Exponential sums with coefficients of certain Dirichlet series

TL;DR

This paper derives bounds for exponential sums involving Dirichlet series coefficients under the generalized Lindel"of Hypothesis and applies these bounds to study Hecke eigenvalues at special primes.

Contribution

It provides new conditional bounds on exponential sums and uses them to analyze Hecke eigenvalues at Piatetski-Shapiro primes.

Findings

Conditional bounds on exponential sums under Lindel"of Hypothesis

Results on squares of Hecke eigenvalues at special primes

Advances in understanding Dirichlet series coefficients

Abstract

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.