On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the d-th digit of certaintranscendental numbers in various bases. These algorithms can be easilyimplemented (multiple precision arithmetic is not needed), require virtuallyno memory, and feature run times that scale nearly linearly with the order ofthe digit desired. They make it feasible to compute, for example, the billionthbinary digit of log (2) or Pi on a modest work station in a few hours run time.We demonstrate this technique by computing the ten billionth hexadecimaldigit of Pi, the billionth hexadecimal digits of Pi^2; log(2) and log2^(2), and theten billionth decimal digit of log(9/10).These calculations rest on the observation that very special types of identitiesexist for certain numbers like Pi, Pi^2, log(2) and log2^(2). These are essentiallypolylogarithmic ladders in an integer base. A number of these identitiesthat we derive in this work appear to be new.