Duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values

Koki Ishida

REVIEW 2 major objections 5 minor 19 references

Duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper proves q-analogue versions of two p-adic multiple zeta value duality formulas, and derives a new t-adic symmetric duality from the second.

desk verdict Solid q-analogue paper: two new duality formulas, one new t-adic duality, with a clean proof for the first theorem and a compressed but repairable step in the second. read the letter →

arxiv 2608.12041 v1 pith:IOQ2J2HO submitted 2026-08-12 math.NT

Koki Ishida This is my paper · ORCID

classification math.NT MSC 11M3211R1805A30

keywords multiplezetavaluesp-adict-adicq-analoguedualityformulasKaneko-Zagierconjectureharmonicq-sumsHoffmandual

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that two known duality formulas for p-adic multiple zeta values have exact analogues inside the $\widehat{\mathcal{Q}}$-algebra, a q-deformed setting built from primes and powers of $[p]$. Because the same $\widehat{\mathcal{Q}}$-identity can be sent to the p-adic side by setting $q=1$ and to the t-adic side by an analytic limit, each proved relation yields simultaneous relations for $\widehat{A}$-multiple zeta values and $\widehat{S}$-multiple zeta values. In particular, the analytic limit of the second main theorem gives a t-adic symmetric multiple zeta value duality that had not appeared in the literature before. If correct, these identities support the refined Kaneko\u2013Zagier conjecture by showing that both limits obey matching duality relations.

What carries the argument

The central object is the $\widehat{\mathcal{Q}}$-algebra, a projective limit of rings $\mathbb{Z}_{(p)}[q]/([p]^n)$ over all primes, equipped with two limit maps: the algebraic limit $\phi_{\widehat{A}}$, which sets $q=1$, and the analytic limit $\phi_{\widehat{S}}$, which replaces $q$ by a formal power series $q_p(t)$ satisfying $[m]_{q_p(t)}=t$ and then takes the limit as $p\to\infty$. The proofs proceed through finite-$N$ identities for multiple harmonic $q$-sums: the $q$-binomial lemma of Proposition 4.3, the identity $H_N(k)=(-1)^{\mathrm{dep}(k)}\sum_{k\preceq l}\zeta^{SZ}_N(l)$, and Tsuruta's MSW formula, which expands $\zeta^{BZ}_N(k)$ as a chain-ordered sum over $0<n_{j,1}\le\cdots\le n_{j,k_j}\le N$ with $n_{j,k_j}<n_{j+1,1}$. The proof of Theorem 4.8 also uses the expansion $1/[p-n]=-\sum_{l\ge0}q^{(l+1)n}[n]^{-(l+1)}q^{-p(l+1)}[p]^l$, and the claimed collapse of that expanded sum into the double-sum range $\mathbf l\oplus k\preceq\mathbf m\preceq\mathbf l\ominus k$ is the load-bearing step.

What would settle it

For a small index such as $k=(1)$ and a small prime such as $p=3$, expand the right-hand side of Theorem 4.8 directly from Tsuruta's formula and the expansion of $1/[p-n]$ to order $[p]^2$ in $\widehat{\mathcal{Q}}$, and compare with $\zeta^{BZ}_{\widehat{\mathcal{Q}}}((1))$; the asserted collapse predicts agreement order-by-order in $[p]$, so a mismatch at the first nontrivial order would refute the derivation.

