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The Nicolas Criterion for the Riemann Hypothesis

EasyChair Preprint 7204

5 pagesDate: December 11, 2021

Abstract

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if $X_{n} > 0$ holds for all primes $p_{n} > 2$. For every prime number $p_{k} > 2$, $X_{k} > 0$ is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes $p_{n} > 2$. In this way, we demonstrate that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, Nicolas inequality, Riemann hypothesis, monotonicity, prime numbers

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@booklet{EasyChair:7204,
  author    = {Frank Vega},
  title     = {The Nicolas Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 7204},
  year      = {EasyChair, 2021}}
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