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VOL. 49 | 2007 New approach to probabilistic number theory - compactifications and integration
Karl-Heinz Indlekofer

Editor(s) Shigeki Akiyama, Kohji Matsumoto, Leo Murata, Hiroshi Sugita


For primes $p$ let $$ A_p := \{n : n \in \mathbb{N},\ p|n\} $$ be the set of all natural numbers divisible by $p$. In his book "Probabilistic methods in the Theory of Numbers" (1964) J. Kubilius applies finite probabilistic models to approximate independence of the events $A_p$. His models are constructed to mimic the behaviour of (truncated) additive functions by suitably defined independent random variables.

Embedding $\mathbb{N}$, endowed with the discrete topology, in the compact space $\beta \mathbb{N}$, the Stone-Čech compactification of $\mathbb{N}$, and taking $\overline{A_p} := \mathrm{clos}_{\beta \mathbb{N}} A_p$ leads to independent events $\overline{A_p}$. This observation is a motivation for a general integration theory on $\mathbb{N}$ which can be used in various topics of Probabilistic Number Theory. In this paper we present a short compendium of Probabilistic Number Theory concerning the distribution of arithmetical functions. The new model is applied to the result of Erdös and Wintner about the limit distribution of additive functions and to the famous result of Szemeredi in combinatorical number theory. Further applications are given with respect to spaces of limit periodic and almost periodic functions and recent results on q-multiplicative functions.


Published: 1 January 2007
First available in Project Euclid: 27 January 2019

zbMATH: 1228.11125
MathSciNet: MR2405602

Digital Object Identifier: 10.2969/aspm/04910133

Primary: 11K65
Secondary: 11N64

Rights: Copyright © 2007 Mathematical Society of Japan


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