Download PDFOpen PDF in browserAnother Criterion for the Riemann HypothesisEasyChair Preprint no. 6812, version 27 pages•Date: October 10, 2021AbstractLet's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}\log \log xB)$, where $B \approx 0.2614972128$ is the MeisselMertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x)  x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}\log \log \theta(x)B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$ or when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} \leq 1$ is always satisfied for some positive constant $C$ independent of $x$. Keyphrases: Chebyshev function, Nicolas inequality, prime numbers, Riemann hypothesis
