[Paper Review] Alternating Euler sums and special values of Witten multiple zeta function attached to so(5)

Jianqiang Zhao|arXiv (Cornell University)|Mar 3, 2009

Advanced Mathematical Identities14 references4 citations

TL;DR

This paper establishes that special values of the Witten multiple zeta function associated with the Lie algebra 𝔰𝔬(5) at nonnegative integers are expressible as finite rational linear combinations of alternating Euler sums of the same weight and depth at most two, with exceptions when only the last one or two arguments are nonzero, requiring ζ(w−1). The results generalize known relations for multiple zeta values and provide explicit rational coefficient formulas for symmetric cases like ζₛₒ(₅)(2m,2m,2m,2m).

ABSTRACT

In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w>2 is a finite rational linear combination of alternating Euler sums of weight w and depth at most two, except when the only nonzero argument is one of the two last variables in which case $ζ(w-1)$ is needed.

Motivation & Objective

• To characterize the special values of the Witten multiple zeta function attached to the Lie algebra 𝔰𝔬(5) at nonnegative integers.
• To investigate whether these special values can be expressed in terms of known special functions, particularly alternating Euler sums.
• To extend known results on parity relations and reducibility of multiple zeta values to the 𝔰𝔬(5) setting.
• To provide explicit rational coefficient formulas for symmetric cases such as ζₛₒ(₅)(2m,2m,2m,2m), resolving Zagier's conjecture on their relation to π^{8m}.

Proposed method

• Define the Witten multiple zeta function for 𝔰𝔬(5) as a 4-variable sum over positive integers with denominators involving m^{s₁}n^{s₂}(m+n)^{s₃}(m+2n)^{s₄}.
• Use integral representations and convergence criteria based on Mordell-Tornheim zeta functions to ensure convergence for nonnegative integer arguments.
• Apply regularization techniques and double shuffle relations for multiple polylogarithms at roots of unity to simplify alternating Euler sums.
• Reduce the original sum to linear combinations of alternating Euler sums using algebraic identities and known relations such as ζ({3}ⁿ) = 8ⁿζ({̄2,1}ⁿ).
• Use the theory of multiple zeta values and Bernoulli numbers to derive closed-form expressions for symmetric cases.
• Verify numerical consistency of formulas up to 10⁻¹⁰⁰ precision and cross-check with known results from Komori, Matsumoto, and Tsumura.

Experimental results

Research questions

• RQ1 Can the special values of the Witten multiple zeta function for 𝔰𝔬(5) at nonnegative integers be expressed in terms of alternating Euler sums?
• RQ2 What is the structure of the rational linear combinations that express these special values, and are there exceptions to the general pattern?
• RQ3 How do these results relate to Zagier’s conjecture that ζₛₒ(₅)(2m,2m,2m,2m) is a rational multiple of π^{8m}?
• RQ4 Can the rational coefficients in Zagier’s formula be explicitly determined using alternating Euler sum identities?
• RQ5 What role do Bernoulli numbers and combinatorial sums play in the closed-form expression for symmetric special values?

Key findings

• All convergent special values of the 𝔰𝔬(5) Witten zeta function at nonnegative integers are finite ℚ-linear combinations of alternating Euler sums of weight w and depth at most two.
• The only exceptions occur when s₁ = s₂ = s₃ = 0 or s₁ = s₂ = s₄ = 0, in which case ζ(w−1) is required in addition to alternating Euler sums.
• For symmetric arguments (2m,2m,2m,2m), the value is exactly c(m)·π^{8m} with c(m) ∈ ℚ, and the paper provides an explicit formula for c(m) in terms of Bernoulli numbers and combinatorial sums.
• The formula for c(m) is given by c(m) = 2^{8m−3}/(8m)! × Σ_{ν=0}^{m} B_{2ν}B_{8m−2ν} (8m choose 2ν) × [sum of signed binomial coefficients], which matches known results from Komori et al.
• Numerical verification confirms the correctness of the derived identities to within 10⁻¹⁰⁰, supporting the algebraic derivations.
• The paper resolves Zagier’s original conjecture by providing a closed-form rational coefficient for ζₛₒ(₅)(2m,2m,2m,2m), explicitly linking it to Bernoulli numbers and binomial coefficients.