Polylogarithmic ladders , hypergeometric series and the ten millionth digits of ζ ( 3 ) and ζ ( 5 )

We develop ladders that reduce ζ(n) := ∑ k>0 k −n, for n = 3, 5, 7, 9, 11, and β(n) := ∑ k≥0(−1)(2k+ 1)−n, for n = 2, 4, 6, to convergent polylogarithms and products of powers of π and log 2. Rapid computability results because the required arguments of Lin(z) = ∑ k>0 z /k satisfy z = 1/16, with p = 1, 3, 5. We prove that G := β(2), π, log 2, ζ(3), π, log 2, log 2, ζ(5), and six products of powers of π and log 2 are constants whose dth hexadecimal digit can be computed in time = O(d log d) and space = O(log d), as was shown for π, log 2, π and log 2 by Bailey, Borwein and Plouffe. The proof of the result for ζ(5) entails detailed analysis of hypergeometric series that yield Euler sums, previously studied in quantum field theory. The other 13 results follow more easily from Kummer’s functional identities. We compute digits of ζ(3) and ζ(5), starting at the ten millionth hexadecimal place. These constants result from calculations of massless Feynman diagrams in quantum chromodynamics. In a related paper, hep-th/9803091, we show that massive diagrams also entail constants whose base of super-fast computation is b = 3. ) [email protected]; http://physics.open.ac.uk/ d̃broadhu