What computer methods are used to quickly calculate the $\zeta$-function (if any)?

Question

So I can think of how I could compute $\zeta(\sigma + it)$ in principle.

We can take $\zeta(\sigma + it)$ for $\sigma>1$ by the usual $\sum_{n} n^{-(\sigma + it)}$ summation. We can use the Dirichlet eta function to compute $\zeta(\sigma + it)$ for $\sigma > 0$, and this is justified by the identity theorem of complex analysis. Now to generalize the zeta function to all $\sigma$, we can apply the reflection formula, and now we know what $\zeta$ is everywhere, at least in principle.

However, surely the numerical methods used to compute the $\zeta$-function using a computer are different. Is there anything analogous to Ramanujan's formula for $\pi$, but for the $\zeta$-function? If there is more than one algorithm, then any are welcome. References to papers would be appreciated.

Answer 1

Morally, you can get really far applying variants of the approximate functional equation or the Riemann Siegel formula.

See for example J. Arias de Reyna, “High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I”, Mathematics of Computation 80 (2011), 995-1009. This forms the fundamental implementation for example in mpmath and arb.

D. J. Platt, “Isolating some non-trivial zeros of zeta”, Mathematics of Computation 86 (2017), 2449-2467 (which uses a form of the approximate functional equation). This is what was used to compute the largest complete collection of zeta zeros thus far known.

An approximate functional equation can be written down for general $L$-functions and applies approximately as well. Implementations that apply to generic $L$-functions (including $\zeta(s)$) using the approximate functional equation include lcalc (written by Rubinstein, included in sagemath), the pari-gp $L$-function system (also included in sage, but available separately), and Dokchitser's algorithm (written in pari-gp, also included in sage). The approximate functional equation itself is described in many places, included the classical treatises on the zeta function or more generally in Analytic Number Theory by Iwaniec and Kowalski.

See also Numerical Algorithms for Number Theory by Belabas and Cohen for some descriptions and implementations of a variety of related topics.