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Approximation of Arithmetical Functions by Additive Ones

Janos Galambos
Proceedings of the American Mathematical Society
Vol. 39, No. 1 (Jun., 1973), pp. 19-25
DOI: 10.2307/2038982
Stable URL: http://www.jstor.org/stable/2038982
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Approximation of Arithmetical Functions by Additive Ones
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Abstract

Let εp(n) = 1 or 0 according as p ∣ n or not. Since the functions εp(n) - 1/p are quasi-orthogonal on the integers 1, 2, ⋯, 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.

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