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On a conjecture of Erdös about additive functions


Author: Karl-Heinz Indlekofer
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 23-31
MSC (2010): Primary 11N37, 11N60, 11K65
DOI: https://doi.org/10.1090/S0094-9000-2015-00932-7
Published electronically: January 26, 2015
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Abstract: For a real-valued additive function $ f\colon \mathbb{N} \rightarrow \mathbb{R}$ and for each $ n \in \mathbb{N}$ we define a distribution function

$\displaystyle F_{n}(x):=\frac {1}{n}\char93 \{m\leq n\colon f(m)\leq x\}. $

In this paper we prove a conjecture of Erdös, which asserts that in order for the sequence $ F_{n}$ to be (weakly) convergent, it is sufficient that there exist two numbers $ a<b$ such that $ \lim _{n\rightarrow \infty }(F_{n}(b)-F_{n}(a))$ exists and is positive.

The proof is based upon the use of the Stone-Čech compactification $ \beta \mathbb{N}$ of $ \mathbb{N}$ to mimic the behavior of an additive function as a sum of independent random variables.


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Additional Information

Karl-Heinz Indlekofer
Affiliation: Department of Mathematics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
Email: k-heinz@math.upb.de

DOI: https://doi.org/10.1090/S0094-9000-2015-00932-7
Keywords: Probabilistic number theory, additive functions
Received by editor(s): January 29, 2013
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society