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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 449.10033

**Autor: ** Diamond, Harold G.; Erdös, Paul

**Title: ** Multiplicative functions whose values are uniformly distributed in (0,oo). (In English)

**Source: ** Proc. Queen's Number Theory Conf. 1979, Queen's Pap. Pure Appl. Math. 54, 329-378 (1980).

**Review: ** A positive valued arithmetic function f: **N** ––> \Bbb r^+ which tends to infinity as n ––> oo has values uniformly distributed in (0,oo) if there exists a positive constant d in **R**^+ such that for y ––> oo**sum**_{f(n) \leq y}1 ~ dy. The number d will be called the density of values. Under certain conditions the uniform distribution of the values of a multiplicative function f is equivalent to the behavior of F(s) = **sum**_{n = 1}^{oo}f(n)^{-s} near s = 1. There is also a connection between the uniform distribution of the values of the multiplicative function f in (0,oo) and the existence of a positive mean values of the arithmetic function h(n) = n/f(n). The cases d = 0 and d = oo are included in a similar way. Several examples show that some of the theorem fail if the condition are weakened.

**Reviewer: ** D.Leitmann

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

11A25 Arithmetic functions, etc.

11K65 Arithmetic functions (probabilistic number theory)

**Keywords: ** arithmetic function; uniform distribution; density of values

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