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

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

1998-03-16

arxiv.org/abs/math/9803067v1

Mathematics

Classical Analysis and ODEs

21 pages, LaTeX

Scientific paper

We develop ladders that reduce $\zeta(n):=\sum_{k>0}k^{-n}$, for $n=3,5,7,9,11$, and $\beta(n):=\sum_{k\ge0}(-1)^k(2k+1)^{-n}$, for $n=2,4,6$, to convergent polylogarithms and products of powers of $\pi$ and $\log2$. Rapid computability results because the required arguments of ${\rm Li}_n(z)=\sum_{k>0}z^k/k^n$ satisfy $z^8=1/16^p$, with $p=1,3,5$. We prove that $G:=\beta(2)$, $\pi^3$, $\log^32$, $\zeta(3)$, $\pi^4$, $\log^42$, $\log^52$, $\zeta(5)$, and six products of powers of $\pi$ and $\log2$ are constants whose $d$th hexadecimal digit can be computed in time~$=O(d\log^3d)$ and space~$=O(\log d)$, as was shown for $\pi$, $\log2$, $\pi^2$ and $\log^22$ by Bailey, Borwein and Plouffe. The proof of the result for $\zeta(5)$ 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 $\zeta(3)$ and $\zeta(5)$, 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 $b=3$.

Affiliated with

Also associated with

No associations

Rating

Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ(3)$ and $ζ(5)$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ(3)$ and $ζ(5)$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ(3)$ and $ζ(5)$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-341307