An analogue of Ramanujan's sum with respect to regular integers (mod $r$)

TL;DR

This paper introduces a new analogue of Ramanujan's sum based on regular integers modulo r, exploring its properties and similarities to the classical sum, expanding the understanding of modular arithmetic structures.

Contribution

The paper defines a novel Ramanujan-like sum for regular integers modulo r and investigates its properties, providing new insights into modular arithmetic and number theory.

Findings

The analogue exhibits properties similar to classical Ramanujan's sum.

The sum satisfies certain orthogonality relations.

It offers potential applications in number theory and modular analysis.

Abstract

An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^2x\equiv a\pmod{r}$. In this paper we introduce an analogue of Ramanujan's sum with respect to regular integers (mod $r$) and show that this analogue possesses properties similar to those of the usual Ramanujan's sum.