Some remarks on regular integers modulo $n$

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.