Properties of counterexample to Robin hypothesis

Let $G(n)=\sigma (n)\mathrm{/}(n\log \log n)$. Robin made hypothesis that $G(n)<e^{\gamma }$ for all integer $n>5040$. If there exists counterexample to Robin hypothesis, then there must exist finite number of counterexamples $n>5040$ such that $G(n)$ attains largest value. This article studies various properties of such number.