1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Proof. The method of Burgess consists of artificially adding summation points to

the above sum -much in the spirit of a fugue where repetition of the theme with

a slight shift creates the harmony. For the purpose of elegance, one introduces the

piecewise linear function g that is 1 on [1, NJ and zero outside [O, N + l]. Note that

[J(y) = L g(x)e(-xy)dx « min(N,y-^1 ,y-^2 ), L l§(t)ldt ( log3N.

One has (by the action of Z on itself),

Sx(N) = L x(n)g(n) = L x(n + ab)g(n +ab),

n n

for all a, b )! 1. Setting AB = N, A, B )! 1, then

[A][B]S = L LLX(n+ab)g(n+ab)

= L x(a) L L x(an + b)g(n +ab)

:s; L L / L j ~[J(~)e(~t)x(an + b)e( bt)dt/,

a::;;A lnl::;;N b::;;B

by Fourier inversion. Hence

[A][B]S « log NL L / L e( bto)x(an + b)/

a::;;A lnl::;;N b::;;B

for some t 0 E R. For u E F q, we set

v(u) = l{l (a (A, lnl ( X, an= u(q)}I,

and then

[A][B]S « logX L v(u)/ L e(bt 0 )x(u + b)/

u(q) b::;;B

( )

1-1/r ( ) 1/2r ( I 1 2r) 1/2r

( L v(u) L v(u)^2 L L e(bt 0 )x(u + b).

u(q) u(q) u(q) b::;;B

One has

L v(u) = [A]N,

u(q)

and, for AN< q/2, one has

L v(u)^2 = l{l ( ai, a2 (A, ln1I, ln2I ( N, a2n1 = ain2}I « AN(log AN)^3 ,

u(q)

since "m1 = m2(q)" and "m1 = m2" are equivalent when lm 1 I, lm 21 < q/2. Hence

we conclude that

(AN)^1 _..!_

S « (log q)l+fr

2

r

AB

· ( L IL x((u +bi)· .. (u + br))x((u + b~) ... (u + b~)) /)fr.

b1,. .. ,br::;;B u(q)

b~, ... ,b~~B

At this point we use Weil's bound for algebraic exponential sums: