On Robin’s criterion for the Riemann Hypothesis

Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.