Arithmetic sums and products of infinite multiple zeta-star values

TL;DR

This paper explores the arithmetic properties of infinite multiple zeta-star values, including sums and products with restricted indices, and proposes conjectures inspired by continued fractions and Cantor sets.

Contribution

It investigates the algebraic structure of infinite multiple zeta-star values and introduces new conjectures relating to their sums, products, and algebraic points.

Findings

Real numbers greater than 1 can be uniquely represented as infinite multiple zeta-star values.

The paper proposes conjectures on algebraic points and arithmetic operations of these values.

Connections to continued fractions and Cantor sets are explored.

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.