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An Explicit Expression for MZVs in Terms of Symmetric MZVs

T0 review · 3 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Symmetric zeta values generate all multiple zeta values, new proof shows

desk verdict Simpler proof of Yasuda's theorem that SMZVs generate MZVs, plus new depth-3 structural results and an explicit algorithm. The proof structure is sound; the main risk is in elided generating-function calculations, not the duality dependence. read the letter →

arxiv 2607.05795 v1 pith:2V7LDAPH submitted 2026-07-07 math.NT

Katsumi Kina This is my paper · ORCID
classification math.NT
keywords multiplevalueszetasymmetricmzvstermsalgorithmargument
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 proves that every multiple zeta value — a number built from nested infinite sums of fractions — can be written as a linear combination of symmetric multiple zeta values (SMZVs) and products of lower-weight MZVs. SMZVs are a special class of zeta values introduced by Kaneko and Zagier, defined by a symmetrization procedure that makes them well-behaved modulo ζ(2). The author gives a simpler proof than the original one by Yasuda, replacing Yasuda's auxiliary real numbers with the coefficients ζ₂(k) that arise naturally from multitangent functions — higher-depth analogues of the cotangent function. The key link is a theorem of Hirose connecting these multitangent coefficients to SMZVs via Hoffman duality. The proof proceeds by showing that the generating function for regularized MZVs is cyclically invariant modulo SMZVs and products, then applying an algebraic lemma to extract vanishing derivatives, which forces every MZV into the span of SMZVs. Beyond the proof, the paper provides an explicit algorithm for computing such decompositions (worked out for depths one and two), and studies the structure of the space spanned by depth-three SMZVs and their finite analogues, proving that this space equals the space of depth-two values for even weights.

What carries the argument

Multitangent functions Ψ_k(z) and their ζ₂(k) coefficients; Hoffman duality connecting multitangent coefficients to SMZVs (Hirose's theorem); generating functions for regularized MZVs and their cyclic invariance modulo SMZVs and products; a differential-algebraic lemma (Lemma 2.6) extracting vanishing partial derivatives from homogeneous polynomial identities

What would settle it

If a weight k MZV were found that cannot be expressed as a Q-linear combination of ∗-SMZVs of weight k, the main theorem would fail. Concretely, the algorithm in Section 3, when run on such an MZV, would produce an inconsistent system.

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Extended reading notes

Core claim

The space of all multiple zeta values of any given weight k is exactly the space spanned by the ∗-symmetric multiple zeta values of that weight. The proof replaces Yasuda's ζ^{♮,F}(k) numbers with the ζ₂(k) coefficients of multitangent functions, connected to SMZVs via Hirose's duality theorem. The argument works through generating function identities: cyclic invariance of the MZV generating function modulo SMZVs and products, combined with a differential lemma, forces all MZVs into the SMZV span. The paper also gives an algorithmic procedure for explicit decomposition and proves that for even weights, the space of triple SMZVs (and their finite analogues) coincides with the space of depth-2

Load-bearing premise

The proof relies on duality relations among MZVs (via Hirose's theorem connecting multitangent coefficients to SMZVs), whereas Yasuda's original proof used only extended double shuffle relations. If duality does not follow from double shuffle in a given formal setting, the proof does not go through there.

