On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function

TL;DR

This paper investigates the convergence properties of Ramanujan expansions when extended to real variables and introduces a new expansion for the divisor function that is both continuous and locally interpolates the original function.

Contribution

It demonstrates that absolute convergence in the integer case does not imply convergence in the real variable generalization and presents a novel Ramanujan expansion for the divisor function.

Findings

Absolute convergence does not ensure convergence in the real variable extension.

A new Ramanujan expansion for the divisor function is constructed.

The new expansion is continuous, absolutely convergent, and locally interpolates the divisor function.

Abstract

We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$.