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On a conjecture of Erdős about additive functions. (English) Zbl 1361.11058

Theory Probab. Math. Stat. 89, 23-31 (2014); translation from Teor. Jmovirn. Mat. Stat. 89, 21–29 (2013).
Summary: For a real-valued additive function \(f: \mathbb{N} \to \mathbb{R}\) and for each \(n \in \mathbb{N}\) we define a distribution function
\[ F_n(x):= \frac{1}{n} \#\{m \leq n : f(m) \leq x \}. \]
In this paper we prove a conjecture of Erdős, 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.

MSC:

11N37 Asymptotic results on arithmetic functions
11N60 Distribution functions associated with additive and positive multiplicative functions
11K65 Arithmetic functions in probabilistic number theory
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