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**Zentralblatt MATH**

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

**Zbl.No: ** 478.10027

**Autor: ** Bateman, Paul T.; Erdös, Paul; Pomerance, Carl; Straus, E.G.

**Title: ** The arithmetic mean of the divisors of an integer. (In English)

**Source: ** Analytic number theory, Proc. Conf., Temple Univ./Phila. 1980, Lect. Notes Math. 899, 197-220 (1981).

**Review: ** [For the entire collection see Zbl 465.00008.]

This paper establishes the following interesting and deep results about the arithmetic function A, defined by A(n) = \sigma(n)/d(n), i.e. A(n) is the arithmetic mean of the divisors of n: If N(x) denotes the number of integers n with n \leq x and A(n) not an integer, then N(x) = x\exp**(**-(1+o(1))2\sqrt{log2}\sqrt{log log x}**)**, (1)

**sum**_{n \leq x}A(n) ~ cx^{2}(log x)^{- ½}, with c an explicity given constant, (2)

**sum**_{A(n) \leq x}1 ~ \lambda x log x, again with \lambda an explicity given constant. (3) Another teorem, in connection with (1), is the following: Denote for every positive real number \beta the number **prod**_{pa||n}p^{[\alpha\beta]} by < n^{\beta}>. Then for any \epsilon between 0 and 2, the set of integers n for which < d(n)^{2-\epsilon}>/sigma(n) has asymptotic density 1, the set of n for which < d(n)^{2+\epsilon}>/\sigma(n) has asymptotic density 0, and the set of n for which d(n)^{2}/\sigma(n) has asymptotic desity ½. The proofs are long and complicated, with applications of results from various parts of number theory. To mention only a few: sieve methods, the generalized Erdös-Kac theorem and Tauberian theorems of Delange.

**Reviewer: ** H.Jager

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

11N05 Distribution of primes

**Keywords: ** divisor function; sum of divisor function; arithmetic mean of divisors; asymptotic density

**Citations: ** Zbl.465.00008

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