[Paper Review] A short note on a conjecture of Okounkov about a q-analogue of multiple zeta values

Henrik Bachmann, Ulf Kuehn|arXiv (Cornell University)|Jul 25, 2014

Advanced Mathematical Identities6 references4 citations

TL;DR

This paper investigates Okounkov's conjecture on a q-analogue of multiple zeta values by establishing a structural isomorphism between Okounkov's q-MZVs and the algebra of generating functions for multiple divisor sums. It proves that the q-MZV algebra is closed under multiplication and provides numerical evidence supporting a refined conjecture on its Hilbert series, linking it to quasi-modular forms and a free polynomial algebra with explicit generating functions.

ABSTRACT

In [Ok] Okounkov studies a specific $q$-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs $q$-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it.

Motivation & Objective

• To investigate the algebraic structure of Okounkov's q-analogue of multiple zeta values (q-MZVs).
• To compare Okounkov's q-MZVs with the algebra of generating functions for multiple divisor sums defined in [BK1].
• To propose and provide numerical evidence for a refined conjecture on the dimension and Hilbert series of the q-MZV algebra.
• To explore the role of the q-derivative operator d/dq in preserving the q-MZV algebra.

Proposed method

• Define q-analogues of multiple zeta values using polynomials Q_s(t) that satisfy a specific product identity (2.3), ensuring closure under multiplication.
• Establish that Okounkov's q-MZV algebra is isomorphic to the subalgebra MD^# of generating functions for multiple divisor sums.
• Use explicit formulas from [BK1] to show that q d/dq Z(k) lies within the q-MZV algebra, supporting the conjecture that d/dq is a derivation.
• Derive a conjectural Hilbert series for the graded algebra of q-MZVs by combining the known Hilbert series of quasi-modular forms with a new algebra A.
• Perform extensive numerical computations using PARI/GP to compute lower bounds for the dimensions of filtered subspaces Fil^{W,L}_{k,l} q-MZV up to weight 21 and length 11.
• Compare these lower bounds with predictions from the refined conjecture to validate its consistency.

Experimental results

Research questions

• RQ1 Is Okounkov's q-analogue of multiple zeta values closed under multiplication, and does it coincide with the algebra of generating functions for multiple divisor sums?
• RQ2 Does the q-derivative operator d/dq preserve the q-MZV algebra, implying it acts as a derivation on this space?
• RQ3 Can the Hilbert series of the graded q-MZV algebra be expressed as a product of the Hilbert series of quasi-modular forms and a new algebra A with a specific rational generating function?
• RQ4 Do the numerical dimensions of filtered subspaces Fil^{W,L}_{k,l} q-MZV match the predictions of the refined conjecture up to weight 21 and length 11?
• RQ5 Is the q-MZV algebra isomorphic to a free graded polynomial algebra tensored with a specific algebra A?

Key findings

• The q-MZV algebra is isomorphic to the subalgebra MD^# of generating functions for multiple divisor sums, proving closure under multiplication.
• The q-derivative operator q d/dq maps elements of the q-MZV algebra back into itself, supporting Okounkov's conjecture that d/dq is a derivation.
• Numerical computations show that the lower bounds for dim Fil^{W,L}_{k,l} q-MZV match the predictions of the refined conjecture (3.2) and (3.3) up to weight 21 and length 11.
• The Hilbert series of the q-MZV algebra is predicted to be the product of the Hilbert series of quasi-modular forms and a rational function with explicit poles, consistent with the conjectured structure.
• The refined conjecture (3.2) implies both Okounkov's original conjecture (3.1) and the Hilbert series formula (3.1), providing a coherent algebraic framework.
• No increase in dimension is observed when adjoining derivatives of q-MZV elements up to weight 19, supporting the conjecture that the algebra is closed under d/dq.