Yamamoto's interpolation of finite multiple zeta and zeta-star values

Hideki Murahara; Masataka Ono

REVIEW 2 major objections 5 minor 47 references

Yamamoto's interpolation of finite multiple zeta and zeta-star values

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single polynomial parameter $t$ interpolates between finite multiple zeta values and their zeta-star variants, and the paper proves the cyclic-sum, Bowman–Bradley, and weighted-sum relations hold at every $t$.

desk verdict Useful interpolation of finite MZV relations; the headline cyclic sum theorem for S-type depends on an unpublished result, but the rest is solid and the paper deserves review after that dependency is addressed. read the letter →

arxiv 1908.09307 v3 pith:QYIL3NIR submitted 2019-08-25 math.NT

Hideki Murahara , Masataka Ono This is my paper · ORCID

classification math.NT MSC 11M3205A19

keywords Multiplezeta(-star)valuesInterpolatedzetaFiniteSymmetricCyclicsumformulaBowman-BradleytypeWeightedt-interpolated

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 defines, for each index $(k_1,\dots,k_r)$ and each $F=A$ or $S$ (the two standard finite analogues of multiple zeta values), a polynomial $\zeta^t_F(k_1,\dots,k_r)$ that interpolates between the finite multiple zeta value at $t=0$ and the finite multiple zeta-star value (the variant allowing equal arguments) at $t=1$. Its central claim is that three major families of relations—the cyclic sum formula, the Bowman–Bradley type formula, and the weighted sum formula—hold for the whole one-parameter family, not just at the two endpoints. A sympathetic reader should care because this turns relations that were proved separately for finite values and finite star values into single polynomial identities, just as already happens for classical multiple zeta values. The proofs are coefficient-wise: each coefficient of $t^n$ is identified with a known $t=0$ relation, so the contribution is a transfer mechanism that carries endpoint identities through the interpolation.

What carries the argument

The load-bearing object is the $t$-index: for an index $\mathbf{k}=(k_1,\dots,k_r)$, write $\mathbf{k}_t$ for the sum over all ways to replace each $\square$ in $k_1\square k_2\square\cdots\square k_r$ by a comma or a plus, weighted by $t^{\#\text{plus}}$; applying the finite zeta map $\zeta^t_F$ to $\mathbf{k}_t$ gives the interpolated value, and the coefficient of $t^m$ isolates the depth-$m$ contributions. For the cyclic sum proof, the key construction is the cyclic index $C_m(\mathbf{k})$, obtained by the same summation with $m$ pluses and a cyclic wrap-around that merges the last and first entries; Proposition 2.2 shows the $t^m$-coefficient of the difference between the two sides of the cyclic sum formula equals $\zeta_F(F^0(C_m(\mathbf{k})))$, reducing every $t$-degree to the known $t=0$ cyclic sum. For the Bowman–Bradley family the key mechanism is the recurrence of Proposition 3.2, which expresses $(n+1)B^{(n+1)}_{l,m}[a]$, the coefficient of $t^{n+1}$ in the $t$-index of the shuffle sum $B_a$, as a combination of $B^{(n)}$ terms with shifted arguments, so that the $n=0$ theorem for finite values propagates to all $n$. For the weighted sum formula, the auxiliary element $H(k,r,n)=F(k,r,n)+S'(k,r,n)+G'(k,r,n)$ together with the duality map $\varphi$ plays the central role: Proposition 4.4 shows $H+\varphi(H)$ is either $0$ or a multiple of $(\{1\}^k)$, and both are killed by $\zeta_F$, using the sum formula and the Ohno-type relation.

What would settle it

Compute both sides of Theorem 1.2 for the index $\mathbf{k}=(2,3)$ as polynomials in $t$ in the A-valued setting at a small prime, say $p=7$: reduce the coefficients of $t^0,t^1,t^2$ modulo $7$ and test whether the difference vanishes. A nonzero coefficient at any $t$-degree would refute the claimed family; conversely, matching all coefficients for a handful of small indices and primes would support it.

