1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 243

4.1. Around Weyl's shift

The shifting method was introduced by Weyl in his proof of ( 4 .1). More generally,

this method provides non-trivial bounds for exponential sums of the form

Y:,J(N) = L e(f(m)),

mE[l,NJ

where f is a sufficiently regular function (i.e. well-approximable by polynomials

for instance). Assuming first that f is a polynomial, one obtains by squaring the

above sum

[Y:,1(N)[^2 = L e(f(m) - f(n)) = L L e(f(l + n) - f(n)).

m,nE[l,N] lll<N O<n~N

O<l+n~N

Now f(l + x) - f(x ) is a polynomial in x of degree reduced by one. One can then

continue the above process until one reaches a sum for a polynomial of degree I to

which one applies either the trivial bound or the geometric series summation; one

can then get a non-trivial bound for the original sum [Y:, 1 (N)[. More generally, this

method applies also to functions f that are well-approximable by polynomials, an

example being f(n) = -(it/2n) logn, which is the case occurring for(.

4.1.1. Burgess's method
Burgess's method is a variant of the Weyl shifting technique but in a purely arith-
metic context and with significant differences^2 • In this section, we prove ( 4.2)
when q is a prime number by an elegant variant of Burgess's original argument due
to Friedlander/Iwaniec. By the approximate functional equation for L(x, s) it is
sufficient to give a bound for the character sum

~ x(n) n

Sv(x) = L...,, nl/2 V( 1;2)

n;;,1 q

where V(x) is a smooth function decaying rapidly as x---> +oo. Integrating by part
it is sufficient to bound the sum

S(x, N) = L x(n)

n~N

non-trivially for q^112 -^0 ~ N < q, for some fixed 6 > 0.

Theorem 4.3. (Burgess) For all r ~ 1, one has for N < q

S(x,N) «r Nl-l/rqTrf(logq)1+3/2r.

Remark 4.1. Since ~ti ,...., ir when r is large, the above bound is non-trivial as
long as N ~ q^1!^4 +o for any fixed 6 > 0. Hence x(n) has oscillations as n ranges
over very short intervals (of size up to q^114 ).

(^2) A closer analog of Weyl's shift is the work of Graham-Ringrose on characters with highly factorized
moduli [GR].