Editorial extensions

If this is right

  • Any multiple zeta value can be algorithmically decomposed into symmetric MZVs and products, making the SMZV basis computationally accessible for low-depth cases.
  • The identification of the triple-SMZV space with the depth-2 space for even weights gives a concrete structural result that can be checked against dimension conjectures involving cusp forms.
  • The algorithm produces explicit formulas with controlled coefficients (in 1/k·Z) and depth structure, which could serve as a tool for numerical experimentation and relation-hunting among MZVs.
  • The partial equivalence of two definitions of Z_A(r,s) (Theorem A.1) advances the Kaneko–Zagier program of establishing a full isomorphism between finite MZVs and SMZVs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This paper provides a simpler proof of Yasuda's theorem (Theorem 1.3) that the space of multiple zeta values (MZVs) of weight k is generated by *-symmetric MZVs (SMZVs). The proof replaces Yasuda's auxiliary real numbers with the coefficients of multitangent functions, leveraging Hirose's result (Theorem 2.3) connecting SMZVs to these coefficients via Hoffman duality. The core argument (Theorem 2.4 implies Theorem 1.3 via weight induction) proceeds through generating function manipulations in Section 2.2. Based on this proof, Section 3 provides an algorithm for expressing MZVs in terms of SMZVs, explicitly carried out for depths 1 and 2. Sections 4 and 5 give results on depth-2 and depth-3 SMZVs and finite MZVs (FMZVs), including a formula for Z_S(k1, k2) and a proof that the space of triple SMZVs equals the space of depth-2 SMZVs for even weight. Appendix A partially addresses the equivalence of two definitions of Z_A(r,s).

Significance. The paper provides a genuine simplification of Yasuda's proof, replacing the technically involved construction of real numbers with the more structured framework of multitangent functions. The explicit algorithm in Section 3, along with the worked examples for depths 1 and 2, adds practical value. The results in Sections 4 and 5 on the structure of triple SMZV/FMZV spaces, particularly Theorem 5.7 and the depth-2 formula in Theorem 4.3, are substantive contributions. The partial result in Appendix A on the equivalence of definitions of Z_A(r,s) is a useful incremental step. The author honestly acknowledges the limitation regarding duality dependence in the Remark after Lemma 2.6.

major comments (3)
  1. §2.2, the identity for ∂²A₂/∂X₁∂Xᵣ: This identity is stated as following from 'a straightforward calculation together with (1.2) and (2.1),' but it is the sole bridge between the generating-function framework and Theorem 2.3 (Hirose's result). The calculation involves expanding A₂ (defined via a sum of products of Z-functions), matching against the explicit formula (2.1) for ζ₂, and tracking signs through stuffle regularization. Given its load-bearing role, this calculation should be expanded or verified in an appendix. The skeptic's note correctly identifies this as the primary correctness risk.
  2. §2.2, the identity A₁(X₁,…,Xᵣ) = 0: This is stated to follow from the shuffle-antipode relation and 'a straightforward calculation.' Since A₁ = 0 is used to derive equation (2.3), which is essential for the proof of Theorem 2.5, the elision here is also load-bearing. Providing the intermediate steps would strengthen the verification.
  3. §5, Proposition 5.4: In the proof, equation (5.6) is derived from Lemma 4.1, but the sign convention and the factor (1 - δ_{r,a}) need careful verification. The transition from (5.6) to (5.7) involves a sum over r from b-1 to k-2, and the subsequent manipulation using Bernoulli number identities is intricate. A minor error in the binomial coefficient tracking could propagate into the formulas (5.4) and (5.5), which are used in Theorem 5.7.
minor comments (7)
  1. §1, p.1: The notation ζ^X_S(k) in (1.2) uses a superscript X, but later the text refers to 'X-symmetric multiple zeta value ζ^X_S(k)'. Consistency in superscript placement would improve readability.
  2. §2.2, p.4: The generating function Z_S(X₁,…,Xᵣ) is defined with a sum from i=0 to r, but the terms Z(X₁,…,Xᵢ) and Z(-Xᵣ,…,-Xᵢ₊₁) need clearer specification of what happens at the boundary cases i=0 and i=r (e.g., Z() = 1?).
  3. §3.1, p.7: In the derivation of (3.1), the step from the generating function expansion to the coefficient comparison could benefit from one intermediate line showing the binomial expansion explicitly.
  4. §4, p.10: In Lemma 4.1, the sum over i from 1 to k₂-1 uses binomial coefficients, but the case k₂=1 (where the sum is empty) should be explicitly noted to confirm the 'in particular' statement.
  5. §5, p.13: In Proposition 5.4, the Bernoulli number B_{r-b+1} appears with r ranging from b-1, so B_0 = 1 is used. This should be stated for clarity.
  6. Appendix A, p.17: The definition of ėZ_A(a,b) uses Z'_A(k-1), which involves B_{2p-1-k}. The range of k for which this is well-defined (given the Bernoulli number vanishing properties) should be specified.
  7. References: Reference [6] (Hirose) is listed as a 2024 arXiv preprint. If published by the time of acceptance, the reference should be updated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. All three major comments request expanded verification of calculations that were elided as 'straightforward.' We agree with all three requests and will add the missing details in the revised manuscript.

