Fast evaluation of multiple zeta sums

Abstract

We show that the multiple zeta sum: ζ ( s 1 , s 2 , . . . , s d ) = ∑ n 1 > n 2 > . . . > n d 1 n 1 s 1 n 2 s 2 . . . n d s d , \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 s_{i} with s 1 > 1 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 d one may resolve D D good digits of ζ \zeta in O ( D log ⁡ D / log ⁡ log ⁡ D ) O(D \log D / \log \log D) arithmetic operations, with the implied big- O O constant depending only on the set { s 1 , . . . , s d } \{s_{1},...,s_{d}\} .