New Bound on the Chebyshev Function and the Riemann Hypothesis

EasyChair Preprint no. 7306, version 5

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7 pagesDate: January 18, 2022

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. 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, prime numbers, Riemann hypothesis