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The factorial moments of additive functions with rational argument

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Multiplicative number theory

Published online by Cambridge University Press:  09 April 2009

J. Šiaulys  and
G. Stepanauskas
J. Šiaulys
Affiliation:
Department of Probability Theory and Number Theory Vilnius UniversityNaugarduko 24 03225 VilniusLithuania e-mail: jonas.siaulys@mif.vu.lt
G. Stepanauskas
Affiliation:
Department of Mathematical Informatics Vilnius UniversityNaugarduko 24 03225 VilniusLithuania e-mail: gediminas.stepanauskas@maf.vu.lt
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Abstract

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We consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

Keywords

additive arithmetic functionfactorial momentlimit distributionFarey fraction.

MSC classification

Secondary: 11K65: Arithmetic functions 11N37: Asymptotic results on arithmetic functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 3 , December 2006 , pp. 425 - 440
Copyright
Copyright © Australian Mathematical Society 2006

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