[Paper Review] Hyper-Mahler measures via Goncharov-Deligne cyclotomy

Yajun Zhou|arXiv (Cornell University)|Oct 31, 2022

Molecular spectroscopy and chirality4 citations

TL;DR

This paper evaluates hyper-Mahler measures $ m_k(1+x_1+x_2) $ and $ m_k(1+x_1+x_2+x_3) $ for $ k > 1 $ in closed form using Goncharov–Deligne periods—$ bQ $-linear combinations of multiple polylogarithms at roots of unity. It establishes that these measures are expressible as rational linear combinations of special values of multiple polylogarithms and Hoffman’s multiple zeta values, with explicit formulas derived via Brown’s iterated integrals and computational tools like HyperInt and MultipleZetaValues packages.

ABSTRACT

The hyper-Mahler measures $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k( 1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb Q$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $ 3$, $4$, $6$, $8$, $10$ and $12$.

Motivation & Objective

• To derive closed-form expressions for hyper-Mahler measures $ m_k(1+x_1+x_2) $ and $ m_k(1+x_1+x_2+x_3) $ for all integers $ k > 1 $.
• To express these measures as $ bQ $-linear combinations of Goncharov–Deligne periods, i.e., special values of multiple polylogarithms at cyclotomic points.
• To connect the hyper-Mahler measures to known constants such as $ ext{Li}_4(1/2) $, $ heta_3 $, and $ inom{2n}{n} $-weighted series via algebraic and analytic techniques.
• To provide algorithmic and computational verification of these identities using the HyperInt and MultipleZetaValues packages.
• To explore the structure of infinite series related to these measures and represent them as Goncharov–Deligne periods of various levels.

Proposed method

• Utilizes Goncharov’s multiple polylogarithms $ ext{Li}_{a_1, u,a_n}(z_1, u,z_n) $ evaluated at third roots of unity $ ho = e^{2 au i/3} $, with convergence ensured by excluding $ (a_1,z_1) = (1,1) $.
• Applies Brown’s homotopy-invariant iterated integrals on $ rak{M}_{0,n} $, the moduli space of genus-zero curves with $ n $ marked points, to relate moments of random walks to multiple polylogarithmic identities.
• Employs Panzer’s HyperInt package and Au’s MultipleZetaValues package to compute and verify identities involving multiple polylogarithms and multiple zeta values.
• Derives integral representations of zeta Mahler measures via Euler’s integral formula and beta function identities, extending analytic continuation to $ s o 0 $.
• Uses logarithmic regularization and fibrating techniques under differentiation to handle divergent terms and maintain algebraic consistency in $ bQ $-linear spans.
• Applies $ bQ $-linear span arguments to show that derivatives of zeta Mahler measures at $ s=0 $ lie in $ rac{rak{Z}_{k+1}(6)}{ au i} $, linking them to cyclotomic multiple zeta values.

Experimental results

Research questions

• RQ1 Can hyper-Mahler measures $ m_k(1+x_1+x_2) $ and $ m_k(1+x_1+x_2+x_3) $ for $ k > 1 $ be expressed in closed form using special values of multiple polylogarithms?
• RQ2 To what extent can these measures be represented as $ bQ $-linear combinations of Goncharov–Deligne periods?
• RQ3 How do these hyper-Mahler measures relate to known constants such as $ ext{Li}_4(1/2) $, $ heta_3 $, and $ inom{2n}{n} $-weighted series?
• RQ4 Can the structure of infinite series related to these measures be systematically reduced to Goncharov–Deligne periods of specific levels?
• RQ5 What is the role of iterated integrals on $ rak{M}_{0,n} $ in connecting random walk moments to multiple zeta values and polylogarithms?

Key findings

• The hyper-Mahler measure $ m_1(1+x_1+x_2) $ is given by $ rac{3}{2 au} ext{Im}ig( ext{Li}_2( ho)ig) $, where $ ho = e^{2 au i/3} $, confirming a known result via Goncharov–Deligne periods.
• For $ k=2 $, $ m_2(1+x_1+x_2) $ is expressed as a $ bQ $-linear combination involving $ ext{Im}ig( ext{Li}_{2,1}( ho,1)ig) $, $ ext{Im}ig( ext{Li}_2( ho)ig) $, and rational multiples of $ au^2 $.
• The measure $ m_2(1+x_1+x_2+x_3) $ is shown to equal $ rac{12 heta_{-3,1}}{ au^2} + rac{ au^2}{20} $, where $ heta_{-3,1} $ is an alternating double sum related to $ ext{Li}_4(1/2) $.
• The paper derives that $ rac{d^k}{ds^k} W_3(s)ig|_{s=0} ig/ au i ig floor rac{rak{Z}_{k+1}(6)}{ au i} $, showing that derivatives of zeta Mahler measures lie in the space of cyclotomic multiple zeta values of level 6.
• The $ k=4 $ case for $ m_4(1+x_1+x_2) $ is explicitly computed as a $ bQ $-linear combination of $ ext{Im}ig( ext{Li}_{4,1}( ho,1)ig) $, $ ext{Im}ig( ext{Li}_{3,2}( ho,1)ig) $, $ ext{Im}ig( ext{Li}_{2,1,1,1}( ho,1,1,1)ig) $, and rational terms including $ rac{5923 au^4}{2^4 imes 3^4 imes 5 imes 13} $.
• The paper confirms that $ ext{Re}ig( ext{Li}_{3,1}( ho,1)ig) $, $ ext{Im}ig( ext{Li}_{2,1}( ho,1)ig) $, and $ ig[ ext{Im}( ext{Li}_2( ho))ig]^2 $ appear in the $ k=4 $ expression, demonstrating the role of real and imaginary parts in higher-order measures.