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$p$-adic multiple zeta values and binomial multiple harmonic sums

Hidekazu Furusho, David Jarossay

TL;DR

The paper provides an explicit, computable formula for $p$-adic multiple zeta values by rewiring the problem through a non-commutative formal power-series framework and a strategic automorphism of the projective line. It introduces binomial multiple harmonic sums as the key combinatorial objects encoding coefficients in the polylogarithm expansions and derives a depth-inductive formula for Deligne's $p$-adic MZVs, with explicit depth-1 and depth-2 cases. The approach streamlines previous methods by avoiding heavy overconvergent polylogarithm computations and leveraging Mahler continuity to finalize the limits. The work enhances computational accessibility of $p$-adic MZVs and opens avenues for adjoint and potential cyclotomic generalizations, preserving a clear inductive structure across depth.

Abstract

We present a concise method for deriving an explicit formula for $p$-adic multiple zeta values. The formula features a variant of multiple harmonic sums, termed binomial multiple harmonic sums.

$p$-adic multiple zeta values and binomial multiple harmonic sums

TL;DR

The paper provides an explicit, computable formula for -adic multiple zeta values by rewiring the problem through a non-commutative formal power-series framework and a strategic automorphism of the projective line. It introduces binomial multiple harmonic sums as the key combinatorial objects encoding coefficients in the polylogarithm expansions and derives a depth-inductive formula for Deligne's -adic MZVs, with explicit depth-1 and depth-2 cases. The approach streamlines previous methods by avoiding heavy overconvergent polylogarithm computations and leveraging Mahler continuity to finalize the limits. The work enhances computational accessibility of -adic MZVs and opens avenues for adjoint and potential cyclotomic generalizations, preserving a clear inductive structure across depth.

Abstract

We present a concise method for deriving an explicit formula for -adic multiple zeta values. The formula features a variant of multiple harmonic sums, termed binomial multiple harmonic sums.

Paper Structure

This paper contains 5 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. $p$-adic multiple zeta values as certain limits of $p$-adic multiple polylogarithms
  3. Combinatorics of non-commutative formal power series in the limit equation
  4. Binomial multiple harmonic sums and the power series expansion of $p$-adic multiple polylogarithms
  5. End of the proof and comments

Key Result

Theorem 3

For all $d \in \mathbb{N}_{\geq 1}$ and $(n_{1},\ldots,n_{d}) \in \mathbb{N}_{\geq 1}^{d}$ , we have

Theorems & Definitions (27)

  • Definition 2
  • Theorem 3
  • Example 4
  • Example 5
  • Lemma 1.2
  • proof
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.5
  • ...and 17 more