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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)=σ(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 nx and A(n) not an integer, then

N(x)=xexp-(1+o(1))2log2loglogx,(1)
nx A(n)cx 2 (logx) -1/2 ,withcanexplicitygivenconstant,(2)
A(n)x 1λxlogx,againwithλanexplicitygivenconstant·(3)

Another teorem, in connection with (1), is the following: Denote for every positive real number β the number p a ||n p [αβ] by n β . Then for any ε between 0 and 2, the set of integers n for which d(n) 2-ε /sigma(n) has asymptotic density 1, the set of n for which d(n) 2+ε /σ(n) has asymptotic density 0, and the set of n for which d(n) 2 /σ(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:
11N37Asymptotic results on arithmetic functions
11N05Distribution of primes