Some multiple Dirichlet series of completely multiplicative arithmetic functions

Nabil Tahmi and Abdallah Derbal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 603–616
DOI: 10.7546/nntdm.2022.28.4.603-616
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Authors and affiliations

Nabil Tahmi
Department of Mathematics, ENS of Laghouat and EDPNLHM Laboratory,
ENS of Kouba, Algiers, Algeria

Abdallah Derbal
Department of Mathematics, EDPNLHM Laboratory,
ENS of Kouba, Algiers, Algeria


Let f_r: \mathbb{N}^r \longrightarrow \mathbb{C} be an arithmetic function of r variables, where r\geq 2. We study multiple Dirichlet series defined by

    \begin{equation*} D(f_r,s_1,\ldots,s_r)=\sum\limits_{\substack{n_1,\ldots,n_r=1 \\ (n_1,\ldots,n_r)=1}}^{+\infty}\frac{f_r(n_1,\ldots,n_r)}{n_1^{s_1}\cdots n_r^{s_r}}, \end{equation*}

where f_r(n_1,\ldots,n_r)=f(n_1)\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.


  • Completely multiplicative function
  • Specially multiplicative function
  • Multiple Dirichlet series
  • Eulerian product
  • Riemann zeta function
  • Dirichlet L-function

2020 Mathematics Subject Classification

  • 11M32, 11M06, 11A25


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Manuscript history

  • Received: 22 February 2022
  • Revised: 23 September 2022
  • Accepted: 12 October 2022
  • Online First: 14 October 2022

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Cite this paper

Tahmi, N., & Derbal, A. (2022). Some multiple Dirichlet series of completely multiplicative arithmetic functions. Notes on Number Theory and Discrete Mathematics, 28(4), 603-616, DOI: 10.7546/nntdm.2022.28.4.603-616.

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