read point-by-point responses

  1. Referee: §2.2, the identity for ∂²A₂/∂X₁∂Xᵣ: This identity is stated as following from 'a straightforward calculation together with (1.2) and (2.1),' but it is the sole bridge between the generating-function framework and Theorem 2.3 (Hirose's result). The calculation should be expanded or verified in an appendix.

    Authors: The referee is correct that this identity is load-bearing and that the elision is not appropriate given its role in the proof. We will add a detailed verification as an appendix (or as a subsection of Section 2) in the revised manuscript. The calculation proceeds by expanding the definition of $A_2$ as a sum of products of $Z$-functions, substituting the explicit formula (2.1) for $ζ_2(k)$, and matching coefficients using the definition (1.2) of $ζ_S^X$. The key steps are: (i) expanding each $Z$-factor in $A_2$ as a power series, (ii) collecting terms of bidegree $(k_1-1, k_r-1)$ after taking $∂^2/∂X_1∂X_r$, and (iii) identifying the resulting coefficient with $k_1 k_r ζ_2(k_1+1, k_2, …, k_{r-1}, k_r+1)$ via (2.1), tracking signs through the stuffle regularization. We agree that presenting these steps explicitly will strengthen the paper and allow independent verification. revision: yes

  2. Referee: §2.2, the identity A₁(X₁,…,Xᵣ) = 0: This is stated to follow from the shuffle-antipode relation and 'a straightforward calculation.' Since A₁ = 0 is used to derive equation (2.3), providing the intermediate steps would strengthen the verification.

    Authors: We agree. The vanishing $A_1 = 0$ follows from the shuffle-antipode relation (Proposition 3.3 in [1]) by expanding $A_1$ and matching the resulting sum against the antipode identity. Specifically, expanding $A_1(X_1, …, X_r)$ and collecting the coefficient of $X_1^{k_1-1} ⋯ X_r^{k_r-1}$ yields exactly the left-hand side of the shuffle-antipode relation as displayed in the manuscript. We will include these intermediate steps in the revised version, making the connection to the antipode relation explicit rather than leaving it as a 'straightforward calculation.' revision: yes

  3. Referee: §5, Proposition 5.4: In the proof, equation (5.6) is derived from Lemma 4.1, but the sign convention and the factor (1 - δ_{r,a}) need careful verification. The transition from (5.6) to (5.7) involves a sum over r from b-1 to k-2, and the subsequent manipulation using Bernoulli number identities is intricate. A minor error in the binomial coefficient tracking could propagate into the formulas (5.4) and (5.5), which are used in Theorem 5.7.

