$t$-adic symmetric multiple zeta values for indices with alternating $1$ and $3$, starting with $1$ and ending with $3$
Kento Fujita
TL;DR
The paper proves the conjectured polynomial expression modulo $\pi^2$ for the $t$-adic SMZV with indices $\{1,3\}^n$ (starting with $1$ and ending with $3$). It builds on prior work showing similar polynomials for $\{3,1\}^n$ and $\{1,3\}^n$ in the $t$-adic setting, and extends to the case $m=3$ by decomposing $\zeta^*_{\mathcal{S}_3}(\mathbf{k})$ into $I_0,I_1,I_2$-components via the $I$- and $\sigma$-maps. The proof reduces modulo $\pi^2$, uses a key lemma relating $I_2$ to products of $\zeta^*$ values with indices $4n+1$ and $4n+3$, and assembles an explicit formula showing the $t$-adic SMZV is a polynomial in ordinary zeta values modulo $\pi^2$. The result supports the Kaneko-Zagier philosophy that $t$-adic SMZVs and related $p$-adic finite MZVs satisfy polynomial relations akin to MZVs, at least modulo $\pi^2$ for these indices. The paper thus confirms the conjecture for indices that start with 1 and end with 3, contributing to the understanding of the algebraic structure of $t$-adic SMZVs.
Abstract
Hirose, Murahara, and Saito proved that some $t$-adic symmetric multiple zeta values, for indices in which $1$ and $3$ appear alternately in succession, can be expressed as polynomials in Riemann zeta values, and conjectured similar formulas. In this paper, we prove the conjectured formula for indices that start with $1$ and end with $3$, showing that they also can be expressed as polynomials in Riemann zeta values.