 Disproof of the Riemann Hypothesis

EasyChair Preprint no. 6834, version 2

Versions: 12history
7 pagesDate: October 12, 2021

Abstract

We define the function $\upsilon(x) = \frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} - 1$ for some positive constant $C$ independent of $x$. We prove that the Riemann hypothesis is false when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\upsilon(x) \leq 0$ is always satisfied. We know that the function $\upsilon(x)$ is monotonically decreasing for all sufficiently large numbers $x \geq 13.1$. Hence, it is enough to find a value of $y \geq 13.1$ such that $\upsilon(y) \leq 0$ since for all $x \geq y$ we would have that $\upsilon(x) \leq \upsilon(y) \leq 0$. Using the tool $\textit{gp}$ from the project PARI/GP, we found the first zero $y$ of the function $\upsilon(y)$ in $y \approx 8.2639316883312400623766461031726662911 \ E5565708$ for $C \geq 1$. In this way, we claim that the Riemann hypothesis could be false.

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