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**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 **{**\epsilon_{d} **}** be a sequence of non-negative numbers and f(n) = **sum** **{**\epsilon_{d}: 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: \epsilon_{p} > 0 **}** = oo and for some fixed \beta > 0, 0 \leq \epsilon_{d} \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 \epsilon_{d} = (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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