Robin's inequality for new families of integers

TL;DR

This paper investigates Robin's inequality related to the Riemann Hypothesis, establishing new conditions on the p-adic orders of integers that guarantee the inequality holds, and provides an unconditional bound for large n.

Contribution

It introduces new criteria based on p-adic orders of integers that ensure Robin's inequality, advancing understanding of the inequality's validity for various integer classes.

Findings

Robin's inequality holds if the 2-adic order of n is sufficiently large compared to its odd part.

Robin's inequality is satisfied for integers with bounded p-adic orders for p=3,5,7,11.

An unconditional upper bound for the ratio σ(n)/n is established for n > 5040.

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 $\nu_2(n) \leq 19$, $\nu_3(n) \leq 12$,$\nu_5(n) \leq 7$, $\nu_7(n) \leq 6$, $\nu_{11}(n) \leq 5$ then Robin's inequality is satisfied, where $\nu_p(n)$ is the p-adic order of $n$. In the end we show that $\sigma(n)/n < 1.0000005645 e^{\gamma}\log \log n$ holds unconditionally for $n > 5040$.