Simple and accurate approximations to the Riemann zeta function

Alexey Kuznetsov

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arXiv · 2503.09519

Simple and accurate approximations to the Riemann zeta function

Alexey Kuznetsov

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Abstract

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations.

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Alexey Kuznetsov. 2025-03-12. Simple and accurate approximations to the Riemann zeta function. https://arxiv.org/abs/2503.09519

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