Some explicit estimates for the error term in the prime number theorem

Daniel R. Johnston; Andrew Yang

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arXiv · 2204.01980

Some explicit estimates for the error term in the prime number theorem

Daniel R. Johnston·Andrew Yang

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Abstract

By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all $x\geq 2$ that $|\psi(x)-x|\leq 9.39x(\log x)^{1.515}\exp(-0.8274\sqrt{\log x})$. Our estimates rely heavily on explicit zero-free regions and zero-density estimates for the Riemann zeta-function, and improve on existing bounds for prime-counting functions for large values of $x$.

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Daniel R. Johnston, Andrew Yang. 2022-04-05. Some explicit estimates for the error term in the prime number theorem. https://arxiv.org/abs/2204.01980

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