Ramanujan-Fourier series of certain arithmetic functions of two variables

TL;DR

This paper extends the theory of Ramanujan-Fourier series to functions of two variables, providing convergence conditions and novel examples beyond simple generalizations from one-variable cases.

Contribution

It generalizes Delange's theorem to two-variable functions and offers new examples not derived from trivial extensions.

Findings

Established sufficient conditions for pointwise convergence of two-variable Ramanujan-Fourier series.

Generalized Delange's theorem to functions of two variables.

Presented novel examples of two-variable arithmetic functions with Ramanujan-Fourier series.

Abstract

We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of Ramanujan-Fourier series of arithmetic functions of two variables. We also give several examples which are not obtained by trivial generalizations of results on Ramanujan-Fourier series of functions of one variable.