Watch

Extended reading notes

Core claim

The discovery is that the interpolation parameter $t$ is structurally transparent: every $t^n$-coefficient of the three main identities is forced by the $t=0$ identity applied to a cyclic or shuffle-modified index, so the polynomial family inherits all endpoint relations. Concretely, the paper proves the cyclic sum formula of Theorem 1.2 for indices $(k_1,\dots,k_r)\neq(1,\dots,1)$, the vanishing $\zeta^t_F(B_a)=0$ of the Bowman–Bradley sums of Theorem 3.1, and the weighted sum formula $\sum_{k_1+\cdots+k_r=k}2^{k_r-1}\zeta^t_F(k_1,\dots,k_r)=0$ for odd $r$ of Theorem 4.1, for every $t$ and both $F=A,S$. At $t=0$ these statements reduce to the known finite MZV relations; at $t=1$ they become the corresponding zeta-star relations. The paper also shows that the harmonic, shuffle, duality, and derivation relations extend to the $t$-family, so the entire algebraic calculus of finite values can be run with the parameter left free.

Load-bearing premise

For the S-valued interpolation, the proof of the cyclic sum formula imports the starting case $\zeta_S(F^0(C_m(\mathbf{k})))=0$ from a result cited as unpublished or as a preprint by the same authors; if that S-type cyclic sum is not accepted, the main theorem for $F=S$ is not self-contained.

Editorial extensions

If this is right

At $t=0$ and $t=1$, Theorems 1.2, 3.1, and 4.1 recover the known cyclic-sum, Bowman–Bradley, and weighted-sum formulas for both the A- and S-type finite multiple zeta(-star) values, so each theorem is a simultaneous generalization of four endpoint statements.
The coefficient-wise proof gives, for every $n$ between $0$ and the depth minus one, new relations among finite MZVs and MZSVs that have no separate name in the literature; these intermediate $t^n$-relations are new content even when the endpoints were known.
The algebraic relations of Section 5 (harmonic, shuffle, duality, derivation, symmetric sum, antipode-like, and Hoffman relations) hold for the $t$-family, extending the two-variable algebraic setup of classical interpolated values to the finite setting.
Because the identities are polynomial in $t$, any endpoint relation that can be phrased coefficient-wise automatically transports across the interpolation, offering a template for future finite-value relations.
The weighted-sum formula for odd $r$ has no known analogue in the classical $t$-MZV world, so this part of the interpolation is special to the finite setting.

Reading between the lines

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

If the coefficient-transfer mechanism generalizes, the same $t$-interpolation should work for other finite-value families, such as cyclotomic finite multiple zeta values or truncated $t$-adic symmetric values, with the same three theorems holding in identical polynomial shape.
A concrete research task is to give a fully self-contained proof of $\zeta_S(F^0(C_m(\mathbf{k})))=0$; this would make Theorem 1.2 independent of the unpublished S-type cyclic sum and might reveal an elementary cyclic-sum identity for S-values.
One could treat $t$ as a deformation parameter and evaluate the proved polynomial identities at special algebraic values of $t$, such as roots of unity, to obtain new explicit zero relations in $\mathcal{A}$ and $\mathbb{Z}/\zeta(2)\mathbb{Z}$.

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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

First, the quick verdict: this is a genuinely useful paper for the finite multiple zeta value community. It takes Seki's t-interpolation and proves that the cyclic sum, Bowman–Bradley, and weighted sum formulas hold for all t, for both A- and S-types. That is a real extension: at t=0 and t=1 you recover separate known relations for FMZVs and FMZSVs, and the paper's versions genuinely interpolate. The proofs are mostly by reducing coefficient-by-coefficient to known t=0 results, and the combinatorial counts in Lemma 4.5 and Proposition 3.2 are laid out in enough detail for a careful reader to check. I did not find circularity in the sense of assuming the t-statements; the paper is honest about what is new and what is prior.

The soft spot is real but localized. Theorem 1.2 for F=S uses the cyclic sum formula for S-MZVs, attributed to Hirose–Sato (unpublished) or to [7, Thm 2.4], a preprint by two of the present authors. The A-case is backed by the published Kawasaki–Oyama paper. So the paper's headline theorem is not self-contained for half of its range. That should be fixed: either prove the S-cyclic sum in an appendix or wait for publication of [7]. It is not a fatal flaw, but it is exactly the kind of dependency a referee should press.

Two smaller issues. Theorem 5.2 (harmonic relation) is asserted "in a similar way" to t-MZVs, but no proof is given; if it is a direct check, include it. The symmetric sum and antipode-like relations are also stated without proof, though they appear to be quick. The rest of the paper—shuffle, duality, derivation—is properly proved and well cited.

Bottom line: the paper deserves a serious referee. I would condition acceptance on making the S-type dependency public. Once that is done, this is a solid contribution that people working on finite MZVs will want to cite.