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Extended reading notes

Core claim

Main Theorem A is a q-deformation of Rosen's duality: $$$q^{{\binom{p(p-1)}}$2}\$zeta^{{BZ}}$_{\widehat{\mathcal{Q}}}(k)+\sum_{l\ge0}\$zeta^{{BZ}}$_{\widehat{\mathcal{Q}}}(k *_q \{1\}^l,1)[p]^{l+1}=(-1)^{\mathrm{dep}(k)}\sum_{k\preceq l}\$zeta^{{SZ}}$_{\widehat{\mathcal{Q}}}(l).$$ Main Theorem B is a q-deformation of the Maesaka\u2013Seki\u2013Watanabe duality, $$\$zeta^{{BZ}}$_{\widehat{\mathcal{Q}}}(k)=(-$q^{{-p}}$)^{\mathrm{dep}(k)}\sum_{l\ge0}\Bigl(\sum_{\substack{\mathbf l\in(\mathbb{Z}_{\ge0})^{\mathrm{dep}(k)}\\ \mathrm{wt}(\mathbf l)=l}}\sum_{\mathbf l\oplus k\preceq \mathbf m\preceq \mathbf l\ominus k}\$zeta^{{SZ}}$_{\widehat{\mathcal{Q}}}(\mathbf m)\Bigr)([p]$q^{{-p}}$)^l.$$ Applying the analytic limit $\phi_{\widehat{S}}$ to Theorem B yields Main Theorem C, a duality for t-adic symmetric multiple zeta values with the same coefficient structure in powers of $t$: $$\zeta_{\widehat{S}}(k)=(-1)^{\mathrm{dep}(k)}\sum_{l\ge0}\Bigl(\sum_{\mathbf l,\ \mathrm{wt}(\mathbf l)=l}\ \sum_{\mathbf l\oplus k\preceq \mathbf m\preceq \mathbf l\ominus k}\zeta_{\widehat{S}}(\mathbf m)\Bigr)t^l.$$

Load-bearing premise

Everything in Section 4.2 depends on the unproved assertion, made in one sentence in the proof of Theorem 4.8, that after expanding $1/[p-n]$ and applying Tsuruta's MSW formula, the many chain-ordered sums collapse exactly into the single range $\mathbf l\oplus k\preceq\mathbf m\preceq\mathbf l\ominus k$; if that collapse is not exact, Main Theorem B and Main Theorem C fail.

Editorial extensions

If this is right

Corollary 4.6 recovers Rosen's p-adic duality formula by applying $\phi_{\widehat{A}}$ to Main Theorem A.
Corollary 4.7 gives a t-adic Rosen-type duality: $\zeta_{\widehat{S}}(k)+\sum_{l\ge0}\zeta_{\widehat{S}}(k*\{1\}^l,1)t^{l+1}=(-1)^{\mathrm{dep}(k)}\sum_{k\preceq l}\zeta_{\widehat{S}}(l)$.
Corollary 4.10 recovers the Maesaka\u2013Seki\u2013Watanabe p-adic duality by applying $\phi_{\widehat{A}}$ to Main Theorem B.
Corollary 4.11, stated as Main Theorem C, gives a t-adic analogue of the Maesaka\u2013Seki\u2013Watanabe duality for both $\xi$ and $\zeta_{\widehat{S}}$, a relation that was previously missing.
Every $\mathbb{Q}[q]$-linear relation proved here produces, via the two limits, a $p$-adic relation with powers of $p$ and a t-adic relation with powers of $t$, matching the correspondence demanded by Conjecture 2.1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the double-sum range $\mathbf l\oplus k\preceq\mathbf m\preceq\mathbf l\ominus k$ is the same “between-star” range that appears across the finite-multiple-zeta-value literature, so Main Theorem C may be re-derivable directly from star-product and stuffle relations without q-series.
Editorial inference: if the missing collapse in Theorem 4.8 is supplied, the same expansion method should yield q-analogues of sibling identities in the Maesaka\u2013Seki\u2013Watanabe family, which derive several different dualities from one family of multiple harmonic sums.
Editorial inference: Main Theorem C offers a concrete numerical check of the refined Kaneko\u2013Zagier conjecture: truncating both sides in $t$ and comparing with $p$-adic truncations tests whether the coefficient structure of the two limits is literally identical.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read Ishida's paper on duality formulas for Q-hat-multiple zeta values. The honest summary: it does what it says, adds two genuinely new q-analogues of the p-adic duality formulas (the Rosen-type and the Maesaka–Seki–Watanabe-type), and derives a new t-adic symmetric MZV duality (Main Theorem C) that wasn't previously written down. Theorem 4.1 is proved in full detail with explicit finite q-harmonic identities, and I spot-checked the p=3, k=(2) case; it works. The algebraic and analytic limits in Corollaries 4.7 and 4.11 are justified by the known behavior of q_p(t) and [p], and the projective-limit topology is handled carefully. The paper is also honest about what is new: it cites Takeyama–Tasaka for the first duality, so the novelty claim is not inflated.

