Oscillations of Hecke Eigenvalues at Primes

TL;DR

This paper investigates the behavior of Hecke eigenvalues at prime numbers, focusing on their cancellation properties when twisted with exponential sums of amplitude proportional to the square root of n.

Contribution

It introduces a novel analysis of Hecke eigenvalues at primes involving exponential sums with specific amplitude, advancing understanding of their oscillatory behavior.

Findings

Demonstrates significant cancellation in twisted Hecke eigenvalues at primes.

Provides bounds on exponential sums involving Hecke eigenvalues.

Enhances theoretical understanding of automorphic forms at prime arguments.

Abstract

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.