Referee Report

2 major / 5 minor

Summary. The paper defines, for each index k, a polynomial ζ^t_F(k) in a variable t that interpolates between the finite multiple zeta value (t=0) and the finite multiple zeta-star value (t=1), for both F=A and F=S. The main results are Theorem 1.2 (cyclic sum formula), Theorem 1.3/3.1 (Bowman–Bradley type vanishing), Theorem 1.5/4.1 (weighted sum formula), and a Section 5 collection of harmonic, shuffle, duality, and derivation relations. The proofs expand coefficients in t and reduce them to already known finite-multiple-zeta relations, with some combinatorial counting arguments in Sections 3 and 4.

Significance. Interpolated finite multiple zeta values are a natural finite analogue of Yamamoto's t-MZVs, and the paper shows that several important relation families (cyclic sum, Bowman–Bradley, weighted sum) hold for the interpolating polynomial rather than only at the endpoints t=0 and t=1. Since ζ^0_F is the finite multiple zeta value and ζ^1_F is the finite multiple zeta-star value, each theorem simultaneously carries the corresponding ordinary and star relations. The algebraic setup in Section 5 is clean and likely to be useful, and the proofs are mostly transparent reductions to published results, with explicit combinatorial counting in Lemma 4.5 and Lemma 4.7. The principal caveat is the S-type cyclic-sum dependence discussed in the major comments.

major comments (2)

[Section 2, Theorem 1.2 (F=S case)] The proof of Theorem 1.2 for F=S is not self-contained. The step ζ_F(F^0(C_m(k)))=0 for S-type finite multiple zeta values is justified only by 'Hirose–Sato (unpublished)' or by [7, Theorem 2.4], where [7] is a preprint of Hirose, Murahara, and Ono, two of the present authors. Because Theorem 1.2 is one of the paper's main theorems, the S-type case rests on an unpublished or non-peer-reviewed source. The A-type case is fine, since it cites the published theorem of Kawasaki–Oyama [20, Theorem 1.2]. Please either include a proof of the S-type cyclic sum identity for F^0(C_m(k)), or restrict the claim for F=S to a conditional statement with a clearly identified published reference once [7] or the Hirose–Sato work is available.
[Section 2, Proposition 2.2] Even the algebraic reduction in Proposition 2.2 uses [7, Lemma 6.3] in an essential way, as do equations (3), (4), (5), and (6). Since [7] is an unpublished preprint by two of the present authors, the main theorem for F=S depends on it twice: once for the coefficient identity and once for the t=0 cyclic-sum relation. The paper should state and prove the needed lemma, or at least give a complete proof of Proposition 2.2 without citing the preprint. This is a load-bearing issue for the central claim, not merely a citation-format concern.

minor comments (5)

[General] The reference list contains several unpublished or in-preparation items ([7], [17], [19], [31], [36]); these should be updated before publication, and the phrase 'Hirose–Sato (unpublished)' in Section 2 should be replaced by a stable reference or a proof in the paper.
[Section 3, equation (8)] In the displayed formula for the third sum in equation (8), the words 'remove ci' and 'remove cj' appear in the printed text. This appears to be a typesetting artifact and should be corrected.
[Section 4, Lemma 4.5] The variable i is overloaded: the proof says 'Choose i (1 ≤ i ≤ a_d)' and then uses i both as the position of the dashed line and as the summation index in formula (13). Please rename one of them to avoid confusion.
[Section 4, proof of Theorem 4.1] In the first displayed equation of the proof, 'ζF(H(k,r,n)' is missing a closing parenthesis; it should read 'ζF(H(k,r,n))'.
[Section 5.3, Theorem 5.13] In the proof of Theorem 5.13, the expression 'S^{-t}(yH_t^x)' should be 'S_{-t}(yH_t^x)' or otherwise disambiguated, since the notation S_t is used elsewhere for the automorphism on H_t.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the t-interpolated FMZV relations are reduced coefficientwise to prior t=0 finite-MZV results, with the t-polynomial identities established by internal combinatorial arguments.

