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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 $$N(x)=x\exp\left(-(1+o(1))2\sqrt{\log2}\sqrt{\log\log x}\right),\tag1$$ $$\sum_{n\leq x}A(n)\sim cx^2(\log x)^{-1/2}, \text{ with c an explicity given constant},\tag 2$$ $$\sum_{A(n)\leq x}1\sim\lambda x\log x, \text{ again with } \lambda \text{ an explicity given constant}.\tag 3$$ 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

11N37Asymptotic results on arithmetic functions
11N05Distribution of primes