On Robin's inequality

Let sigma(n) denote the sum of divisors function of a positive intege rn. Robin proved that the Riemann hypothesis is true if and only if the inequality sigma(n)< e(gamma)n log lognholds for every integern > 5040, where gamma is the Euler-Mascheroni constant. In this paper weestablish a new family of integers for which Robin's inequality sigma(n)< e(gamma)n log lognhold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev's theta-function and for some product defined over prime numbers.