The discrete Fourier transform of $r$-even functions

TL;DR

This paper investigates the discrete Fourier transform of r-even functions, revealing how it generalizes Ramanujan sum properties and simplifies understanding of their mean values and Dirichlet series.

Contribution

It provides a comprehensive analysis of the DFT of r-even functions, extending known properties of Ramanujan sums and offering new insights into their structure.

Findings

Generalizes properties of Ramanujan sums

Simplifies derivation of known properties of r-even functions

Analyzes mean values and Dirichlet series of r-even functions

Abstract

We give a detailed study of the discrete Fourier transform (DFT) of $r$-even arithmetic functions, which form a subspace of the space of $r$-periodic arithmetic functions. We consider the DFT of sequences of $r$-even functions, their mean values and Dirichlet series. Our results generalize properties of the Ramanujan sum. We show that some known properties of $r$-even functions and of the Ramanujan sum can be obtained in a simple manner via the DFT.