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

Let $k\ge 2$ be a fixed integer. We define the multiplicative function $D_k(n)=d_k(n)\mathrm{/}{d}_k^{\ast }(n)$, 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 $$\sum _{\begin{array}{c}n\le x\\ n\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)\end{array}}{D}_k(n),$$ where $\gcd (a,q)=1$. This result is a generalization of the study presented in \cite{Derbal 2023}. \noindent