zeta_8

Computes the approximation to the Riemann zeta function, valid in the critical strip (and in other vertical strips) when Im(s)>100

This function computes the approximation zeta_8(s) to the Riemann zeta function, which was described in [1]. This function was tested in the strip 1/2<Re(s)<2 and Im(s)>100. In this strip it provides fast computations of zeta(s) to accuracy of at least 10^{-10} when Im(s)<10000. For Im(s)>100000 the accuracy will decrease due to accumulation of rounding errors in the main sum of the Riemann-Siegel approximation. With every increase of Im(s) by a factor of ten we lose one decimal digit of precision. More details can be found at the end of Section 1 in [1].

This approximation should work well in other vertical strips. For example, in the vertical strip -1<Re(s)<1/2 it produces relative errors of order 10^{-10} when 100<Im(s)<10000.

References:

[1] A. Kuznetsov, "Simple and accurate approximations to the Riemann zeta function", 2025, preprint, https://arxiv.org/abs/2503.09519