Let \(n\) be a positive integer. We denote by \(s(n)\) the sum of the divisors of \(n\). For example, the divisors of \(n=6\) are \(1\), \(2\), \(3\) and \(6\), so \(s(6)=1+2+3+6=12\). Prove that, for all \(n\ge1\), \[\sum_{k=1}^ns(k)=s(1)+s(2)+...+s(n)\le\frac{\pi^2}{12}n^2+\frac{n\log n}{2}+\frac{n}{2}.\]