Title: Extensions of the class of multiplicative functions
Abstract: AbstractWe consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicativefunctions as extensions of the class of multiplicative functions. We apply these concepts to Ra-manujan’s sum and its analogue with respect to regular integers (mod r). Mathematics Subject Classi cation: 11A25, 11L03Keywords: quasimultiplicative function, semimultiplicative function, Selberg multiplicative function,Ramanujan’s sum, regular integer 1 Introduction An arithmetical function f: N !C is said to be multiplicative if f(mn) = f(m)f(n) for all m;n2Nwith (m;n) = 1. These functions play a central role in number theory. The works of E. T. Bell andR. Vaidyanathaswamy are prominent in the history of multiplicative functions, see e.g. [4, 24].Many of the classical arithmetical functions are multiplicative, e.g. the Mobius function, Euler’stotient function and the divisor functions. On the other hand, multiplicative functions have some weakpoints, e.g., they are destroyed by compositions such as cf(n);f(kn);f(k=n);f(n=k);f([k;n]), where[k;n] is the lcm of kand n. This has led to certain extensions of the class of multiplicative functions. Inthis paper we introduce quasimultiplicative, semimultiplicative and Selberg multiplicative functions,see [11, 15, 19]. As a motivation of these concepts we also consider multiplicative properties ofRamanujan’s sum and its analogue with respect to regular integers [9].There are also important subclasses of the class of multiplicative functions in the number theoreticliterature, e.g., the class of rational arithmetical functions, see [12]. We do not consider these classesin this paper.
Publication Year: 2012
Publication Date: 2012-01-01
Language: en
Type: article
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Cited By Count: 5
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