Editing Riemann zeta function (section)

== Numerical algorithms ==
A classical algorithm, in use prior to about 1930, proceeds by applying the [[Euler-Maclaurin formula]] to obtain, for ''n'' and ''m'' positive integers,

:<math>\zeta(s) = \sum_{j=1}^{n-1}j^{-s} + \tfrac12 n^{-s} + \frac{n^{1-s}}{s-1} + \sum_{k=1}^m T_{k,n}(s) + E_{m,n}(s)</math>

where, letting <math>B_{2k}</math> denote the indicated [[Bernoulli number]],

:<math>T_{k,n}(s) = \frac{B_{2k}}{(2k)!} n^{1-s-2k}\prod_{j=0}^{2k-2}(s+j)</math>

and the error satisfies

:<math>|E_{m,n}(s)| < \left|\frac{s+2m+1}{\sigma + 2m + 1}T_{m+1,n}(s)\right|,</math>

with ''&sigma;'' = Re(''s'').<ref>{{cite journal|mr=0961614
|author1-link=Odlyzko|author2-link=Schönhage|last1=Odlyzko|first1= A. M.|last2= Schönhage|first2= A.
|title=Fast algorithms for multiple evaluations of the Riemann zeta function
|journal=Trans. Amer. Math. Soc.|volume= 309 |year=1988|issue= 2|pages= 797–809
|doi=10.2307/2000939|jstor=2000939|doi-access=free}}.
</ref>

A modern numerical algorithm is the [[Odlyzko–Schönhage algorithm]].