1998/03/16 by Broadhurst, D. J.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.math/9803067
We develop ladders that reduce ζ(n):=∑k>0k-n, for n=3,5,7,9,11, and β(n):=∑k≥0(-1)k(2k+1)-n, for n=2,4,6, to convergent polylogarithms and products of powers of π and log2. Rapid computability results because the required arguments of \rm Lin(z)=∑k>0zk/kn satisfy z8=1/16p, with p=1,3,5. We prove that G:=β(2), π3, log32, ζ(3), π4, log42, log52, ζ(5), and six products of powers of π and log2 are constants whose dth hexadecimal digit can be computed in time~=O(dlog3d) and space~=O(log d), as was shown for π, log2, π2 and log22 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.