RETRACTED: New Criterion for the Riemann Hypothesis

09 June 2023, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

There are several statements equivalent to the famous Riemann hypothesis. In 2011, Sol{\'e} and and Planat stated that the Riemann hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >\frac{e^{\gamma}}{\zeta(2)}\cdot \log\theta(q_{n})$ is satisfied for all primes $q_{n} > 3$, where $\theta(x)$ is the Chebyshev function, $\gamma\approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. Using this result, we create a new criterion for the Riemann hypothesis. We prove the Riemann hypothesis is true using this new criterion.

Keywords

Riemann hypothesis
Prime numbers
Chebyshev function
Riemann zeta function

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