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

• Published in 1998
In the collection
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 $b=3$.

### BibTeX entry

@article{Polylogarithmicladdershypergeometricseriesandthetenmillionthdigitsof3and5,
title = {Polylogarithmic ladders, hypergeometric series and the ten millionth  digits of {\$}ζ(3){\$} and {\$}ζ(5){\$}},
abstract = {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{\$}.},
url = {http://arxiv.org/abs/math/9803067v1 http://arxiv.org/pdf/math/9803067v1},
}