Quotients of arithmetical functions under the Dirichlet convolution

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 185–194
DOI: 10.7546/nntdm.2023.29.2.185-194
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Authors and affiliations

Pentti Haukkanen
Faculty of Information Technology and Communication Sciences,
FI-33014 Tampere University, Finland

Abstract

We study existence of a solution of the arithmetical equation f\ast h = g in f, where f\ast h is the Dirichlet convolution of arithmetical functions f and h, and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function \mu and the so-called totients. As applications we also present our results on the arithmetical equation f\ast h = g in the language of Cauchy convolution and further deconvolution in discrete linear systems.

Keywords

  • Arithmetical equation
  • Dirichlet convolution
  • Möbius function
  • Totient function
  • Cauchy convolution
  • Discrete linear system

2020 Mathematics Subject Classification

  • 11A25

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Manuscript history

  • Received: 30 January 2023
  • Revised: 11 March 2023
  • Accepted: 22 March 2023
  • Online First: 3 April 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Haukkanen, P. (2023). Quotients of arithmetical functions under the Dirichlet convolution. Notes on Number Theory and Discrete Mathematics, 29(2), 185-194, DOI: 10.7546/nntdm.2023.29.2.185-194.

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