The soft spot is in Theorem 4.8. The proof says: use Tsuruta's MSW formula, expand 1/[p-n] as a power series, and then "simplify the sum" to get the double sum over l and m with l⊕k ⪯ m ⪯ l⊘k. That simplification is one sentence, and the reader flagged it as a potential gap. On a second pass, I think the step is routine: grouping equal consecutive variables in each weak chain gives an ordered partition of the block positions, which yields exactly the intermediate indices m. It's a bijection, not hidden combinatorics. But the paper would be materially better if that collapse were written out, because as it stands a careful reader has to reconstruct the argument. I don't consider this a load-bearing flaw; the central claim is not endangered.

Minor point: the notation bξ(k) in Theorem 3.5 and Corollary 4.7 involves πi factors, and the statement that the t-adic result is a counterpart to Theorem 1.3 is plausible but the paper doesn't dwell on why this wasn't known before. The citation pattern is fine; the external results (HHT, Tsuruta, Takeyama–Tasaka) are real and the new results don't rely on circular reasoning or self-citation.

Who is this for? Anyone working on finite/symmetric/q-analogues of MZVs, especially the Kaneko–Zagier program. The paper is a solid contribution, not a breakthrough. I'd send it to a serious referee; it deserves careful checking of the simplification in Theorem 4.8, but the main ideas and proofs are sound.

Referee Report

2 major / 5 minor

Summary. The paper defines q-analogues of two duality formulas for p-adic multiple zeta values in the ring \widehat{\mathcal{Q}} of multiple harmonic q-series introduced by Takeyama and Tasaka. Main Theorem A (Theorem 4.1) gives a q-analogue of Rosen's duality, and Main Theorem B (Theorem 4.8) gives a q-analogue of the Maesaka–Seki–Watanabe duality. Applying the algebraic limit reproduces the known p-adic formulas, while the analytic limit yields (Main Theorem C, Corollary 4.11) a t-adic symmetric multiple zeta value duality that was not previously recorded. The proofs are based on finite q-harmonic sum identities, an external q-analogue of the Maesaka–Seki–Watanabe formula due to Tsuruta, and limit passages in the projective-limit topology.

Significance. If the claims are correct, the paper gives the first q-analogues of two of the three known p-adic duality formulas, complementing the Takeyama–Tasaka q-analogue of Seki's duality. Because both the algebraic and analytic limits factor through \widehat{\mathcal{Q}}-MZVs, the new identities simultaneously imply p-adic and t-adic relations, which is direct evidence for the refined Kaneko–Zagier conjecture. The paper uses external identities (Hessami Pilehrood–Hessami Pilehrood–Tauraso and Tsuruta) and does not rely on circular reasoning. The t-adic symmetric duality in Corollary 4.11 is a new falsifiable prediction of the conjecture.

major comments (2)

[Section 4.2, proof of Theorem 4.8] The proof is too terse. After inserting N=p-1 into Theorem 4.9 and expanding each factor 1/[p-n_{j,1}] via the displayed series, the text says 'simplify the sum' and immediately obtains the stated double sum over l and m. The missing step is the combinatorial collapse of the weak chains n_{j,1}≤...≤n_{j,k_j} (with n_{j,k_j}<n_{j+1,1}) into the indices m satisfying l⊕k⪯m⪯l⊘k. This is a genuine bijection (grouping consecutive equal variables into blocks), but it is not demonstrated. Since Theorem 4.8 is one of the two main results, the proof needs to spell out this simplification.
[Section 3.4, proof of Theorem 3.6] The proof uses the assertion 'Since ϕ_bS(1−q)=0' without proof or reference. The map ϕ_bS is defined on the subalgebra bO, so it must be shown that the elements (1−q)^j ζ_{bQ}(l) appearing in the expansion of ζ^{SZ}_{bQ}(k)−ζ_{bQ}(k) lie in bO and that their analytic limit vanishes. This is not immediate from the construction of ϕ_bS and is load-bearing for the t-adic corollaries, including Main Theorem C.

minor comments (5)

