Chapter 1

Mathematical constants and numbers edited by Simon Plouffe Associate Professor LaCIM, University of Quebec at Montreal http://www.lacim.uqam.ca/pi : Plouffe's Inverter [email protected]

The value of Zeta(3) to 1,000,000 decimal digits. the number is defined as sum(1/n^3,n=1..infinity), the sum of inverses of cubes and equals 1.2020569031...

Computed by : Sebastian Wedeniwski ([email protected]) who computed more than 128 million digits using this more efficient formula found by Theodor Amdeberhan and Doron Zeilberger.

/ \ | ------ 3 | | \ n A(n) ((2 n + 1)! (2 n)! n!) | Zeta(3) = 1/24 | ) (-1) ------------------------------ | | / 3 | | ------ (3 n + 2)! ((4 n + 3)!) | \ n >= 0 /

5 4 3 2 with A(n) := 126392 n + 412708 n + 531578 n + 336367 n + 104000 n + 12463 given by Theodor Amdeberhan and Doron Zeilberger (see [1]).

References: ===========

[1] T. Amdeberhan und D. Zeilberger: Hypergeometric Series Acceleration via the WZ Method, Electronic Journal of Combinatorics (Wilf Festschrift Volume) 4 (1997).

[2] B. Haible, T. Papanikolaou: Fast multiprecision evaluation of series of rational numbers, Technical Report TI-97-7, Darmstadt University of Technology, April 1997.

[3] S. Wedeniwski: Piologie - Eine exakte arithmetische Bibliothek in C++, Technical Report WSI 96-35, Tuebingen University, available by anonymous ftp from "ftp://ftp.informatik.uni-tuebingen.de/pub/CA/software/Piologie/" or "ftp://ruediger.informatik.uni-tuebingen.de/Piologie/".