## 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 ––> oosumf(n) \leq y1 ~ 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) = sumn = 1oof(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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