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Laurent type expansion of multiple polylogarithms at integer points

Pawan Singh Mehta, Biswajyoti Saha

TL;DR

This work delivers a comprehensive Laurent-type expansion for depth $r$ multiple polylogarithms around integer points by extending a regularisation framework to arbitrary integer vectors. Key innovation is the explicit formula expressing $\mathrm{Li}_{\mathbf{z}}(s_1,\ldots,s_r)$ in terms of regularised values $\mathrm{Li}^{\mathrm{Reg}}_{(\mathbf{z};\mathbf{a})}$ and a finite collection of tail contributions, with rational coefficient functions captured by an intricate boundary data $(\mathcal{I}(\mathbf{z},\mathbf{a}), H_j^{(i)})$. The paper develops a rigorous asymptotic calculus for sequences of holomorphic and meromorphic germs with variable coefficients, including a robust translation identity for star tails and a system of boundary lemmas, to justify the Laurent inversion. These results extend previous depth‑2 regularisation and provide a unified framework for holomorphy and regularised values at general integer points, with explicit constructions valid for arbitrary depth and root‑of‑unity parameters. The findings have potential implications for analytic continuation, special value evaluations, and the structural understanding of multiple polylogarithms in number theory and related fields.

Abstract

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational functions. The coefficients of these power series are the regularised values of the multiple polylogarithm functions at certain related integer points.

Laurent type expansion of multiple polylogarithms at integer points

TL;DR

This work delivers a comprehensive Laurent-type expansion for depth multiple polylogarithms around integer points by extending a regularisation framework to arbitrary integer vectors. Key innovation is the explicit formula expressing in terms of regularised values and a finite collection of tail contributions, with rational coefficient functions captured by an intricate boundary data . The paper develops a rigorous asymptotic calculus for sequences of holomorphic and meromorphic germs with variable coefficients, including a robust translation identity for star tails and a system of boundary lemmas, to justify the Laurent inversion. These results extend previous depth‑2 regularisation and provide a unified framework for holomorphy and regularised values at general integer points, with explicit constructions valid for arbitrary depth and root‑of‑unity parameters. The findings have potential implications for analytic continuation, special value evaluations, and the structural understanding of multiple polylogarithms in number theory and related fields.

Abstract

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the -aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational functions. The coefficients of these power series are the regularised values of the multiple polylogarithm functions at certain related integer points.
Paper Structure (17 sections, 17 theorems, 150 equations)

This paper contains 17 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Behaviour of the multiple polylogarithms at the integer points of $\overline{U_r(\mathbf{z})}$
  3. Intermediate steps for Theorem \ref{['main-thm-section-2']}
  4. Proof of Theorem \ref{['main-thm-section-2']}
  5. Proof of the intermediate steps
  6. Proof of Lemma \ref{['uni-convg']}
  7. Proof of Proposition \ref{['trans']}
  8. Proof of Lemma \ref{['boundary-1']}
  9. Proof of Lemma \ref{['boundary-2']}
  10. Proof of Lemma \ref{['boundary-3']}
  11. Proof of Lemma \ref{['Taylor-lem']}
  12. Asymptotic expansions of sequences of germs of holomorphic and meromorphic functions with variable coefficients
  13. Asymptotic expansions of sequences of germs of holomorphic functions with variable coefficients
  14. Asymptotic expansions of sequences of germs of meromorphic functions with variable coefficients
  15. Asymptotic expansion of $\mathrm{Li}^{\star}_{(z)}(s)_{\geq N}$
  16. ...and 2 more sections

Key Result

Theorem 1

The multiple Dirichlet series in mp converges on $U_r(\mathbf{z})$. Moreover, the series in mp defines a holomorphic function on $U_r(\mathbf{z})$.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Example 1
  • Example 2
  • Proposition 1
  • Remark 2
  • Theorem 5
  • ...and 25 more