On Nicolas Criterion for the Riemann Hypothesis

EasyChair Preprint 10961, version 2

Abstract

The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 1983, Nicolas stated that the Riemann hypothesis is true if and only if the inequality $\prod_{q\leq x} \frac{q}{q - 1} > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 2$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. In this note, using the Nicolas criterion, we prove that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, Riemann hypothesis, Riemann zeta function, prime numbers

@booklet{EasyChair:10961,
  author    = {Frank Vega},
  title     = {On Nicolas Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 10961},
  year      = {EasyChair, 2023}}