Multiple Zeta Dagger Values in Positive Characteristic

• Multiple Zeta Dagger Values are function field multizeta values in positive characteristic defined via signed, symmetrically-indexed summations that extend classical MZVs.
• They introduce a nontrivial algebraic involution on quotient spaces, revealing duality phenomena and novel structures distinct from classical depth-reversal.
• The framework links Carlitz multiple polylogarithms with p-adic and v-adic zeta constructions, offering new methods for exploring product relations and linear dependencies.

Multiple zeta dagger values (MZDVs) are a new class of function field multizeta values in positive characteristic, introduced to extend the algebraic and combinatorial framework of classical multiple zeta values (MZVs). MZDVs are defined via signed and symmetrically-indexed summations over twisted power sums of monic polynomials, in close parallel to the Carlitz multiple polylogarithms. They play a pivotal role in constructing the first nontrivial algebra involution on a quotient space of MZVs in positive characteristic, revealing duality phenomena distinct from those of the classical depth-reversal and foreshadowing new structures in the function field arithmetic context (Mishiba, 1 Jan 2026).

1. Foundational Definitions

Let $F$ denote the finite field with $q$ elements and $A = F[\theta]$ the corresponding polynomial ring. Let $A_+$ be the set of monic polynomials in $A$. For $d \geq 0$, define $L_d = \prod_{i=1}^d (\theta - \theta^{q^i}) \in A$. Given an $F[L_1]$-subalgebra $R \subset k := F(\theta)$, and the set of index tuples $I = \bigsqcup_{r \geq 0} \mathbb{N}^r$, the Thakur power sums are $S_d(s) = \sum_{a \in A_+, \deg a = d} a^{-s} \in k$ for $s \in \mathbb{N}$.

The multiple zeta value (MZV) in positive characteristic is

$$\zeta_A(\mathbf{s}) = \sum_{d_1 > \cdots > d_r \geq 0} S_{d_1}(s_1) \cdots S_{d_r}(s_r) \in C_\infty,$$

where $\mathbf{s} = (s_1, \ldots, s_r) \in I$.

The multiple zeta dagger value (MZDV) is defined for $\mathbf{s} \in I$ by

$$\zeta_A^\dagger(\mathbf{s}) := {-1}^r \sum_{0 \leq d_1 \leq \cdots \leq d_r} S_{d_1}(s_1) \cdots S_{d_r}(s_r).$$

When $s_i \leq q$ for all $i$, one can write $S_d(s_i) = L_d^{-s_i}$, giving the alternative formula

$$\zeta_A^\dagger(\mathbf{s}) = (-1)^r \sum_{0 \leq d_1 \leq \cdots \leq d_r} \frac{1}{L_{d_1}^{s_1} \cdots L_{d_r}^{s_r}}.$$

A telescoping-sum argument shows $\zeta_A^\dagger(\mathbf{s})$ belongs to $Z_R$, the $R$-span of classical MZVs.

2. Carlitz Multiple Dagger Polylogarithms and Special Values

The Carlitz multiple dagger polylogarithm (CMDPL) for $\mathbf{s} = (s_1, \ldots, s_r)$ and variables $z_1, \ldots, z_r$ is

$$\operatorname{Li}^\dagger_\mathbf{s}(z_1, \ldots, z_r) = (-1)^r \sum_{0 \leq d_1 \leq \cdots \leq d_r} \frac{z_1^{q^{d_1}} \cdots z_r^{q^{d_r}}}{L_{d_1}^{s_1} \cdots L_{d_r}^{s_r}} \in k[[z_1,\ldots,z_r]].$$

The special value at $(1,\ldots,1)$ recovers the MZDV: $$\operatorname{Li}^\dagger_\mathbf{s}(1, \ldots, 1) = \zeta_A^\dagger(\mathbf{s}).$$ When $s_i \leq q$ for all $i$, these values coincide with the classical Carlitz multiple polylogarithm.

