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The arithmetic mean of the divisors of an integer. (English) Zbl 0478.10027
Analytic number theory, Proc. Conf., Temple Univ./Phila. 1980, Lect. Notes Math. 899, 197-220 (1981).
[For the entire collection see Zbl 0465.00008.]
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 $N(x)=x\exp\left(-(1+o(1))2\sqrt{\log2}\sqrt{\log\log x}\right),\tag{1}$ $\sum_{n\leq x}A(n)\sim cx^2(\log x)^{-1/2}, \text{ with c an explicity given constant},\tag{2}$ $\sum_{A(n)\leq x}1\sim\lambda x\log x, \text{ again with } \lambda \text{ an explicity given constant}.\tag{3}$ Another teorem, in connection with (1), is the following: Denote for every positive real number $$\beta$$ the number $$\prod_{p^a||n}p^{[\alpha\beta]}$$ by $$\langle n^{\beta}\rangle$$. Then for any $$\varepsilon$$ between 0 and 2, the set of integers $$n$$ for which $$\langle d(n)^{2-\varepsilon}\rangle/sigma(n)$$ has asymptotic density 1, the set of $$n$$ for which $$\langle d(n)^{2+\varepsilon}\rangle/\sigma(n)$$ has asymptotic density 0, and the set of $$n$$ for which $$d(n)^2/\sigma(n)$$ has asymptotic desity $$1/2$$. 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 Erdős-Kac theorem and Tauberian theorems of Delange.
Reviewer: H.Jager

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11N05 Distribution of primes