Upper bound for sum of divisors function and the Riemann hypothesis

Abstract

Let $\sigma(n)$ denote the sum of divisors function. We prove that if $(2, m) = 1$ and $2m \gt 3^{9}$ then $1^{0} \sigma(2m) \lt \frac{39}{40}e^{\gamma}2m \log \log 2m$, and for all odd integers $m \gt \frac{3^{9}}{2}$, we have $2^{0} \sigma(m) \lt e^{\gamma}m \log \log m$. Moreover, we show that if $\sigma(2m) \lt \frac{3}{4}e^{\gamma}2m \log \log 2m$, for $m \gt m_{0}$ and $(2, m) = 1$, then the inequality $\sigma(2^{\alpha}m) \lt e^{\gamma}2^{\alpha}m \log \log 2^{\alpha}m$ is true for all integers $\alpha \geq 2$ and $m \gt m_{0}$. Robin criterion implies that the Riemann hypothesis is true for these cases.