The distribution functions of σ(n)/n and n/ϕ(n), II

2010/11/18 by Andreas Weingartner, Weingartner, Andreas
Mathematics · #11N25 #11N60 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N25 #msc:11N60

paper · pdf · doi:10.48550/arxiv.1011.4262

11 pages

arxiv created 2010/11/18 · arxiv updated 2010/11/19

Abstract

Let σ(n) be the sum of the positive divisors of n, and let A(t) be the natural density of the set of positive integers n satisfying σ(n)/n ≥ t. We give an improved asymptotic result for log A(t) as t grows unbounded. The same result holds if σ(n)/n is replaced by n/ϕ(n), where ϕ(n) is Euler's totient function.

Related

• Distribution mod p of Euler's totient and the sum of proper divisors
• Distribution of values of general Euler totient function
• The number of preimages of iterates of ϕ and σ
• The distribution of gcd(n,ϕ(n))
• On the natural density of integers n for which σ(kn+r₁) >σ(kn+r₂)