ON THE RAPID COMPUTATION OF VARIOUS POLYLOGARITHMIC CONSTANTS

We give algorithms for the computation of the d-th digit of certain tr anscendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require vir tually no memory, and feature run times that scale nearly linearly wit h the order of the digit desired. They make it feasible to compute, fo r example, the billionth binary digit of log(2) or pi on a modest work station in a few hours run time. We demonstrate this technique by com puting the ten billionth hexadecimal digit of pi, the billionth hexade cimal digits of pi(2), log(2) and log(2)(2), and the ten billionth dec imal digit of log(9/10). These calculations rest on the observation th at very special types of identities exist for certain numbers like pi, pi(2) log(2) and log(2)(2). These are essentially polylogarithmic lad ders in an integer base. A number of these identities that we derive i n this work appear to be new, for example the critical identity for pi : [GRAPHICS]