On the distribution of cubic exponential sums

TL;DR

This paper investigates the asymptotic behavior and sign changes of cubic exponential sums over Eisenstein integers using metaplectic forms, providing new estimates and demonstrating infinite sign oscillations over almost prime moduli.

Contribution

It introduces new average estimates for cubic exponential sums and proves infinitely many sign changes over almost prime integers, advancing understanding of their distribution.

Findings

Established non-trivial average estimates for cubic exponential sums.

Proved that the sign of these sums changes infinitely often.

Demonstrated sign oscillations over almost prime moduli.

Abstract

Using the theory of metaplectic forms,we study the asymptotic behavior of cubic exponential sums over the ring of Eisenstein integers. In the first part of the paper, some non-trivial estimates on average over arithmetic progressions are obtained. In the second part of the paper, we prove that the sign of cubic exponential sums changes infinitely often, as the modulus runs over almost prime integers.