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 $\{s_{1},...,s_{d}\}$.References
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Additional Information
- Richard E. Crandall
- Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202
- Email: crandall@reed.edu
- Received by editor(s): September 30, 1996
- Received by editor(s) in revised form: March 3, 1997
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp. 67 (1998), 1163-1172
- MSC (1991): Primary 11Y60, 11Y65; Secondary 11M99
- DOI: https://doi.org/10.1090/S0025-5718-98-00950-8
- MathSciNet review: 1459385