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arXiv · 2109.02249

Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

Daniel R. Johnston
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Abstract

Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $π(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|π(x)-\text{li}(x)|<\sqrt{x}\log x/8π$ for $2657\leq x\leq 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.

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Daniel R. Johnston. 2022-06-14. Improving bounds on prime counting functions by partial verification of the Riemann hypothesis. https://arxiv.org/abs/2109.02249

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