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 be an arbitrary arithmetical function and let and be sequences of real numbers with with . We give a sufficient condition for to have a limiting distribution. The case when is defined by , where the summation is over all divisors of and is any given arithmetical function, is discussed in more detail. A concrete example is given as an application of our result, in which example is neither additive nor multiplicative. Our method of proof is to approximate by a suitably chosen additive function, as proposed in [4], and then to apply general theorems available for additive functions.

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Additional Information

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 of Kubilius

Article copyright:
© Copyright 1974
American Mathematical Society