Some remarks on regular integers modulo $n$

TL;DR

This paper studies regular integers modulo n, introduces a multidimensional generalization of their counting function, and derives identities and properties involving Bernoulli polynomials, Gamma function, and cyclotomic polynomials.

Contribution

It introduces a multidimensional generalization of the regular integers counting function and establishes new identities involving special functions and polynomials.

Findings

Derived identities for power sums of regular integers

Established analogues of Menon's identity

Investigated maximal orders of related functions

Abstract

An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of Euler's $\phi$ function. We introduce the multidimensional generalization of $\varrho$, which is the analogue of Jordan's function. We establish identities for the power sums of regular integers (mod $n$) and for some other finite sums and products over regular integers (mod $n$), involving the Bernoulli polynomials, the Gamma function and the cyclotomic polynomials, among others. We also deduce an analogue of Menon's identity and investigate the maximal orders of certain related functions.