On accelerated methods to evaluate sums of products of rational numbers

In this paper we consider the problem of fast computation of sums of n-ary products of rational numbers, for large n. We present improvements to the standard binary splitting algorithm which are due to numerous factors, including changing the standard arbitrary precision integer representation to one that is more suitable for such computations, unrolling, and chains of recurrences techniques. For the computation of ζ(3) to 640000 decimal digits, we achieve a speedup factor of 2.65 over the standard binary splitting algorithm, which compares favorably to the ideal case in which the numerator and the denominator can be reduced by their greatest common divisor at no cost. If asymptotically fast multiplication is not available (as in the Java Development Kit), a speedup of an order of magnitude is easily obtained.