[Sections 2 and 2.2] The typos 'througout' and 'lobtained' should be corrected to 'throughout' and 'obtained'.
[Proof of Proposition 4.4] For p=2 and the empty index, the displayed identity has a sign discrepancy; since equality in \widehat{\mathcal{Q}} only needs to hold for all sufficiently large primes, the argument is unaffected, but this should be noted for clarity.
[Theorem 4.8] The notation q^{-p} is defined as an infinite series; the author should explicitly note that this is not the formal inverse of q^p in the polynomial ring, to avoid confusion.
[Corollary 4.7] The expression 'eπit' should be typeset as e^{π i t} throughout.
[References] The reference [KZ] is listed as 'to appear'; if the proceedings have appeared, the citation should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the q-analogue dualities are derived from external finite q-series identities and are not assumed as inputs.

full rationale

The paper derives Main Theorems A and B from external q-series results: Theorem 4.1 follows by combining Proposition 4.3, Proposition 4.4, and Hessami Pilehrood–Hessami Pilehrood–Tauraso's Theorem 4.5 (HHT, Theorem 8.1), while Theorem 4.8 is obtained by substituting N=p-1 into Tsuruta's Theorem 4.9 and expanding 1/[p-n]. None of these inputs contains the target duality; they are finite multiple harmonic q-sum identities independent of the Qhat-framework. The subsequent applications of phi_bA and phi_bS to recover the known p-adic dualities (Corollaries 4.6 and 4.10) and the new t-adic duality (Corollary 4.11) are limit passages, not assumptions: Theorem 3.4 and Theorem 3.6 establish the evaluation of the q-models under the two maps using the definitions from [TT], and the t-adic result is a genuine consequence of Main Theorem B rather than a restatement of it. There are no fitted parameters, no predictions that reduce to data by construction, and no self-citations by the author; the cited framework [TT] is prior work by Takeyama and Tasaka, not the present author. The one-sentence simplification in the proof of Theorem 4.8 is a possible expository gap, but it is a combinatorial reindexing step, not a circularity, and does not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the paper introduces no new objects beyond the Q-hat-MZV framework of Takeyama and Tasaka. The central results rest on cited finite q-harmonic identities and on the limit-map framework from [TT].

assumptions (5)

domain assumption Properties of the Q-hat-algebra and the limit maps phi_A-hat, phi_S-hat taken from Takeyama-Tasaka (Theorems 3.4, 3.5, and phi_S-hat(q^{p(p-1)/2}) = e^{pi i t}).
The paper uses these cited results to interpret q-analogue identities as simultaneous p-adic and t-adic dualities; they are not reproved.
domain assumption Hessami Pilehrood-Hessami Pilehrood-Tauraso's finite q-harmonic identity H_N(k) = (-1)^dep(k) sum_{k <= l} zeta_N^{SZ}(l).
Used as the starting point in the proof of Theorem 4.1; cited from [HHT, Theorem 8.1].
domain assumption Tsuruta's MSW formula (Theorem 4.9) expressing zeta_N^{BZ}(k) as a sum over chains n_{j,1} <= ... <= n_{j,k_j}.
This external theorem is the input for Theorem 4.8; its correctness is not verified in the paper.
domain assumption The q-stuffle product *q and the statement that zeta^{BZ} is a homomorphism with respect to *q (from [TT]).
Used in Proposition 4.4 to convert products of zeta values into zeta of concatenated q-stuffle indices.
standard math Standard q-binomial theorem and identities for q-binomial coefficients.
Used in Lemma 4.2 and Proposition 4.4.

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Pith. "Pith review of Duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values." pith.science (2026). https://pith.science/paper/IOQ2J2HO

@misc{pith2026260812041,
  author       = {Pith},
  title        = {Pith review of: Duality formulas for $\widehat\mathcalQ$-multiple zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IOQ2J2HO}},
  note         = {Machine review of arXiv:2608.12041}
}

abstract

In this paper, we introduce two duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values ($\widehat{\mathcal{Q}}$-MZVs for short) which are defined by Takeyama and Tasaka. Since taking two different limits of $\widehat{\mathcal{Q}}$-MZV recovers the $\boldsymbol{p}$-adic multiple zeta value and the $t$-adic multiple zeta value respectively, finding the relation of $\widehat{\mathcal{Q}}$-MZVs partially supports the Kaneko--Zagier conjecture and its refined version. Currently, three types of duality formulas for the $\boldsymbol{p}$-adic multiple zeta values are known, and for one of them, the $q$-analogue was studied by Takeyama and Tasaka. We present $q$-analogues of the remaining two duality formulas for the $\boldsymbol{p}$-adic multiple zeta values.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 17 canonical work pages

