Download PDFOpen PDF in browserCounterexample of the Riemann HypothesisEasyChair Preprint no. 7306, version 86 pages•Date: February 6, 2022AbstractBy proving the existence of a zerofree region for the Riemann zetafunction, de la Vall{\'e}ePoussin 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
