The Riemann hypothesis and an upper bound of the divisor function for odd integers

Abstract

In [6] Robin showed that the Riemann hypothesis is equivalent to the statement that Robin's inequality sigma (n) < e(gamma) n log log n holds for n >= 5041, where gamma is the Euler-Mascheroni constant. We provide a sharper bound for sigma (n) than Robin's one for integers, by using the ideas of Choie et al. [1], and show that Robin's inequality holds for n not equivalent to 0 (mod 3) with finitely many exceptions.