full rationale

The paper's central claims are polynomial identities in t. For Theorem 1.2, Proposition 2.2 shows the coefficient at t^m of F^t(k) is F^0(C_m(k)); the proof then invokes the t=0 cyclic sum formula for FMZVs (Kawasaki–Oyama for A, Hirose–Sato or [7, Thm 2.4] for S). This is a genuine reduction, not a renaming: the coefficient identity is a nontrivial internal statement (Lemma 2.1, Proposition 2.2) and the t=0 input is a separate prior result. The F=S branch depends on an unpublished or self-authored source, but that affects self-containedness and verification, not circularity, because no equation of the present theorem is defined from that source. Theorem 3.1 is proved by induction whose base is Saito–Wakabayashi's external Bowman–Bradley theorem and whose inductive step is an internal counting identity (Prop 3.2). Theorem 4.1 reduces zeta_F(F(k,r,n))=0 to the sum formula, duality, and Ohno-type relation; these are cited prior theorems, and the proof does not fit any parameter to the target. Section 5 relations are obtained by transporting known t=0 relations through the invertible St operator; this is standard algebraic transfer, not circular. No self-definitional, fitted-input, or renaming pattern is present.

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

The paper introduces no fitted parameters and no empirically motivated entities. It relies on the standard algebraic setup for finite multiple zeta values and on a list of previously established FMZ(S)V relations, treated as background. The t-FMZV object is a new definition rather than a hypothesis, so it is not an invented entity in the physics sense.

assumptions (5)

domain assumption The ring A = prod_p F_p / oplus_p F_p is a Q-algebra and the finite multiple zeta values zeta_A and zeta_S are defined as in Kaneko-Zagier.
Used throughout the paper; introduced in Section 1.2.1.
domain assumption Both F=A and F=S satisfy the same baseline relations: sum formula, cyclic sum formula, duality, shuffle, Ohno-type, and derivation relations.
Each relation is cited to prior published work (e.g. [12], [17], [20], [29], [30], [37], [40]); the paper's theorems interpolate these relations.
domain assumption The S-FMZV cyclic sum formula holds, via Hirose-Sato (unpublished) or [7, Theorem 2.4].
Location: Section 2, proof of Theorem 1.2. This is the least independently verified premise because the cited source is unpublished or a preprint by two of the present authors.
standard math The map S_t is an automorphism of H_t with S_t(H1_t)=H1_t and S_t(yHtx)=yHtx, and Z^t_F = Z_F composed with S_t.
Used throughout Section 5; invariance is cited to Li [23] and follows directly from the definitions.
standard math The duality map phi(x)=x+y, phi(y)=-y satisfies Z_F(w)=Z_F(phi(w)) for w in H1.
Used in Proposition 4.4 and Theorem 5.13; cited to Saito [38] as the known duality relation for FMZVs.

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Pith. "Pith review of Yamamoto's interpolation of finite multiple zeta and zeta-star values." pith.science (2026). https://pith.science/paper/QYIL3NIR

@misc{pith2026190809307,
  author       = {Pith},
  title        = {Pith review of: Yamamoto's interpolation of finite multiple zeta and zeta-star values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QYIL3NIR}},
  note         = {Machine review of arXiv:1908.09307}
}

abstract

We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in particular, prove the cyclic sum formula, the Bowman-Bradley type formula, and the weighted sum formula. The harmonic relation, the shuffle relation, the duality relation, and the derivation relation are also presented.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

On variants of symmetric multiple zeta-star values and the cyclic sum formula

M. Hirose, H. Murahara, and M. Ono, On variants of symmetric multiple zeta-star values and the cyclic sum formula , preprint, arXiv:2001.03832

work page Pith review arXiv 2001

[2] [1]

Bachmann, Y

H. Bachmann, Y. Takeyama and K. Tasaka, Cyclotomic analogues of ﬁnite multiple zeta values , Compositio Math. 154 (2018), 2701–2721

work page 2018

[3] [2]

Bachmann and T

H. Bachmann and T. Tanaka, Rooted tree maps and the derivation relation for multiple ze ta values , Int. J. Number Theory 14 (2018), 2657–2662

work page 2018

[4] [3]

Bowman and D

D. Bowman and D. M. Bradley, The algebra and combinatorics of shuﬄes and multiple zeta va lues, J. Combin. Theory Ser. A 97 (2002), 43–61

work page 2002

[5] [4]

Guo and B

L. Guo and B. Xie, Weighted sum formula for multiple zeta values , J. Number Theory 129 (2009), 2747–2765

work page 2009

[6] [5]

Double shuffle relations for refined symmetric multiple zeta values

M. Hirose, Double shuﬄe relations for reﬁned symmetric multiple zeta v alues, preprint, arXiv:1807.04747v2

work page Pith review arXiv

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Hirose, H

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