On the verification of a Nicolas inequality

Abstract

Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^\gamma \log\log N_x < \dfrac{N_x}{\varphi(N_x)}\,, \end{equation} where $x\ge 2$, $N_x$ denotes the product of the primes less or equal than $x$, $\gamma$ is the Euler constant and $\varphi$ is the Euler totient function. We see that verification of such an inequality depends on the sign of the big-O function in the Mertens estimate for the sum of reciprocals of primes that. Then we analyze the sign of such an error term.