On Quadratic Gauss Sums and Variations Thereof

TL;DR

This paper introduces new series involving sine and cosine of quadratic terms, connects them to Gauss quadratic sums, and derives new closed-form expressions, broadening the understanding of these sums.

Contribution

It presents novel terminating series related to Gauss quadratic sums and generalizes existing results with new closed-form solutions.

Findings

New terminating series involving sine and cosine functions of quadratic terms.

Generalized and derived new closed-form expressions for Gauss quadratic sums.

Connected series to classical Gauss sums, expanding their analytical understanding.

Abstract

A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are generalized.