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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Asymptotic distribution of normalized arithmetical functions


Authors: Paul Erdős and Janos Galambos
Journal: Proc. Amer. Math. Soc. 46 (1974), 1-8
MSC: Primary 10K20
DOI: https://doi.org/10.1090/S0002-9939-1974-0357360-4
MathSciNet review: 0357360
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Abstract: Let $ f(n)$ be an arbitrary arithmetical function and let $ {A_N}$ and $ {B_N}$ be sequences of real numbers with $ 0 < {B_N} \to + \infty $ with $ N$. We give a sufficient condition for $ (f(n) - {A_N})/{B_N}$ to have a limiting distribution. The case when $ f(n)$ is defined by $ f(n) = \Sigma g(d)$, where the summation is over all divisors $ d$ of $ n$ and $ g(d)$ is any given arithmetical function, is discussed in more detail. A concrete example is given as an application of our result, in which example $ f(n)$ is neither additive nor multiplicative. Our method of proof is to approximate $ f(n)$ by a suitably chosen additive function, as proposed in [4], and then to apply general theorems available for additive functions.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0357360-4
Keywords: Arithmetical functions, additive, multiplicative, norming by constants, asymptotic distribution, approximation by additive functions, Erdös-Kac theorem, class $ {\text{H}}$ of Kubilius
Article copyright: © Copyright 1974 American Mathematical Society