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Hyper-Mahler measures via Goncharov-Deligne cyclotomy

Yajun Zhou

TL;DR

The paper develops a unified framework for hyper-Mahler measures $m_k(1+x_1+x_2)$ and $m_k(1+x_1+x_2+x_3)$ by representing them as Goncharov–Deligne periods, i.e. $ \mathbb{Q}$-linear combinations of multiple polylogarithms at cyclotomic points, and situates these within cyclotomic multiple zeta value spaces. It leverages Broadhurst–Brown–Panzer theory, GPL/HPL formalisms, and computational tools (HyperInt, MultipleZetaValues) to prove period inclusions in levels such as 3 and 2, forming explicit decompositions via auxiliary functions and fibration structures. The work also develops a broad array of infinite-series representations (inverse binomial, Sun-type, Wang–Chu-type) that are provably expressible as CMZVs or AMZVs, accompanied by concrete tables and algorithmic reductions. Collectively, these results link hyper-Mahler measures to moduli-space periods and cyclotomic polylogarithms, with implications for number theory and high-energy physics through unified polylogarithm techniques and automated symbolic reductions.

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 $12$.

Hyper-Mahler measures via Goncharov-Deligne cyclotomy

TL;DR

The paper develops a unified framework for hyper-Mahler measures and by representing them as Goncharov–Deligne periods, i.e. -linear combinations of multiple polylogarithms at cyclotomic points, and situates these within cyclotomic multiple zeta value spaces. It leverages Broadhurst–Brown–Panzer theory, GPL/HPL formalisms, and computational tools (HyperInt, MultipleZetaValues) to prove period inclusions in levels such as 3 and 2, forming explicit decompositions via auxiliary functions and fibration structures. The work also develops a broad array of infinite-series representations (inverse binomial, Sun-type, Wang–Chu-type) that are provably expressible as CMZVs or AMZVs, accompanied by concrete tables and algorithmic reductions. Collectively, these results link hyper-Mahler measures to moduli-space periods and cyclotomic polylogarithms, with implications for number theory and high-energy physics through unified polylogarithm techniques and automated symbolic reductions.

Abstract

The hyper-Mahler measures and are evaluated in closed form via Goncharov-Deligne periods, namely -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 , , , , , , and .
Paper Structure (11 sections, 13 theorems, 131 equations, 5 tables)

This paper contains 11 sections, 13 theorems, 131 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Integral representations of $k$-Mahler measures
  3. Broadhurst's formula for zeta Mahler measures
  4. Some integral formulae involving Bessel functions
  5. Period structure of $m_k(1+x_1+x_2)$
  6. Period structure of $m_k(1+x_1+x_2+x_3)$
  7. Some infinite series behind hyper-Mahler measures
  8. Non-linear Euler sums revisited
  9. Infinite series related to $m_k(1+x_1+x_2)$
  10. Infinite series related to $m_k(1+x_1+x_2+x_{3})$
  11. An outlook on $\varepsilon$-expansions of hypergeometric expressions

Key Result

Lemma 2.1

For $n\in\mathbb Z_{>0}$, the integral representation is valid for $s\in\left( \max\{-2,-\frac{n}{2}\} ,0\right)$, where $J_0(x)\colonequals\sum_{\ell=0}^\infty\frac{(-1)^\ell}{(\ell!)^{2}}\left(\frac{x}{2}\right)^{2\ell}$ is the zeroth-order Bessel function of the first kind. $\blacksquare$

Theorems & Definitions (40)

  • Lemma 2.1: Broadhurst's formula for $W_n(s)$
  • Remark 2.2
  • Proposition 2.3: Weber--Schafheitlin integral and its generalizations
  • proof
  • Corollary 2.4: Hankel--Broadhurst representations for $W_3(s)$ and $W_4(s)$
  • proof
  • Theorem 2.5: $\pi i\,m_k(1+x_1+x_2)$ as Goncharov--Deligne periods
  • proof
  • Remark 2.6
  • Theorem 2.7: $\pi i\,m_k(1+x_1+x_2)$ as Goncharov--Deligne periods, reprise
  • ...and 30 more