A study on some generalized multiplicative and generalized additive arithmetic functions

D. Bhattacharjee
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 32–44
DOI: 10.7546/nntdm.2021.27.1.32-44
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Authors and affiliations

D. Bhattacharjee
Department of Mathematics, North-Eastern Hill University
Permanent Campus, Shillong-793022, India

Abstract

In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.

Keywords

• Arithmetic function
• Multiplicative function
• Arithmetic convolution
• Dirichlet convolution
• Narkiewicz’s A-product
• B-product
• Multiplicative B-product
• GA-function
• GM-function

2010 Mathematics Subject Classification

• 11A25

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