8 On a symmetric space attached to polyzeta values

Quickly converging series are given to compute polyzeta numbers ζ(r1, . . . , rk). The formulas involve an intricate combination of (generalized) polylogarithms at 1/2. However, the combinatoric has a very simple geometric interpretation: it corresponds with the map p 7→ p on a certain symmetric space P . Introduction: Let k ≥ 1. For a k-uple (r1, r2, . . . , rk) of positive integers, set ζ(r1, . . . , rk) = ∑ 00(−1)n/n and π/4 = ∑n≥0(−1)n1/(2n+ 1), which converge very slowly. However, we easily notice that: log 2 = − log(1− 1/2) = ∑n>0 2/n. A remarkable series for π has been discovered by Bailey, Borwein and Plouffe [BBP]: π = ∑ n≥0 1/2 [4/(8n+ 1)− 2/(8n+ 4)− 1/(8n+ 5)− 1/(8n+ 6)] Now to evaluate log 2 or π up to the Nth digit, one only needs the first O(N)-terms of the series and therefore log 2 and π can be computed very quickly. The goal of the paper is to provide similar identities for all polyzeta values. To do so, one needs to use the functions Lr1,...,rk(z) = ∑ 0<n1...<nk 1/n1 1 ...n rk k z nk , where r1, . . . , rk are positive integers. By definition, a Q-linear combinations of the functions Lr1,...,rk(z) is called a polylogarithmic function. The obvious identity ζ(r1, . . . , rk) = Lr1,...,rk(1) does not help to quickly evaluate polyzeta values. However, the series defining polylogarithms at 1/2 converges very quickly: to evaluate Lr1,...,rk(1/2) up to the N th digit, one only needs to sum O(N)-terms, and this can be done in polynomial time. This remark suggests the following result: 2000 Mathematics Subject Classification: 11M99, 17B01, 53C35