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arXiv · 2509.11182

On the verification of a Nicolas inequality

Orlando Galdames-Bravo
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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.

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Orlando Galdames-Bravo. 2025-09-14. On the verification of a Nicolas inequality. https://arxiv.org/abs/2509.11182

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