On Robin’s inequality

Abstract

Let sigma (n) denote 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) < textrm{e}^{gamma }n log log n holds for every integer n > 5040, where gamma is the Euler–Mascheroni constant. In this paper we establish a new family of integers for which Robin’s inequality sigma (n) < textrm{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.