Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. Bailey, private communication, 1994.
  • 2. D. Bailey, J. Borwein, and R. Crandall, ``On the Khintchine Constant,'' Math. Comp., 66 (1997), 417-431. MR 97c:11119
  • 3. D. Bailey, J. Borwein and R. Girgensohn, ``Experimental evaluation of Euler sums,'' Experimental Mathematics 3, (1994), 17-30. MR 96e:11168
  • 4. D. Borwein, J. Borwein and R. Girgensohn, ``Explicit evaluation of Euler sums,'' Proc. Edinburgh Math. Soc. 38, (1995), 277-294. MR 96f:11106
  • 5. J. M. Borwein and P. B. Borwein, Pi and the AGM, John Wiley & Sons, 1987. MR 89a:11134
  • 6. J. Borwein and D. Bradley, ``Empirically Determined Apery-like Formulae for $\zeta (4n+3)$,'' manuscript, 1996.
  • 7. J. M. Borwein, D. M. Bradley and D. J. Broadhurst, ``Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$,'' manuscript, 1996.
  • 8. J. M. Borwein and R. Girgensohn, ``Evaluation of Triple Euler Sums,'' manuscript, 1995.
  • 9. D. J. Broadhurst, R. Delbourgo and D. Kreimer, Phys. Lett. B366 (1996), 421.
  • 10. R. E. Crandall, Topics in Advanced Scientific Computation, TELOS/Springer-Verlag, New York, 1996. MR 97g:65005
  • 11. R. E. Crandall and J. P. Buhler, ``On the Evaluation of Euler Sums,'' Experimental Mathematics 3, (1994), 275-285. MR 96e:11113
  • 12. E. A. Karatsuba, ``Fast Calculation of the Riemann Zeta Function $\zeta (s)$ for Integer Values of the Argument $s$,'' Probs. Information Trans. 31 (1995), 353-355. MR 96k:11155
  • 13. C. Markett, ``Triple Sums and the Riemann Zeta Function,'' J. Number Theory, 48 (1994), 113-132. MR 95f:11067
  • 14. D. Zagier, ``Values of zeta functions and their applications,'' preprint, Max-Planck Institut, 1994.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11Y60, 11Y65, 11M99

Retrieve articles in all journals with MSC (1991): 11Y60, 11Y65, 11M99

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