Fast evaluation of multiple zeta sums

Crandall, Richard

doi:10.1090/S0025-5718-98-00950-8

Fast evaluation of multiple zeta sums
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by Richard E. Crandall PDF

Math. Comp. 67 (1998), 1163-1172 Request permission

Abstract:

We show that the multiple zeta sum: \begin{equation*}\zeta (s_{1}, s_{2}, ..., s_{d}) = \sum _{n_{1} > n_{2} > ... > n_{d}} {{\frac {1 }{{n_{1}^{s_{1}} n_{2}^{s_{2}} ... n_{d}^{s_{d}}}}}},\end{equation*} for positive integers $s_{i}$ with $s_{1}>1$, can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one-dimensional summation. In particular, for any dimension $d$ one may resolve $D$ good digits of $\zeta$ in $O(D \log D / \log \log D)$ arithmetic operations, with the implied big-$O$ constant depending only on the set $\{s_{1},...,s_{d}\}$.

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