On Robin’s criterion for the Riemann Hypothesis

Abstract

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.

On Ramanujan's prime counting inequality

“…For instance, the present author [7,Theorem 1.3] proved that the inequality σ(n) n < e γ log 2 n + 0.0094243e γ log 2 2 n holds unconditionally for every integer n > 5040. The initial motivation to write this paper is based on a recent paper by Aoudjit, Berkane, and Dusart [4] concerning another upper bound for σ(n)/n. Balazard [11, p. 257] showed that for every positive integer N , one has…”