3. Algebraic Structure, Product Rules, and Linear Relations

Indices assemble into the free $R$-module $h_R^1$ with basis $I$, equipped with two principal algebraic products. The $q$-shuffle product $*^\zeta$ on $h^1_R$ reflects multiplication of MZVs, while the harmonic product $*$ corresponds to products of polylogarithm or dagger values. Both product structures yield the familiar product-to-series expansions: $$\begin{aligned} \zeta_A(P) \cdot \zeta_A(Q) &= \zeta_A(P *^\zeta Q), \ \operatorname{Li}_P(1) \cdot \operatorname{Li}_Q(1) &= \operatorname{Li}_{P*Q}(1), \ _{P}(1) \cdot _{Q}(1) &= _{P*Q}(1). \end{aligned}$$ There exists a bilinear box-plus operation $\boxplus$ and a family of $R$-linear relations $A^\zeta(\mathbf{s};m;\mathbf{n})$ spanning all $R$-linear dependencies among MZVs, as well as $A^{\operatorname{Li}}$ for polylogarithms, parametrized by indices $\mathbf{s}, \mathbf{n} \in I$ and $m \geq 1$. These relations generate all linear relations among the respective families, with

$$\zeta_A(A^\zeta(\mathbf{s};m;\mathbf{n})) = 0, \quad \operatorname{Li}(A^{\operatorname{Li}}(\mathbf{s};m;\mathbf{n})) = 0.$$

4. Quotient Algebra and Nontrivial Involution

Setting $Z_R$ as the graded $R$-algebra of positive-characteristic MZVs, $\zeta_A(q-1)$ is a nonzero-divisor in $Z_R$. Forming the quotient space $\bar Z_R := Z_R / (\zeta_A(q-1)Z_R)$, there is a uniquely determined and nontrivial $R$-algebra involution

$\iota: \bar Z_R \to \bar Z_R,$

characterized by its action on special values: $$\iota(\operatorname{Li}_\mathbf{s}(1) \bmod \zeta_A(q-1)) = \zeta_A^\dagger(\mathbf{s}) \bmod \zeta_A(q-1).$$ Equivalently, $$\iota(\zeta_A(\mathbf{s}) \bmod \zeta_A(q-1)) = \zeta_A^\dagger(\mathbf{s}) \bmod \zeta_A(q-1)$$. This involution satisfies $\iota^2 = \operatorname{Id}$ and $\iota \neq \operatorname{Id}$, as demonstrated by the explicit computation $$\iota(\operatorname{Li}_{(1)}(1)) = -\operatorname{Li}_{(1)}(1) \neq \operatorname{Li}_{(1)}(1)$$.

The involution arises from the fact that $\{\operatorname{Li}_\mathbf{s}(1)\}$ and $\{\zeta_A^\dagger(\mathbf{s})\}$ satisfy the same product and $R$-linear relations in the quotient, permitting the construction of a unique algebra isomorphism between them.

5. Explicit Examples and Low-Weight Behavior

Representative computations exemplify the use and behavior of MZDVs and the involution:

• Weight 1 (depth 1):

$$\zeta_A(1) = \sum_{d \geq 0} S_d(1), \quad \zeta_A^\dagger(1) = -\sum_{d \geq 0} S_d(1) = -\zeta_A(1).$$

Therefore, in the quotient, $\iota(\zeta_A(1)) = -\zeta_A(1)$.

• Weight 2: For many cases, $\iota(\zeta_A(2)) \equiv -\zeta_A(2)$.
• Weight 3 (depth 1 or 2): For $(1,2)$ the involution produces nontrivial exchanges between depth decompositions modulo $\zeta_A(q-1)$.

These computations demonstrate the nontrivial nature of the involution and the interaction between classical and dagger MZVs.

6. Theoretical Significance and Broader Implications

The involution $\iota$ solves a positive-characteristic analogue of the open problem of finding nontrivial automorphisms on the $\mathbb{Q}$-algebra of classical MZVs—a problem that, classically, has only the depth-reversal involution (e.g., $\zeta(2) \mapsto -\zeta(2)$). The new involution demonstrates that Carlitz multizeta values and their dagger analogs, though constructed distinctly, share identical $R$-linear and algebraic relations in the quotient, underlining an unexpected duality structure beyond depth-reversal.

Furthermore, the quotient $\bar Z_R$ is conjecturally isomorphic to the algebra of $v$-adic MZVs (Chang–Chen–Mishiba), suggesting that $\iota$ may descend to a $p$-adic involution in the arithmetic of function fields. This connection opens avenues to new families of functional equations for Carlitz multiple dagger polylogarithms and relationships with Anderson $t$-module periods. A plausible implication is the existence of hidden dualities and automorphisms in higher-depth algebraic structures among positive-characteristic multizeta values, with relevance to analogs of the Drinfeld associator, $p$-adic zeta values, and Galois symmetries in function field settings (Mishiba, 1 Jan 2026).