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Mathematics of Computation

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Fast evaluation of multiple zeta sums

Author: Richard E. Crandall
Journal: Math. Comp. 67 (1998), 1163-1172
MSC (1991): Primary 11Y60, 11Y65; Secondary 11M99
MathSciNet review: 1459385
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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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Additional Information

Richard E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202

Received by editor(s): September 30, 1996
Received by editor(s) in revised form: March 3, 1997
Article copyright: © Copyright 1998 American Mathematical Society