The aliquot constant (Q2890801)

Let \(s(n)\) denote the sum of the proper divisors of a natural number \(n\): NEWLINE\[NEWLINE s(n)=\sum_{{d|n}\atop{1\leq d<n}}d. NEWLINE\]NEWLINENEWLINENEWLINEThe sequence of numbers \( n,s(n),s^2(n),s^3(n),\dots \) is called an aliquot sequence with a starting value \(n\).NEWLINENEWLINEIn the paper, it is shown that the limit NEWLINE\[NEWLINE \lim_{N\rightarrow\infty}\frac1{N}\sum_{n=1}^N\log\frac{s(2n)}{2n}=\lambda NEWLINE\]NEWLINE exists and that this limit does not exceed \(-0.03\). It follows from this that, the geometric mean of \(s(2n)/2n\), the growth factor of the function \(s\) on even numbers, in the long run tends to be less than \(1\). In some sense, this fact may be taken as probabilistic evidence that aliquot sequences with an even starting value tend to be bounded.