arXiv subjects

Taiki Watanabe

Publications and source records attributed to Taiki Watanabe.

3 recordsLinked to original sources

arXiv2026

A unified proof of conjectures on the spaces of multiple $q$-zeta values

We prove two conjectures on the spaces generated by multiple $q$-zeta values. More precisely, we show that the spaces $Z_q^{\mathrm{o}}$ and $Z_{q,1}^{\mathrm{o}}$ already generate the larger spaces $Z_q$ and $Z_{q,1}$, respectively. Our result is stronger than the equality of $\mathbb{Q}$-vector spaces: for every generator in the larger spaces, we construct an explicit expression with integer coefficients in terms of the smaller generating families. We first establish these formulas at the finite level, where suitable finite $q$-analogues admit recursive descriptions through generating series, and then pass to the infinite limit.

math.NT↗

arXiv2025

The $\mathbb{Z}$-module of multiple zeta values is generated by ones for indices without ones

We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $ζ(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified multiple harmonic sums that partially satisfy the relations among multiple zeta values and to determine the structure of the space generated by them.

math.NT↗

arXiv2024

Deriving two dualities simultaneously from a family of identities for multiple harmonic sums

We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values.

math.NT↗