Asymptotics and formulas for cubic exponential sums

TL;DR

This paper derives asymptotic expansions and formulas for cubic exponential sums, especially useful when the cubic coefficient is restricted, aiding in numerical approximation and bounding of these sums.

Contribution

It generalizes previous quadratic results to cubic sums, providing new asymptotic formulas and insights for numerical and theoretical analysis.

Findings

Derived asymptotic expansions for cubic exponential sums

Provided formulas for numerical approximation of cubic sums

Established upper bounds for certain cases

Abstract

Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.