Finite Trigonometric Character Sums Via Discrete Fourier Analysis

TL;DR

This paper uses elementary discrete Fourier analysis to prove various classical and new identities involving finite sums of characters and trigonometric functions, connecting to Ramanujan's theta function identities.

Contribution

It introduces elementary proofs for many identities previously proven by complex analysis, expanding understanding of finite character sums with trigonometric functions.

Findings

Proved several classical identities using elementary methods.

Derived new identities involving character sums and trigonometric functions.

Connected identities to Ramanujan's theta function identities.

Abstract

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt-Zaharescu's and some new identities by elementary methods of discrete Fourier Analysis.