Problem #WSP-000344

Problem

Let $\sigma (n)$ be the sum of the divisors of $n$. For example, $\sigma (12)=1+2+3+4+6+12=28$. We use $\gamma$ to denote the Euler-Mascheroni constant - one way to define this is as $\gamma :=\underset{n\to \mathrm{\infty }}{lim}(\sum _{k=1}^n\frac{1}{n}-\log n)$.

Prove that $\sigma (n)<e^{\gamma }n\log \log n$ for all integers $n>5040$.