[Paper Review] Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

Kam Cheong Au|arXiv (Cornell University)|Jan 5, 2022

Advanced Mathematical Identities4 citations

TL;DR

This paper develops a systematic method to express multiple polylogarithm values at algebraic arguments as colored multiple zeta values (CMZVs), leveraging iterated integrals and algebraic number theory. It proves that specific polylogarithmic values—such as those at the golden ratio or rational arguments—lie in CMZV spaces of bounded level, enabling new identities and nonstandard relations, with applications to Apéry-like series and classical polylogarithm identities.

ABSTRACT

By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset $S\subset \mathbb{P}^1(\mathbb{C})$ and studying their transformation properties under rational functions, we show that multiple polylogarithms evaluated at roots of unity (cyclotomic multiple zeta values, CMZVs) can be equivalently expressed in terms of iterated integrals involving certain non-roots of unity. We apply this theory to elucidate previously unknown $\mathbb{Q}$-linear relations among CMZVs: they come from nontrivial solutions of certain $S$-unit equations in the function field of $\mathbb{P}^1(\mathbb{C})$, thereby attaining the motivic dimension for low level and weight. We introduce a datamine of CMZVs that appears to be the first rigorous compilation of this kind in the literature. In addition, we formulate several nontrivial Galois descent conjectures for multiple polylogarithms and present applications to certain Apéry-type infinite series.

Motivation & Objective

• To systematically map values of multiple polylogarithms at algebraic arguments into the space of colored multiple zeta values (CMZVs).
• To uncover nonstandard relations among CMZVs that are not captured by known algebraic or functional identities.
• To provide a uniform, computable framework for proving classical polylogarithm identities and Apéry-like series.
• To extend the applicability of CMZV theory beyond roots of unity to include algebraic numbers such as (\sqrt{5}-1)/2 and rational values.
• To develop a Mathematica package for explicit computation of CMZV values and iterated integrals at specified algebraic points.

Proposed method

• Utilizes iterated integrals on the complex projective line with logarithmic 1-forms to represent multiple polylogarithms.
• Applies pullback of differential forms along paths to express polylogarithmic values as integrals over [0,1] with algebraic base points.
• Employs shuffle product identities and path composition rules to decompose and recombine iterated integrals into linear combinations.
• Reduces the problem to computing iterated integrals at algebraic arguments, which are then expressed as Q-linear combinations of basis elements in CMZV spaces.
• Uses the structure of the fundamental group of the punctured complex line to classify and compute relations among CMZVs.
• Leverages a custom Mathematica package to compute and verify CMZV values and iterated integrals explicitly for specific arguments and weights.

Experimental results

Research questions

• RQ1 For which algebraic arguments x is the generalized polylogarithm Li_{s1,...,sk}(x) expressible as a Q-linear combination of CMZVs of bounded level?
• RQ2 What nonstandard algebraic relations exist among CMZVs that are not derivable from standard functional or shuffle relations?
• RQ3 Can classical polylogarithm identities, such as Coxeter's ladder, be derived systematically using iterated integrals and CMZV algebra?
• RQ4 To what extent can Apéry-like series involving harmonic numbers and central binomial coefficients be proven using this framework?
• RQ5 What are the limits of this method when applied to higher-weight or higher-level CMZV spaces, particularly beyond level 12?

Key findings

• The multiple polylogarithm Li_{s1,...,sn}(x1,...,xn) lies in CMZV^5_w when at most two xi are equal to the golden ratio (\sqrt{5}-1)/2 and the rest are 1.
• The generalized polylogarithm Li_{s1,...,sn}(z) is in CMZV^6_w for z = -1/2, 1/3, 1/4, 1/9, and in CMZV^8_w for z = 1 - \sqrt{2}, (4 - 3\sqrt{2})/8.
• For z = -1/2, if at most three arguments are -1/2, the value lies in CMZV^12_w, demonstrating a level jump with increasing algebraic complexity.
• The method successfully proves Coxeter's ladder identity: Li_2(ρ^20) = 2Li_2(ρ^10) + 15Li_2(ρ^4) - 10Li_2(ρ^2) + π²/5 with ρ = (\sqrt{5}-1)/2.
• The paper provides alternative proofs for several Apéry-like series, including ∑ (-1)^{n-1}(10H_n - 3/n)/(n^3 binom(2n,n)) = π⁴/30 and ∑ (-102H_n + 3H_{2n} + 28/n)/(n^4 binom(2n,n)) = -55π²ζ(3)/18.
• The method is limited for weights above 4 or levels beyond 12 due to the combinatorial explosion of CMZV bases and lack of computational tools for higher-level spaces.