Possible Counterexample of the Riemann Hypothesis

EasyChair Preprint no. 7306, version 4

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7 pagesDate: January 16, 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. 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. Note that, the von Koch asymptotic formula uses the Big $O$ notation, where $f(x) = O(g(x))$ means that there exists a positive real number $M$ and a real number $y$, such that $|f(x)| \leq M \times g(x)$ for all $x \geq y$. However, no matter how big we get the real number $y \geq 10^{8}$, the another positive real number $M$ could always prevail over the value of $\frac{1}{\log \log \log x}$ for sufficiently large numbers $x \geq y$.

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