On Robin's inequality

TL;DR

This paper investigates Robin's inequality related to the Riemann hypothesis, establishing new integer families satisfying the inequality and providing an unconditional upper bound for the sum of divisors function using prime number approximations.

Contribution

The paper introduces a new family of integers satisfying Robin's inequality and derives an unconditional upper bound for the sum of divisors function.

Findings

New family of integers satisfying Robin's inequality

Unconditional upper bound for the sum of divisors function

Use of Chebyshev's θ-function and prime product approximations

Abstract

Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq 5041$, where $\gamma$ is the Euler-Mascheroni constant. In this paper we establish a new family of integers for which Robin's inequality $\sigma(n) < e^{\gamma}n \log \log n$ hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev's $\vartheta$-function and for some product defined over prime numbers.