On r-Regular Integers (mod nr)

Abstract

Let ρr(nr) denote the number of positive r-regular integers (modnr) that are less than or equal to nr; in this paper, we investigate some arithmetic properties of certain functions related to r-regular integers (modnr). Then, we study the average orders and the extremal orders of ρr(nr) in connection with the divisor function and the generalized Dedekind function. Moreover, we also introduce an analogue of Cohen–Ramanujan’s sum with respect to r-regular integers (modnr) and show some basic properties of this function.