Arithmetical functions associated with conjugate pairs of sets under regular convolutions

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 656–665
DOI: 10.7546/nntdm.2022.28.4.656-665
Full paper (PDF, 179 Kb)

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Authors and affiliations

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

Abstract

Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.

Keywords

• Conjugate pair
• Regular convolution
• Möbius function
• Totient function
• Inversion formula

2020 Mathematics Subject Classification

• 11A25

References

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Manuscript history

• Received: 8 September 2022
• Accepted: 17 October 2022
• Online First: 24 October 2022