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)

sumn \leq xA(n) ~ cx2(log x)- ½, with c an explicity given constant,    (2)

sumA(n) \leq x1 ~ \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 prodpa||np[\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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