The arithmetic mean of the divisors of an integer. (English) Zbl 0478.10027

Bateman, Paul T.; Erdős, Paul; Pomerance, Carl; Straus, E. G.

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

Keywords:

divisor function; sum of divisor function; arithmetic mean of divisors; asymptotic density

Citations:

Zbl 0465.00008