Robin's inequality for new families of integers

Abstract

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality $\sigma (n):={\sum }_{p|n}p<e^{\gamma }n\log \log n$ 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 ${\nu }_2(n)\le 19$, ${\nu }_3(n)\le 12$,${\nu }_5(n)\le 7$, ${\nu }_7(n)\le 6$, ${\nu }_{11}(n)\le 5$ then Robin's inequality is satisfied, where ${\nu }_p(n)$ is the p-adic order of $n$. In the end we show that $\sigma (n)/n<1.0000005645e^{\gamma }\log \log n$ holds unconditionally for $n>5040$.

arXiv comment6 pages