Riemann_zeta

approximates the Riemann zeta function in the entire complex plane

Riemann_zeta(s) approximates the Riemann zeta function in the entire complex plane. The input s can be a scalar or vector.

For |Im(s)|<100 the approximation is correct to around 13 decimal digits;

For |Im(s)|<1000 the approximation is correct to around 12 decimal digits;

For |Im(s)|<10000 the approximation is correct to around 11 decimal digits;

For larger values of |Im(s)| the accuracy will continue to decrease in a similar way: with every increase of Im(s) by a factor of ten we lose approximately one decimal digit of precision. More details can be found at the end of Section 1 in [1].

When |Im(s)|>200 and -4<Re(s)<5 we use the approximation zeta_8(s) developed in [1]. For other values of s we use either Euler-Maclaurin formula or direct summation zeta(s)=\sum_{n=1}^{\infty} n^{-s}.

References:

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