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 σ(n)<eγnlog⁡log⁡n holds for n≥5041, where γ is the Euler–Mascheroni constant. We provide a sharper bound for σ(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≢0(mod3) with finitely many exceptions.