On the distribution of numbers of the form \(\sigma(n)/n\) and on some related questions
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 \(g(- \infty)=0\), \(g(\infty)=1\). 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 \(c_1\) so that for \(x>t\) \(F \left(x;a,a+{1 \over t} \right) <c_1x/\ln t\), where apart from the constant \(c_1\) the inequality is best possible. Further from the author's work can be derived some best possible estimates for \(g \left(c+{1 \over t} \right) -g(t)\) 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 \(\varphi\)-function and sharpens some earlier known results.
scientific article; zbMATH DE number 3455353