Possible Counterexample of the Riemann Hypothesis

EasyChair Preprint 7306, version 3

Abstract

Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where $\theta(x)$ is the Chebyshev function. A precise version of this was given by Schoenfeld: He found under the assumption that the Riemann hypothesis is true that $\theta(x) < x + \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. On the contrary, we prove if there exists some real number $x \geq 10^{8}$ such that $\theta(x) > x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $\theta(x) < x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$ for all $x \geq 10^{8}$.

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

@booklet{EasyChair:7306,
  author    = {Frank Vega},
  title     = {Possible Counterexample of the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 7306},
  year      = {EasyChair, 2022}}