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Bounded Additive Relation and Application to Finite Multiple Zeta Values

Tomoki Mihara

Abstract

We formulate an algebraic problem to find a generating system of a finite subset of an Abelian group with respect to linear relations whose coefficients are bounded by a constant, and recall MITM algorithm for the problem. As an application of MITM algorithm for the Abelian group \begin{eqnarray*} \mathbb{Z}/106700590455862347842907841856033238416352421 \mathbb{Z} \end{eqnarray*} combined with Chinese remainder algorithm, we give a table of expected linear relations of finite multiple zeta values of weight $10$.

Bounded Additive Relation and Application to Finite Multiple Zeta Values

Abstract

We formulate an algebraic problem to find a generating system of a finite subset of an Abelian group with respect to linear relations whose coefficients are bounded by a constant, and recall MITM algorithm for the problem. As an application of MITM algorithm for the Abelian group \begin{eqnarray*} \mathbb{Z}/106700590455862347842907841856033238416352421 \mathbb{Z} \end{eqnarray*} combined with Chinese remainder algorithm, we give a table of expected linear relations of finite multiple zeta values of weight .
Paper Structure (9 sections, 14 equations, 7 figures, 14 tables, 7 algorithms)

This paper contains 9 sections, 14 equations, 7 figures, 14 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Convention
  3. Bounded Additive Relation Problem
  4. MITM Algorithm
  5. Dynamic MITM Algorithm
  6. Numerical Analysis of Finite Multiple Zeta Values
  7. Dynamic Programming
  8. Chinese Remainder Algorithm
  9. Experiment

Figures (7)

  • Figure : MITM algorithm
  • Figure : Expected time complexity for $D^{\text{\rm R}}$
  • Figure : Dynamic MITM algorithm to compute a minimal generator system of $S \subset M$ over $\vec{c}$
  • Figure : Parallel computation of the mod $p$ harmonic sums for a tree $T$ of indices
  • Figure : Pre-computation of for mod $p_{\ell}$ harmonic sums of weight $w$ for each $\ell \in \mathbb{N}_{< L}$ for a $\vec{p} = (p_{\ell})_{\ell=0}^{L-1}$
  • ...and 2 more figures

Theorems & Definitions (3)

  • Conjecture 1: Dimension conjecture
  • Example 2.1
  • Example 5.1