On averages of sums over regular integers modulo $n$

TL;DR

This paper derives formulas for partial sums of weighted averages involving regular integers modulo n and the gcd-sum function, with broad applications in number theory.

Contribution

It introduces new formulas for sums over regular integers modulo n involving gcd and arbitrary arithmetic functions, expanding analytical tools.

Findings

Formulas for partial sums of weighted averages over regular integers modulo n

Applications to various problems in number theory

Enhanced understanding of gcd-sum functions in modular arithmetic

Abstract

In this paper, we provide formulas for partial sums of weighted averages over regular integers modulo $n$ of the $\gcd$-sum function with any arithmetic function. Many interesting applications of the results are also given.