New Insights into Integrals Involving Gaussian Sums

TL;DR

This paper investigates elementary integrals connected to Gaussian sums, deriving identities and evaluations through classical calculus to illuminate the relationship between analysis and arithmetic structures.

Contribution

It introduces new elementary techniques for evaluating integrals involving Gaussian sums, emphasizing accessible methods over advanced machinery.

Findings

Derived several new identities involving Gaussian sum integrals

Evaluated nontrivial definite integrals related to Gaussian sums

Highlighted the interplay between analysis and arithmetic in these integrals

Abstract

In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals that emerge naturally in this context. The approach is mostly elementary, avoiding the need for advanced machinery, and aims to shed light on the rich interplay between analysis and arithmetic structures arising from these integrals.