Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 275.10028
Autor: Erdös, Paul; Galambos, Janos
Title: Asymptotic distribution of normalized arithmetical functions. (In English)
Source: Proc. Am. Math. Soc. 46, 1-8 (1974).
Review: Let f(n) be an arbitrary arithmetical function and let AN and BN > 0 be sequences of real numbers such that BN ––> +oo with N. Let N \nuN(n: ...) denote the number of those integers n \leq N for which the property stated in the dotted space holds. Our aim in the present paper is to determine sequences AN and BN for which \nuN(n: f(n)-AN < xBN) = F(x)+o(1) for all continuity points of a distribution function F(x). We use the method proposed by the second named author [Proc. Amer. math. Soc. 39, 19-25 (1973; Zbl 246.10039)]. That is, we construct additive functions GN(n) which are close in quadratic mean to f(n) and for which a relation of the form \nuN(n: GN(n)-A^*N < xBN) = F(x)+o(1) is known to hold. From A^*N and GN(n) we then determine AN. A general theorem of the above nature is proved in the first part while in the second one we discuss in more detail the case when f(n) = sumd | n g(d) with a given arithmetical function g(d). Notice that if g(d) = 0 for all d except when d is the power of a prime number then f(n) is additive while if g(d) is multiplicative, then so is f(n). The most natural extension of the distribution theory of additive and multiplicative functions would therefore be for f(n) of the form given above (perhaps with some restrictions on g(d)). We obtain a sufficient condition for the existence of a limit law F(x) for (f(n)-AN)/BN with specific AN and BN. An example is given for illustration of the method and of the result.
Reviewer: P.Erdös
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
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