Dual Ramanujan-Fourier series

TL;DR

This paper introduces and studies dual Ramanujan-Fourier series, extending classical theorems to this new form and exploring their properties within number theory.

Contribution

It defines dual Ramanujan-Fourier series and extends existing theorems like Lucht's and Delange's to analyze these series.

Findings

Extension of Lucht's theorem to dual series

Extension of Delange's theorem to dual series

New results on properties of dual Ramanujan-Fourier series

Abstract

Let $c_q(n)$ be the Ramanujan sums. Many results concerning Ramanujan-Fourier series $f(n)=\sum_{q=1}^\infty a_q c_q (n)$ are obtained by many mathematicians. In this paper we study series of the form $f(q)=\sum_{n=1}^\infty a_n c_q (n)$, which we call dual Ramanujan-Fourier series. We extend Lucht's theorem and Delange's theorem to this case and obtain some results.