On the Mean Value of $D_k(n)$ in Arithmetic Progressions

TL;DR

This paper derives an asymptotic formula for the average value of a multiplicative function related to divisor functions over arithmetic progressions, extending previous work in the field.

Contribution

It introduces a generalized asymptotic formula for the sum of the function D_k(n) over arithmetic progressions, broadening the understanding of divisor-related functions.

Findings

Established an asymptotic formula for the sum of D_k(n) in arithmetic progressions.

Generalized previous results to a wider class of multiplicative functions.

Extended the analysis of divisor functions in the context of arithmetic progressions.

Abstract

Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of distinct prime divisors of $n$. We establish an asymptotic formula for the sum \[ \sum_{\substack{n \le x \\ n \equiv a \pmod q}} D_k(n), \] where $\gcd(a,q)=1$. This result is a generalization of the study presented in \cite{Derbal 2023}. \noindent