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

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

Johnston, Daniel R.; Yang, Andrew

Journal of mathematical analysis and applications, 11/2023, Volume: 527, Issue: 2

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By combining and improving recent techniques and results, we provide explicit estimates for the error terms |π(x)−li(x)|, |θ(x)−x| and |ψ(x)−x| appearing in the prime number theorem. For example, we show for all x≥2 that |ψ(x)−x|≤9.39x(log⁡x)1.515exp⁡(−0.8274log⁡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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