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On the limit distributions for some sets of additive arithmetic functions
Last modified: 2014-01-29
Abstract
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\title{On the limit distributions for some sets of additive arithmetic functions}\author{Gediminas Stepanauskas}\address{Department of Mathematical Computer Science \\ Vilnius University \\ Naugarduko 24 \\ 03225 Vilnius, Lithuania}\email{gediminas.stepanauskas@mif.vu.lt}
\author{Jonas \v Siaulys}\address{Department of Mathematical Analysis \\ Vilnius University \\ Naugarduko 24 \\ 03225 Vilnius, Lithuania}\email{jonas.siaulys@mif.vu.lt}
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The limit behaviour of distributions \((1/[x])\sum\nolimits_{n\le x\atop f_x(n)-\alpha(x)<u}1 \)was consi\-de\-red in the probabilistic number theory very often.There were considered various classes of additive functions \(f_x(n)=\sum\nolimits_{p^r||n}f_x(p^r) \) with different centeringfunctions $\alpha(x)$. But the class of examined additivefunctions $f_x$ was of a special expression:\begin{equation}\label{pipi2}f_x(n)=h(n)/\beta(x),\end{equation}where $f$ is some additive function and $\beta(x)$ is someunboundedly increa\-sing function. In the books \cite{Elliott} and \cite{Kubilius}, and works cited there, one canfind almost all classical results and their historical context.
An object of our talk is strongly additive functions taking thevalues $0$ or $1$ on the set of primes and maybe depending on $x$(therefore we call usually $f_x$ by a set of additive functions), and in general case it isimpossible to express $f_x$ by relation (\ref{pipi2}). We willdiscuss about the weak convergence of distributions\[(1/[x])\sum\nolimits_{n\le x\atop f_x(n)<u}1\] for the set of strongly additive functions $f_x$ as $x\to \infty$and about the class of possible limit laws.
\begin{thebibliography}{9}\bibitem{Elliott} P.D.T.A.~Elliott, \emph{Probabilistic Number Theory, I},Springer-Verlag, New York, 1979; \emph{Probabilistic NumberTheory, II}, Springer-Verlag, New York, 1980.
\bibitem{Kubilius} J.~Kubilius, \emph{Probabilistic Methods in the Theory of Numbers}, Providence, Amer. Math. Soc. Transl. of Math. Monographs, \textbf{11}, 1964.\end{thebibliography}
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\usepackage[text={130mm,200mm}]{geometry}\pagestyle{empty}
\begin{document}
\title{On the limit distributions for some sets of additive arithmetic functions}\author{Gediminas Stepanauskas}\address{Department of Mathematical Computer Science \\ Vilnius University \\ Naugarduko 24 \\ 03225 Vilnius, Lithuania}\email{gediminas.stepanauskas@mif.vu.lt}
\author{Jonas \v Siaulys}\address{Department of Mathematical Analysis \\ Vilnius University \\ Naugarduko 24 \\ 03225 Vilnius, Lithuania}\email{jonas.siaulys@mif.vu.lt}
\maketitle
\thispagestyle{empty}
The limit behaviour of distributions \((1/[x])\sum\nolimits_{n\le x\atop f_x(n)-\alpha(x)<u}1 \)was consi\-de\-red in the probabilistic number theory very often.There were considered various classes of additive functions \(f_x(n)=\sum\nolimits_{p^r||n}f_x(p^r) \) with different centeringfunctions $\alpha(x)$. But the class of examined additivefunctions $f_x$ was of a special expression:\begin{equation}\label{pipi2}f_x(n)=h(n)/\beta(x),\end{equation}where $f$ is some additive function and $\beta(x)$ is someunboundedly increa\-sing function. In the books \cite{Elliott} and \cite{Kubilius}, and works cited there, one canfind almost all classical results and their historical context.
An object of our talk is strongly additive functions taking thevalues $0$ or $1$ on the set of primes and maybe depending on $x$(therefore we call usually $f_x$ by a set of additive functions), and in general case it isimpossible to express $f_x$ by relation (\ref{pipi2}). We willdiscuss about the weak convergence of distributions\[(1/[x])\sum\nolimits_{n\le x\atop f_x(n)<u}1\] for the set of strongly additive functions $f_x$ as $x\to \infty$and about the class of possible limit laws.
\begin{thebibliography}{9}\bibitem{Elliott} P.D.T.A.~Elliott, \emph{Probabilistic Number Theory, I},Springer-Verlag, New York, 1979; \emph{Probabilistic NumberTheory, II}, Springer-Verlag, New York, 1980.
\bibitem{Kubilius} J.~Kubilius, \emph{Probabilistic Methods in the Theory of Numbers}, Providence, Amer. Math. Soc. Transl. of Math. Monographs, \textbf{11}, 1964.\end{thebibliography}
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