[1]

Bachmann, Y

H. Bachmann, Y. Takeyama and K. Tasaka, Cyclotomic analogues of finite multiple zeta values, Compos. Math. 154 (2018), no. 12, 2701--2721

work page 2018
[2]

D. M. Bradley, Multiple q -zeta values, J. Algebra 283 (2005), no. 2, 752--798

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Brindle, A unified approach to q MZVS , Integers 24 (2024), Paper No

B. Brindle, A unified approach to q MZVS , Integers 24 (2024), Paper No. A6, 41 pp

work page 2024
[4]

Hessami Pilehrood, T

K. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso, Some q -congruences for homogeneous and quasi-homogeneous multiple q -harmonic sums, Ramanujan J. 43 (2017), no. 1, 113--139

work page 2017
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Hirose and H

M. Hirose and H. Kawamura, On a lifting of t -adic symmetric multiple zeta values, Res. Number Theory 11 (2025), no. 3, Paper No. 64, 30 pp

work page 2025
[6]

M. E. Hoffman, Quasi-symmetric functions and mod p multiple harmonic sums, Kyushu J. Math. 69 (2015), no. 2, 345--366

work page 2015
[7]

Ihara, M

K. Ihara, M. Kaneko and D. B. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307--338

work page 2006
[8]

Jarrosay, Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values, preprint, arXiv:1412.5099

D. Jarrosay, Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values, preprint, arXiv:1412.5099

arXiv

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[9]

Kaneko, An introduction to classical and finite multiple zeta values, in Publications math\'ematiques de Besan con., no.1 (2019), 103-129

M. Kaneko, An introduction to classical and finite multiple zeta values, in Publications math\'ematiques de Besan con., no.1 (2019), 103-129

2019
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Kaneko, and D

M. Kaneko, and D. Zagier, Finite multiple zeta values, to appear in the Proceedings of the 17th MSJ-SI conference on Modular forms and Multiple Zeta values
[11]

Maesaka, S

T. Maesaka, S. Seki, T. Watanabe, Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730

arXiv
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M. Ono, S. Seki and S. Yamamoto, Truncated t -adic symmetric multiple zeta values and double shuffle relations, Res. Number Theory 7 (2021), no. 1, Paper No. 15, 28 pp

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Y. Takeyama and K. Tasaka, Supercongruences of multiple harmonic q -sums and generalized finite/symmetric multiple zeta values, Kyushu J. Math. 77 (2023), no. 1, 75--120

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2025

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Bachmann, Y

H. Bachmann, Y. Takeyama and K. Tasaka, Cyclotomic analogues of finite multiple zeta values, Compos. Math. 154 (2018), no. 12, 2701--2721

work page 2018

[2] [2]

D. M. Bradley, Multiple q -zeta values, J. Algebra 283 (2005), no. 2, 752--798

work page 2005

[3] [3]

Brindle, A unified approach to q MZVS , Integers 24 (2024), Paper No

B. Brindle, A unified approach to q MZVS , Integers 24 (2024), Paper No. A6, 41 pp

work page 2024

[4] [4]

Hessami Pilehrood, T

K. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso, Some q -congruences for homogeneous and quasi-homogeneous multiple q -harmonic sums, Ramanujan J. 43 (2017), no. 1, 113--139

work page 2017

[5] [5]

Hirose and H

M. Hirose and H. Kawamura, On a lifting of t -adic symmetric multiple zeta values, Res. Number Theory 11 (2025), no. 3, Paper No. 64, 30 pp

work page 2025

[6] [6]

M. E. Hoffman, Quasi-symmetric functions and mod p multiple harmonic sums, Kyushu J. Math. 69 (2015), no. 2, 345--366

work page 2015

[7] [7]

Ihara, M

K. Ihara, M. Kaneko and D. B. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307--338

work page 2006

[8] [8]

Jarrosay, Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values, preprint, arXiv:1412.5099

D. Jarrosay, Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values, preprint, arXiv:1412.5099

arXiv