## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probability and Number Theory — Kanazawa 2005, S. Akiyama, K. Matsumoto, L. Murata and H. Sugita, eds. (Tokyo: Mathematical Society of Japan, 2007), 133 - 170

### New approach to probabilistic number theory - compactifications and integration

#### Abstract

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.

#### Article information

**Dates**

Received: 10 January 2006

Revised: 6 December 2006

First available in Project Euclid:
27 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1548550897

**Digital Object Identifier**

doi:10.2969/aspm/04910133

**Mathematical Reviews number (MathSciNet)**

MR2405602

**Zentralblatt MATH identifier**

1228.11125

**Subjects**

Primary: 11K65: Arithmetic functions [See also 11Nxx]

Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions

#### Citation

Indlekofer, Karl-Heinz. New approach to probabilistic number theory - compactifications and integration. Probability and Number Theory — Kanazawa 2005, 133--170, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04910133. https://projecteuclid.org/euclid.aspm/1548550897