Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593592 | Journal of Number Theory | 2015 | 16 Pages |
Abstract
Let ω(n)ω(n) (resp. Ω(n)Ω(n)) denote the number of prime divisors (resp. with multiplicity) of a natural number n . In 1917, Hardy and Ramanujan proved that the normal order of ω(n)ω(n) is loglognloglogn, and the same is true of Ω(n)Ω(n); roughly speaking, a typical natural number n has about loglognloglogn prime factors. We prove a similar result for ω(s(n))ω(s(n)), where s(n)s(n) denotes the sum of the proper divisors of n : For all n≤xn≤x not belonging to a set of size o(x)o(x),|ω(s(n))−loglogs(n)|<ϵloglogs(n),|ω(s(n))−loglogs(n)|<ϵloglogs(n), and the same is true for Ω(s(n))Ω(s(n)).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Lee Troupe,