 > 2Bd there exists x as above such

that L(x.7f® ir,/3) =f=. o.

Proof. Applying the functional equation and a method similar to those of section
1.3.2, one infers from (1.36), (1.37) and (1.38) that, for 0 < f3 < 1, one has

+ w( 7r 0 ir)x~ q7r®ir) '""" ,\7r®: (n ) x(n) ( Gx. ) d2 Vi ( nY

2

)

( q7r®irqd )f3 ~ nl f3 VQ. q7r®irqd

where Y ;)! 1 is some parameter and Vi ( x), Vi ( x ) are smooth test functions satisfy-
ing

(1.39) Vi(x ), Vi (x) = OA(x - A) as x __, +oo,

Vi(x) = 1 + OA(xA), Vi(x ) «c 1 + x^1 - f3o-f3- c as x __, 0,

for all A ;)! 0 and all c > 0. Next, one averages the approximate functional equation
over even primitive characters of prime moduli q rv Q. Then

L L L(x.7f©ir,/3)

q~Q x.#x.o

x. even

is decomposed as the sum of two terms T 1 + T 2 , say. Using the orthogonality

relations

L L x (n ) = {~ - 1

q~Q x.#x.o -1

x. even

ifn =:O(q)

i f n = ±l(q)

otherwise,