[Paper Review] Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta(3)$ and $\zeta(5)$

David Broadhurst|arXiv (Cornell University)|Mar 16, 1998

Advanced Mathematical Identities17 references7 citations

TL;DR

This paper establishes that the 10 millionth hexadecimal digits of ζ(3) and ζ(5) can be computed in O(d log³ d) time and O(log d) space using polylogarithmic ladders and hypergeometric series. It proves that ζ(3), ζ(5), and 17 related constants—including π, log 2, Catalan’s constant, and products of powers of π and log 2—are in the SC∗ class, enabling super-fast digit computation via functional identities of polylogarithms and novel hypergeometric generating functions.

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 ${ m Li}_n(z)=\sum_{k>0}z^k/k^n$ 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 $b=3$.

Motivation & Objective

• To determine whether ζ(3), ζ(5), and related constants like Catalan’s constant and powers of π and log 2 belong to the SC∗ class of constants with super-fast digit computation.
• To extend the Bailey-Borwein-Plouffé (BBP) method beyond π, log 2, π², and log²2 to higher zeta values and Euler sums.
• To develop polylogarithmic ladders and hypergeometric generating functions that reduce multiple zeta and Dirichlet beta values to rapidly computable combinations of π, log 2, and their powers.
• To compute the 10 millionth hexadecimal digit of ζ(3) and ζ(5) using these new representations.

Proposed method

• Derives polylogarithmic ladders using Kummer’s functional identities and special values of z with z⁸ = 1/16ᵖ for p = 1, 3, 5, reducing ζ(n) and β(n) to convergent polylogarithms.
• Applies functional relations of Liₙ(z) to express ζ(3), ζ(5), and related constants as combinations of polylogarithms evaluated at roots of unity and algebraic numbers.
• Uses hypergeometric series to generate Euler sums and proves their connection to zeta values via contour integrals and Pochhammer symbol identities.
• Employs symbolic computation (Reduce) to automate derivation of functional identities at orders n ≤ 11.
• Validates integer relations using high-precision PSLQ and symbolic algebra, ensuring completeness of the derived identities.
• Computes digits via modular arithmetic and series acceleration, using 312-bit precision and 0.3 MB memory on a 333 MHz machine.

Experimental results

Research questions

• RQ1 Can ζ(3) and ζ(5) be computed in logarithmic space and almost linear time, like π and log 2, via a BBP-type formula?
• RQ2 Do combinations of πⁿ, logⁿ2, Catalan’s constant, and ζ(n) for n = 3, 5, 7, 9, 11 belong to the SC∗ class of constants with fast digit extraction?
• RQ3 What hypergeometric generating functions underlie the Euler sums that appear in the evaluation of ζ(5)?
• RQ4 Can functional identities of polylogarithms be systematically used to reduce multiple zeta and Dirichlet beta values to SC∗-computable forms?
• RQ5 Is there a hypergeometric representation of the generator series for non-eulerian sums that matches the singularities and rational asymptotics observed in the data?

Key findings

• The 10 millionth hexadecimal digit of ζ(3) is CDA018F4E167F435B2AB045FB045A42F86BED12EF82BE2E1C6ECD305E92C5E4B.
• The 10 millionth hexadecimal digit of ζ(5) is F7A15E1277F7B2C04106F04B05C48AC71ACECAB14D555FDA6E5E1EC299535511.
• All 18 constants—including π, log 2, π³, log³2, ζ(3), π⁴, log⁴2, log⁵2, ζ(5), and their products—are in the SC∗ class, with dth hexadecimal digit computable in O(d log³ d) time and O(log d) space.
• A new formula for Catalan’s constant G is derived as G = 3 S₂,₁(1,−1,1,0,−1,1,−1,0) − 2 S₂,₃(1,1,1,0,−1,−1,−1,0), enabling its fast digit computation.
• The paper proves that hypergeometric series with Pochhammer symbols generate Euler sums whose singularities match trigonometric functions, confirming the structure of the generating series.
• The computation of ζ(5)’s 10 millionth digit was achieved in 19 hours on a 333 MHz DecAlpha 600 using only 0.3 MB of memory, demonstrating the efficiency of the method.