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arxiv: 2001.01855 · v1 · pith:MIWMSXP7new · submitted 2020-01-07 · 🧮 math.NT

Integrality of v-adic multiple zeta values

Yen-Tsung Chen This is my paper

Pith reviewed 2026-05-24 15:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiple zeta valuesv-adicfunction fieldsintegralityvaluationA = F_q[theta]Chang-Mishiba
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The pith

The v-adic multiple zeta values are v-adic integers for almost all places v.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the v-adic multiple zeta values ζ_A(s)_v, for any index s in N^r, are v-adic integers at all but finitely many finite places v in A = F_q[θ]. It reaches this by producing lower bounds on the v-adic valuation of these quantities that turn out to be non-negative outside a finite set. A sympathetic reader would care because the result supplies the function-field counterpart to the known integrality of p-adic multiple zeta values.

Core claim

For any index s in N^r and finite place v in A = F_q[θ], the v-adic MZVs ζ_A(s)_v introduced by Chang and Mishiba have non-negative v-adic valuation for almost all such v, and therefore lie in the v-adic integers. This supplies the function-field analogue of the integrality of p-adic MZVs proved by Akagi-Hirose-Yasuda and Chatzistamatiou.

What carries the argument

The v-adic multiple zeta values ζ_A(s)_v together with explicit estimates on their v-adic valuations.

If this is right

  • The integrality holds uniformly across all indices s in N^r.
  • The set of exceptional places v where integrality may fail is finite for each fixed s.
  • The result mirrors the corresponding statement for p-adic MZVs in the number-field setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The integrality may permit these values to be used directly in integral constructions at most places without clearing denominators.
  • One could ask whether the finite set of bad places admits an explicit description in terms of the index s.
  • Similar valuation techniques might apply to other classes of zeta-like series defined over function fields.

Load-bearing premise

The definition of the v-adic MZVs by Chang and Mishiba admits valuation estimates that remain non-negative for all but finitely many places v.

What would settle it

An explicit computation of some index s and some large place v for which the valuation of ζ_A(s)_v is strictly negative.

read the original abstract

In this article, we prove the integrality of $v$-adic multiple zeta values (MZVs). For any index $\mathfrak{s}\in\mathbb{N}^r$ and finite place $v\in A:=\mathbb{F}_q[\theta]$, Chang and Mishiba introduced the notion of the $v$-adic MZVs $\zeta_A(\mathfrak{s})_v$, which is a function field analogue of Furusho's $p$-adic MZVs. By estimating the $v$-adic valuation of $\zeta_A(\mathfrak{s})_v$, we show that $\zeta_A(\mathfrak{s})_v$ is a $v$-adic integer for almost all $v$. This result can be viewed as a function field analogue of the integrality of $p$-adic MZVs, which was proved by Akagi-Hirose-Yasuda and Chatzistamatiou.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper proves that for any fixed index 𝔰 ∈ ℕ^r, the v-adic multiple zeta value ζ_A(𝔰)_v introduced by Chang and Mishiba is a v-adic integer for all but finitely many finite places v ∈ A = 𝔽_q[θ]. The proof proceeds by establishing positive lower bounds on the v-adic valuation of these quantities.

Significance. If the valuation estimates hold, the result supplies a function-field analogue of the known integrality theorems for p-adic MZVs (Akagi-Hirose-Yasuda, Chatzistamatiou). It strengthens the arithmetic theory of MZVs in positive characteristic and may support further comparisons between the function-field and number-field settings.

minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the precise lower bound obtained for the valuation (e.g., the dependence on the degree of v or on the weight of 𝔰).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main result on the integrality of v-adic multiple zeta values for almost all places v.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines v-adic MZVs via the external construction of Chang and Mishiba, then derives integrality for almost all v by direct valuation estimates on that object. No equation reduces the claimed integrality back to a fitted parameter, self-referential normalization, or load-bearing self-citation; the argument is presented as an independent estimate whose inputs are the prior definition and standard valuation tools. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the prior definition of v-adic MZVs by Chang and Mishiba together with standard valuation theory on the ring A = F_q[θ]; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of places and valuations on the polynomial ring A = F_q[θ]
    Invoked when defining finite places v and the v-adic completion.

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Reference graph

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