Arithmetic sums and products of infinite multiple zeta-star values

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.

💡 Research Summary

This paper conducts a detailed investigation into the arithmetic properties of sets comprised of infinite multiple zeta-star values. Multiple zeta-star values, denoted ζ⋆(k1, k2, …), are a variant of multiple zeta values where non-strict inequalities are allowed in the summation indices. The authors establish that every real number greater than 1 can be uniquely represented as such an infinite series via a bijective “zeta-star correspondence” map η.

The core analysis focuses on two families of restricted index sets: D_q, consisting of index sequences where all terms are bounded above by q (q≥2), and T_p, consisting of sequences where all terms are bounded below by p (p≥2). The set η(D_q) is a Cantor-like set of Lebesgue measure zero.

The first main result, Theorem 1.1, reveals that despite being a thin set, η(D_q) exhibits rich additive and multiplicative structure. Specifically, the sumset η(D_q) + η(D_q) equals the closed interval