Proposed Proof of the Riemann Hypothesis

EasyChair Preprint no. 6967

3 pagesDate: November 1, 2021

Abstract

For every prime number $q_{n}$, we define the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This is known as the Nicolas inequality. The Nicolas criterion states that the Riemann hypothesis is true if and only if the Nicolas inequality is satisfied for all primes $q_{n} > 2$. We prove indeed that the Nicolas inequality is satisfied for all primes $q_{n} > 2$. In this way, we show that the Riemann hypothesis is true.

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