Robin's inequality for new families of integers

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma (n):={\sum }_{p\mathrm{\mid }n}p<e^{\gamma }n\log \log n$ is satisfied for $n>5040$, where $\gamma$ denotes the Euler-Mascheroni constant. We show that if the 2-adic order of n is big enough in comparison to the odd part of n then Robin’s inequality is satisfied. We also show that if an positive integer $n$ satisfies either ${\nu }_2(n)\le 19$, $\nu_3(n) \leq 12$,$\nu_5(n) \leq 7$,$\nu_7(n) \leq 6$,$\nu_{11}(n) \leq 5$ then Robin’s inequality is satisfied, where ${\nu }_p(n)$ is the p-adic order of $n$. In the end we show that $\sigma (n)\mathrm{/}n<1.0000005645e^{\gamma }\log \log n$ holds unconditionally for $n>5040$.