On the Mean Value of $D_k(n)$ in Arithmetic Progressions
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910020935876608 |
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| author | Karras, Meselem |
| author_facet | Karras, Meselem |
| contents | 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 $d_k^*(n) = k^{ω(n)}$ is its unitary analogue, where $ω(n)$ 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Mean Value of $D_k(n)$ in Arithmetic Progressions Karras, Meselem Number Theory 11A25, 11N37 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 $d_k^*(n) = k^{ω(n)}$ is its unitary analogue, where $ω(n)$ 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 |
| title | On the Mean Value of $D_k(n)$ in Arithmetic Progressions |
| topic | Number Theory 11A25, 11N37 |
| url | https://arxiv.org/abs/2602.12371 |