Characterizations of L-additive functions via generalized arithmetic convolutions

Champak Talukdar, Debashis Bhattacharjee and Helen K. Saikia
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 15–22
DOI: 10.7546/nntdm.2026.32.1.15-22
Full paper (PDF, 189 Kb)

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Authors and affiliations

Champak Talukdar
Department of Mathematics, Behali Degree College
Assam, India
Department of Mathematics, Gauhati University
Assam, India

Debashis Bhattacharjee
Retd. Prof., Department of Mathematics, North Eastern Hill University
Meghalaya, India

Helen K. Saikia
Department of Mathematics, Gauhati University
Assam, India

Abstract

This paper investigates the properties of L-additive functions within the algebraic frameworks of two generalized arithmetic convolutions: the K-convolution and Narkiewicz’s A-convolution. We establish the foundational algebraic context for these operations by citing the established conditions for their associativity and commutativity. Our main results provide rigorous characterization theorems for completely additive and L-additive functions, which manifest as Leibniz-type rules that these functions satisfy with respect to the convolutions. Furthermore, we provide insightful, non-trivial examples using classical arithmetic functions to illustrate the mechanics of these characterizations, thereby demonstrating the utility of the generalized convolution framework in the study of arithmetic derivatives and their generalizations.

Keywords

• Arithmetic function
• L-additive function
• Arithmetic derivative
• K-convolution
• Narkiewicz’s A-convolution
• Completely additive function

2020 Mathematics Subject Classification

• 11A25

References

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Manuscript history

• Received: 8 November 2025
• Revised: 9 February 2026
• Accepted: 15 February 2026
• Online First: 18 February 2026

Copyright information

Ⓒ 2026 by the Authors.
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).