Mean values with cubic characters

Let n≥3 be a fixed integer and let F be a global field containing the n-th roots of unity. In this paper we study the collective behavior of the n-th order twists of a fixed Hecke L-series for F. To do so, we introduce a double Dirichlet series in two complex variables (s, w) which is a weighted sum of the twists, and obtain its meromorphic continuation. We also study related sums of n-th order Gauss sums. These objects together satisfy a nonabelian group of functional equations in (s, w) of order 32.

Double Dirichlet series and the n -th order twists of Hecke L -series

In this work we study the moments of central values of Hecke L-functions associated with cubic characters, and establish quantitative non-vanishing result for the L-values.

/pdf/on-hecke-l-series-associated-with-cubic-characters-2y9evke44k.pdf

On Hecke L -series associated with cubic characters

Combining sieve methods with automorphic form theory and techniques from l-adic cohomology, we prove that the sign of Kloosterman sums Kl(1, 1; n) changes infinitely often as n ranges over the squarefree integers having all their prime factors larger than n 1/23.9 .

http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n3-p01.pdf

Sur le changement de signe des sommes de Kloosterman

Let ƒ(x) be a polynomial with integral coefficients and let, for . If ƒ is a cubic polynomial then it is expected that . In this paper, we consider the special case ƒ(x) = Ax 3 + Bx and propose a precise formula for k(ƒ). This conjecture represents a refined version of the classical Kummer conjecture.

/pdf/the-asymptotic-distribution-of-exponential-sums-i-1fyeb5xnem.pdf

S.J. Patterson

Papers

The first moment of cubic exponential sums