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Arithmetic sums and products of infinite multiple zeta-star values

Jiangtao Li, Siyu Yang

Abstract

Multiple zeta-star values are variants of multiple zeta values which allow equality in the definition. Similar to the theory of continued fractions, every real number which is greater than $1$ can be realized as an unique infinite multiple zeta-star values in a natural way. In this paper, we investigate the arithmetic sums and products of infinite multiple zeta-star values with restricted indices. Moreover, inspired by the theory of continued fractions and Cantor set, we propose a series of conjectures concerning the algebraic points and arithmetic sums and products of infinite multiple zeta-star values with certain indices.

Arithmetic sums and products of infinite multiple zeta-star values

Abstract

Multiple zeta-star values are variants of multiple zeta values which allow equality in the definition. Similar to the theory of continued fractions, every real number which is greater than can be realized as an unique infinite multiple zeta-star values in a natural way. In this paper, we investigate the arithmetic sums and products of infinite multiple zeta-star values with restricted indices. Moreover, inspired by the theory of continued fractions and Cantor set, we propose a series of conjectures concerning the algebraic points and arithmetic sums and products of infinite multiple zeta-star values with certain indices.

Paper Structure

This paper contains 4 sections, 8 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The order structure of multiple zeta-star values
  3. Arithmetic sums of binary expression
  4. Sums and products of infinite depth multiple zeta-star values

Key Result

Theorem 1.1

For $q\geq 2$, we have: $(i)$ Here $(ii)$ Here $(iii)$ $(iv)$

Figures (2)

  • Figure 1: interval of $T_i$
  • Figure 2: first stage of $\overline{\eta(\mathcal{T}_2)}$ subdivision

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 3.2
  • ...and 4 more