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

TL;DR

The paper develops eight polylogarithmic ladders to order $n=11$ that express $\zeta(n)$ and related sums in terms of convergent ${\rm Li}_n(z)$ with special arguments, enabling rapid digit extraction and placing many constants in the $SC^*$ class. It provides explicit, largely functional-relations-based constructions for $n=1$ to $11$, including proving $\zeta(3)$ and $\zeta(5)$ belong to $SC^*$ and deriving new representations for $G$, $\pi^3$, $\log^3 2$, and related quantities. A hypergeometric approach is developed to prove $\zeta(5) \in SC^*$ and to connect these results with Euler sums from quantum field theory, while substantial data and partial progress are provided for higher-order cases, including an impasse at the $p=5$ ladder. Finally, the digits of $\zeta(3)$ and $\zeta(5)$ are computed starting at the ten-millionth hexadecimal place, demonstrating the practical feasibility and significance of these analytic constructions for high-precision constants.

Abstract

We develop ladders that reduce $ζ(n):=\sum_{k>0}k^{-n}$, for $n=3,5,7,9,11$, and $β(n):=\sum_{k\ge0}(-1)^k(2k+1)^{-n}$, for $n=2,4,6$, to convergent polylogarithms and products of powers of $π$ and $\log2$. Rapid computability results because the required arguments of ${\rm Li}_n(z)=\sum_{k>0}z^k/k^n$ satisfy $z^8=1/16^p$, with $p=1,3,5$. We prove that $G:=β(2)$, $π^3$, $\log^32$, $ζ(3)$, $π^4$, $\log^42$, $\log^52$, $ζ(5)$, and six products of powers of $π$ 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 $π$, $\log2$, $π^2$ and $\log^22$ 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 $b=3$.