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

Abstract

Let $k\ge 2$ be a fixed integer. We define the multiplicative function $D_k(n)=d_k(n)/d_k^{\ast }(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^{\ast }(n)=k^{\omega (n)}$ 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

arXiv comment8 pages