Counterexample of the Riemann Hypothesis

EasyChair Preprint no. 7306, version 8

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6 pagesDate: February 6, 2022

Abstract

By proving the existence of a zero-free region for the Riemann zeta-function, de la Vall{\'e}e-Poussin was able to bound $\theta(x) = x + O(x \times \exp(-c_{2} \times \sqrt{\log x}))$, where $\theta(x)$ is the Chebyshev function and $c_{2}$ is a positive absolute constant. Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. We prove when $\theta(x) = x + \Omega(\sqrt{x} \times \log^{2} x)$, then the Riemann hypothesis is false.

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