1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 209

one finds that

= L q ;

1

L >-1r:~ ( n) Vi (;) - L L >-1r:~ ( n) Vi (;)

q~Q n=:±l(q) q~Q n;EO,±l(q)

The contribution from n = 1 in the first sum above is

~ q-lVi(_!._) = ~ q-l +0A(Q2y-A)

L..,2 y L..,2

q~Q q~Q

for any A?: 0. The sum over n = l(q), n =I-1 contributes as an error term

~ q - 1 ~ >.7r®ir(l + dq) v; ( 1 + dq) Q ~ A1r®ir(n)n" !Vi ( n )I

L.., 2 L.., (1 + dq)/3^1 Y « L.., n/3^1 y

q~Q d~l n

,,,,,, ......._,C,7r QY1- 13 +,, '

for any i:: > O; here we have used that >.1r®ir(n) ?: 0 and that the number of repre-

sentations of n of the form n = 1 + dq with d, q?: 1 is O,,(n"), and also (1.39). The

remaining terms in T 1 are bounded similarly and we find that

T1 = ~ L.., q - 2 1 + O(QY^1 - /3+^0 ).

q~Q

The term T 2 contains the following moments of Gauss sums:

(1.40)

This sum is zero if qln and otherwise equals

q-l. ~

• 2

-{Kld2(r; q) + Kld2(-r; q)} - (-1)

mod q) and Kld2 (r; q) denotes the hyper-Kloosterman sum

~ X1 + X2 + · · · + Xr

L.., e( ).

x1x2... xd2-=r(q) q

The latter sum was bounded by «d q d

2

2-

1
by Deligne as a consequence of his reso-
lution of the Weil Conjectures [De4]. Hence

Md2(n) «d q ~ 2 •

This bound is the key saving and shows considerable oscillation of the root numbers
of L(x.1r © ?T, s). Using this bound, the inequalities {3 0 < 1, /3 > 0, and the bounds
for Vi in (1.39), one obtains

T2 = ~ w(7r®:) ~ Md2(n)>.1r®~(n)x(n)(Gx)d 2 Vi( nY 2)

qrv L..,Q ( q1r®irqd )/3 n:::;:::; ~1 nl /3 y'q q1r®irqd