Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 291.10040
Autor: Erdös, Paul
Title: On the distribution of numbers of the form \sigma (n)/n and on some related questions. (In English)
Source: Pac. J. Math. 52, 59-65 (1974).
Review: An arithmetic function f is said to have a distribution function, if for any c the density g(c) of integers satisfying f(n) < c exists and g(- oo) = 0, g(oo) = 1. Let f(n) = \sigma (n)/n, where \sigma is the sum of divisors function. Then the distribution function g is known to exist, is continuous and monotonic but purely singular. Let F(x; a,b) be the number of integers n \leq x satisfying a \leq \sigma (n)/n < b. The author proves the theorem: There is an absolute constant c1 so that for x > t F (x; a,a+ 1/t ) < c1x/ ln t, where apart from the constant c1 the inequality is best possible. Further from the author's work can be derived some best possible estimates for g (c+ 1/t ) -g(t) for the case of \sigma (n)/n. The author also refers to the relevant problems of abundant numbers and of amicable pairs of numbers. Further he deals with the case where \sigma is replaced by Euler's \phi function and sharpens some earlier known results.
Reviewer: S.M.Kerawala
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
11A25 Arithmetic functions, etc.
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