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

Abstract

An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^{2}x\equiv a(\mathrm{mod}r)$. 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.