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

Source abstract

Let σ ( n ) \sigma (n) be the sum of the positive divisors of n n . We show that the natural density of the set of integers n n satisfying σ ( n ) / n ≥ t \sigma (n)/n\ge t is given by exp ⁡ { − e t e − γ ( 1 + O ( t − 2 ) ) } \exp \left \{ -e^{t \, e^{-\gamma }} \left (1+O\left ({t^{-2}}\right )\right ) \right \} , where γ \gamma denotes Euler’s constant. The same result holds when σ ( n ) / n \sigma (n)/n is replaced by n / φ ( n ) n/\varphi (n) , where φ \varphi is Euler’s totient function.