On the verification of a Nicolas inequality
Orlando Galdames-Bravo
TL;DR
The paper investigates the Nicolas inequality $e^\gamma \log\log N_x < N_x/\varphi(N_x)$ by expressing it through $N_x$, $\varphi_x$, and the Mertens error term in $\sum_{p\le x} 1/p$. It derives conditional claims: if the Mertens error term is eventually bounded below by a positive multiple of $1/\log x$, then the inequality holds; if it is bounded above by a nonpositive multiple, then it fails on infinitely many $x$, with the oscillation of $\theta(x)-x$ playing a central role. The analysis rewrites the inequality using $\log\varphi_x = \theta(x) - \sum_{n\ge1} P_x(n)/n$ and studies the sums $P_x(n)$ to connect to the Mertens constant $b$ and Euler's constant $\gamma$, showing that current estimates cannot fix the sign. The work highlights that the underlying error terms oscillate similarly to classical arithmetical functions, complicating a decisive resolution.
Abstract
Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^γ\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 $x$, $γ$ 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.