    Authors: We appreciate the referee's careful attention to the sign conventions and binomial coefficient tracking in this proof. We have re-examined the derivation and confirm that the sign $(-1)^r$ and the factor $(1 - δ_{r,a})$ in (5.6) are correct: the sign comes from the $(-1)^{k_1+k_3}$ factor in Lemma 4.1 with $k_1 = r$, $k_3 = 1$, and the Kronecker delta excludes the trivial term $a = r$ where the binomial coefficient $¥binom{a}{r}$ would give a degenerate contribution. The transition from (5.6) to (5.7) involves multiplying both sides of (5.6) by $¥frac{1}{k-b}¥binom{a}{b}(k-a)B_{r-b+1}$ and summing over $r$ from $b-1$ to $a-1$, then swapping the order of summation. We will add these intermediate steps explicitly in the revised manuscript, including the justification for the summation range change and the application of the Bernoulli number identity (5.9), so that the binomial coefficient tracking can be verified step by step. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the proof chain rests on independently derived external results and the paper's own algebraic manipulations are non-circular.

full rationale

The paper proves Theorem 1.3 (Z_k = Z_{S,*}^k) via Theorem 2.4 → Theorem 2.5, using three load-bearing external results: Theorem 2.1 (Kaneko–Zagier [12]), Theorem 2.2 (Bouillot [3]), and Theorem 2.3 (Hirose [6]). None of these are self-citations by the author Kina. The key bridge identity ∂²A₂/∂X₁∂Xᵣ = Σ k₁kᵣ ζ₂(...)X... is derived from the definitions of A₂ and Z together with Bouillot's explicit formula (2.1), and then Theorem 2.3 (Hirose) is applied to connect ζ₂-coefficients to SMZVs. The shuffle-antipode relation (A₁ = 0) is cited from Bachmann [1]. Lemma 2.6 is a purely algebraic lemma (cf. Yasuda [17]) with a self-contained proof. The subsequent algebraic manipulations (2.2)–(2.12) are variable substitutions and additions that the reader's skeptic analysis independently verified. The definition of Z_A(r,s) in (5.13) is defined by direct analogy with the proven formula of Theorem 4.3 for Z_S(k₁,k₂); this is a definition of a new object for FMZVs, not a circular derivation of the target result. Theorem A.1 then proves a non-trivial property of this defined object. No step reduces to its own inputs by construction, no prediction is a renamed fit, and no self-citation chain is load-bearing.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced; all formulas are parameter-free. No invented entities. The axioms are standard domain assumptions in the theory of multiple zeta values (EDSR, duality) and cited results (Bouillot, Hirose). The reliance on duality relations beyond EDSR is the key structural assumption distinguishing this proof from Yasuda's.

assumptions (4)
  • domain assumption Extended double shuffle relations (EDSR) hold for MZVs
    Used implicitly throughout the generating function manipulations; the shuffle-antipode relation (Proposition 3.3 in [1]) is used to show A1=0 in Section 2.2.
  • domain assumption Hoffman duality relations hold for MZVs
    Theorem 2.3 (Hirose [6]), which connects ζ_X^S((↓k↓)∨) to ζ₂(k), depends on duality. This is the key difference from Yasuda's proof, as noted in the Remark after Lemma 2.6.
  • standard math Bouillot's decomposition of multitangent functions (Theorem 2.2)
    Provides the expression for ζ_a(k) coefficients used to connect A2 to Z∨ via Theorem 2.3.
  • domain assumption Kaneko-Zagier conjecture on FMZV-SMZV correspondence (Conjecture 1.2)
    Motivates the study of SMZVs and FMZVs in parallel; the paper verifies that structural results transfer between S and A settings.

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Pith. "Pith review of An Explicit Expression for MZVs in Terms of Symmetric MZVs." pith.science (2026). https://pith.science/paper/2V7LDAPH

@misc{pith2026260705795,
  author       = {Pith},
  title        = {Pith review of: An Explicit Expression for MZVs in Terms of Symmetric MZVs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2V7LDAPH}},
  note         = {Machine review of arXiv:2607.05795}
}
read the original abstract

We provide a simpler proof of the fact, originally proved by Seidai Yasuda, that symmetric multiple zeta values generate the entire space of multiple zeta values. Furthermore, based on this argument, we present an algorithm for expressing multiple zeta values in terms of symmetric multiple zeta values and products of multiple zeta values. Moreover, we give some results on symmetric and finite multiple zeta values of depth three.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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