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Approximations of arithmetical functions by additive ones

Author: János Galambos
Journal: Proc. Amer. Math. Soc. 39 (1973), 19-25
MSC: Primary 10K20
MathSciNet review: 0311614
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Abstract: Let $ {\varepsilon _p}(n) = 1$ or 0 according as $ p\vert n$ or not. Since the functions $ {\varepsilon _p}(n) - 1/p$ are quasi-orthogonal on the integers $ 1,2, \cdots ,N$ with the relative frequency as measure, the theory of orthogonal expansions suggests an approximation of arbitrary arithmetical functions by strongly additive ones. In the present note, the approximating additive functions are determined and a sufficient condition is given for an arithmetical function to have an asymptotic distribution. Examples are given to illustrate the result.

References [Enhancements On Off] (What's this?)

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Keywords: Arithmetical functions, additive, multiplicative, limiting distribution, asymptotic mean value, approximation, quasi-orthogonal functions, Erdös theorem, Turán-Kubilius inequality
Article copyright: © Copyright 1973 American Mathematical Society

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