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Approximation of Arithmetical Functions by Additive Ones
Proceedings of the American Mathematical Society
Vol. 39, No. 1 (Jun., 1973), pp. 19-25
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2038982
Page Count: 7
You can always find the topics here!Topics: Number theoretic functions, Approximation, Additivity, Integers, Mathematical theorems, Asymptotic value, Mathematical functions, Coefficients, Prime numbers
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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.
Proceedings of the American Mathematical Society © 1973 American Mathematical Society