Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  274.10043
Autor:  Erdös, Paul; Hall, R.R.
Title:  On the distribution of values of certain divisor functions. (In English)
Source:  J. Number Theory 6, 52-63 (1974).
Review:  Let {\epsilond } be a sequence of non-negative numbers and f(n) = sum {\epsilond: d | n }. The authors investigate under what circumstances there exists a continuous distribution function F(c) such that F(c) ––> 0 as c ––> oo and for each fixed c, card {n < x: f(n) > c } ~ F(c). They show that it is sufficient that sum {1/p: \epsilonp > 0 } = oo and for some fixed \beta > 0,

0 \leq \epsilond \leq 2- log log d-(1+\beta)(2 log log d. log log log log d)^{½}.     (1)

The authors also obtain the result F(c- \delta)-F(c) << (log 1/ \delta)- ½ uniformly for all c and \delta < ½ in the special cases \epsilond = (log d)- \alpha, (\alpha > log 2, d \geq 2) or when (1) holds with equality on the right. The conditions \alpha > log 2, \beta > 0 are best possible in their contexts.
Classif.:  * 11N37 Asymptotic results on arithmetic functions

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