Discrete Ramanujan-Fourier Transform of Even Functions (mod $r$)

TL;DR

This paper introduces a Discrete Ramanujan-Fourier Transform for even functions modulo r, expressing it through Ramanujan's sum using a linear algebraic approach, enriching the analysis of arithmetical functions.

Contribution

It defines and develops a novel Discrete Ramanujan-Fourier Transform for even functions mod r, linking Fourier analysis with Ramanujan sums through linear algebra.

Findings

Transform can be expressed in terms of Ramanujan's sum

Provides a new analytical tool for even functions mod r

Bridges Fourier analysis with number-theoretic sums

Abstract

An arithmetical function $f$ is said to be even (mod r) if f(n)=f((n,r)) for all n\in\Z^+, where (n, r) is the greatest common divisor of n and r. We adopt a linear algebraic approach to show that the Discrete Fourier Transform of an even function (mod r) can be written in terms of Ramanujan's sum and may thus be referred to as the Discrete Ramanujan-Fourier Transform.