On Robin's inequality

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 $n\ge 5041$, 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.

v3: A typo in Theorem 1.4 is fixed