On certain combination of colored multizeta values

正特征下的着色多重 zeta 值

Colored multizeta values in positive characteristic

该研究考察正特征下 Shen-Shi 着色多重 zeta 值，建立非零性、和混洗关系、$t$-motivic 解释，并证明不同权重的值在 $\overline{k}$ 上线性无关。

Special relations between multizeta values and parity results

We study relations between multizeta values for function fields introduced by D. Thakur. The F_p-span of Thakur's multizeta values is an algebra (Thakur. Shuffle relations for function field multizeta values). In particular, the product ζ(a)ζ(b) is a linear combination of multizeta values. In this paper, several of the conjectures formulated by the author and by D. Thakur for small values or for special families of a about how to write ζ(a)ζ(b) as an F_p-linear combination of multizeta values, are proved. Also, the parity conjecture formulated by Thakur is proved.

Relations between multizeta values in characteristic p

We study relations between the multizeta values for function fields introduced by D. Thakur. The product ζ(a)ζ(b) is a linear combination of multizeta values. For q=2, a full conjectural description of how the product of two zeta values can be described as the sum of multizetas was given by Thakur. The recursion part of this recipe was generalized by the author. In this paper, the main conjecture formulated by the author, as well as some conjectures of Thakur are proved. Moreover, for general q, we prove closed formulas as well as a recursive recipe to express ζ(a)ζ(b) as a sum of multizeta va…

Some conjectures and results about multizeta values for F_q[t]

In this paper, we explain several conjectures about how a product of two Carlitz-Goss zeta values can be expressed as a F_p-linear combination of Thakur's multizeta values, generalizing the q=2 case dealt by D. Thakur in Relations between multizeta values for F_q[t]. In contrast to the classical sum shuffle, ζ(a)ζ(b)=ζ(a+b)+ζ(a,b)+ζ(b,a), the identities we get are much more involved. Hundreds of instances of these conjectures have been proved and we describe the proof method and the evidence.

A $t$-motivic interpretation of shuffle relations for multizeta values

Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $ζ_A(r)$ and $ζ_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $ζ_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(θ)^{r+s}$ gives a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[θ])$ constructed with respect to the …

Finite multiple zeta values associated with 2-colored rooted trees

We define finite multiple zeta values (FMZVs) associated with some combinatorial objects, which we call 2-colored rooted trees, and prove that FMZVs associated with 2-colored rooted trees satisfying certain mild assumptions can be written explicitly as $\mathbb{Z}$-linear combinations of the usual FMZVs. Our result can be regarded as a generalization of Kamano's recent work on finite Mordell-Tornheim multiple zeta values. As an application, we will give a new proof of the shuffle relation of FMZVs, which was first proved by Kaneko and Zagier.