Exponential sums with Dirichlet coefficients of L-functions

TL;DR

This paper develops conditional estimates for exponential sums involving Dirichlet coefficients of L-functions, including Hecke eigenvalues, and applies these to improve results on primes and eigenvalues under the Riemann Hypothesis.

Contribution

It introduces new conditional bounds for exponential sums with L-function coefficients, extending previous work and applying these to prime-related problems under RH.

Findings

Conditional estimates for exponential sums over all integers and primes.

Improved bounds on Hecke eigenvalues at Piatetski-Shapiro primes.

Enhanced results assuming the Riemann Hypothesis for symmetric square L-functions.

Abstract

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.