FAST EVALUATION OF MULTIPLE ZETA-SUM

We show that the multiple zeta sum: [GRAPHICS] for positive integers s (i) with s(1) > 1, can always be written as a finite sum of products o f rapidly convergent series. Perhaps surprisingly, one may develop fas t 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 ze ta 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)}.