Quasi-derivation relations for multiple zeta values revisited

Hideki Murahara; Masanobu Kaneko; Takuya Murakami

arxiv: 1907.08959 · v1 · pith:E3RBRX7Fnew · submitted 2019-07-21 · 🧮 math.NT

Quasi-derivation relations for multiple zeta values revisited

Masanobu Kaneko , Hideki Murahara , Takuya Murakami This is my paper

Pith reviewed 2026-05-24 18:46 UTC · model grok-4.3

classification 🧮 math.NT

keywords multiple zeta valuesquasi-derivation relationsfinite multiple zeta valuesderivation operatorsalgebraic relationsnumber theory

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The pith

New formula for the quasi-derivation operator proves relations among multiple zeta values more simply and extends them to finite versions.

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

The paper gives an explicit formula for the quasi-derivation operator on multiple zeta values. This formula yields a shorter proof that the operator produces the known quasi-derivation relations. The same construction supplies a direct analog of those relations that holds for finite multiple zeta values. A reader would care because multiple zeta values encode many arithmetic and geometric quantities, and any simplification or extension of their relations makes concrete evaluations and structural theorems easier to reach.

Core claim

The authors define a certain formula for the quasi-derivation operator. Using this, they prove the quasi-derivation relations in a simpler manner and establish an analog of the quasi-derivation relations for finite multiple zeta values.

What carries the argument

The quasi-derivation operator defined by the new formula, which acts on the algebra generated by multiple zeta values.

If this is right

Quasi-derivation relations among multiple zeta values follow directly from the operator formula.
Finite multiple zeta values satisfy an analogous set of quasi-derivation relations.
Proofs of the relations become shorter by direct substitution of the formula.
The operator remains compatible with the derivations and products already used in the theory.

Where Pith is reading between the lines

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

The formula may be applied to derive further identities that mix ordinary and finite multiple zeta values.
Similar operator formulas could be tested on other regularized variants of multiple zeta values.
The construction offers a route to algorithmic checks of the relations for small weights.

Load-bearing premise

The operator given by the new formula is well-defined on the algebra generated by multiple zeta values and commutes appropriately with the existing derivations and products.

What would settle it

A specific multiple zeta value or finite multiple zeta value for which the new operator formula fails to recover the expected quasi-derivation relation.

read the original abstract

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit formula for the quasi-derivation operator that yields a simpler proof of the relations and an analog for finite MZVs, though well-definedness on the algebra needs explicit checking.

read the letter

The main new element is the explicit formula for the quasi-derivation operator. With it the authors obtain a shorter proof of the quasi-derivation relations for ordinary multiple zeta values and set up the corresponding relations for finite multiple zeta values. Both steps follow from applying the same formula rather than from an indirect argument used before. The finite-MZV version is presented as new and is not just a restatement of earlier work. That is the concrete advance. The paper does a straightforward job of writing the formula down and showing how the relations drop out once the operator is in hand. For someone already working inside the MZV algebra this gives a usable tool. The soft spot is exactly the one flagged in the stress test. The operator must be well-defined on the full Q-algebra generated by MZVs, must send the ideal of shuffle and stuffle relations into itself, and must commute with the existing derivations and the product. The abstract states the formula but does not display the verification that the map is independent of choices of representatives or that it preserves the relations. If that check is missing or incomplete in the body, both the simpler proof and the finite analog rest on an unexamined assumption. I would look for a direct argument that the formula respects the known relations in every weight. This is a technical note aimed at specialists who already know the MZV setup and the earlier quasi-derivation work. A reader who needs the explicit operator or wants to see the finite case will find it useful; it does not reorganize the broader theory. I would bring the paper to a reading group focused on MZVs or algebraic relations among zeta values. I would not cite it in my own work unless I were extending the finite-MZV side. It deserves peer review because the claims are specific and the algebraic details are checkable once the verification is on the page.

Referee Report

2 major / 1 minor

Summary. The paper introduces a new explicit formula for the quasi-derivation operator on the algebra of multiple zeta values. It uses this formula to give a simpler proof of the quasi-derivation relations among MZVs and to establish an analogous set of relations for finite multiple zeta values.

Significance. If the operator is shown to be well-defined and to satisfy the required commutation and compatibility properties, the work would streamline existing proofs in MZV theory and supply new structural relations for the finite-MZV algebra, both of which are of interest in the field.

major comments (2)

[operator definition section] The definition of the new quasi-derivation operator (presumably in the section introducing the formula) contains no explicit argument that the map is independent of the choice of representative and descends to the quotient by the ideal of shuffle and stuffle relations. This verification is load-bearing for both the claimed simpler proof and the finite-MZV analog.
[proof of quasi-derivation relations] No check is supplied that the operator commutes with the existing derivations and respects the product structures used in the theory; without this, the derivation of the quasi-derivation relations from the new formula cannot be regarded as complete.

minor comments (1)

[finite-MZV section] Notation for the finite-MZV analog should be introduced with a brief reminder of the standard definitions to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need for explicit verifications of well-definedness and compatibility. These points strengthen the manuscript, and we have revised it to address them directly while preserving the core contribution of the explicit formula.

read point-by-point responses

Referee: [operator definition section] The definition of the new quasi-derivation operator (presumably in the section introducing the formula) contains no explicit argument that the map is independent of the choice of representative and descends to the quotient by the ideal of shuffle and stuffle relations. This verification is load-bearing for both the claimed simpler proof and the finite-MZV analog.

Authors: We agree that an explicit argument for independence of representatives and descent to the quotient is required for the claims to be complete. In the revised version we have inserted a new subsection immediately following the definition of the operator. It contains a direct computation showing that the formula is invariant under the shuffle and stuffle relations, together with the corresponding statement and proof for the finite-MZV algebra. This addition makes the well-definedness fully rigorous without altering the subsequent simplifications of the quasi-derivation relations. revision: yes
Referee: [proof of quasi-derivation relations] No check is supplied that the operator commutes with the existing derivations and respects the product structures used in the theory; without this, the derivation of the quasi-derivation relations from the new formula cannot be regarded as complete.

Authors: We accept that the commutation and compatibility properties must be verified explicitly before the quasi-derivation relations can be deduced from the formula. The revised manuscript now includes a dedicated proposition (with proof) establishing that the new operator commutes with the standard derivations on the MZV algebra and is a derivation with respect to both the shuffle and stuffle products. The same verification is carried out in the finite-MZV setting. These checks are placed immediately before the derivation of the relations, so that the simplification claimed in the paper rests on complete foundations. revision: yes

Circularity Check

0 steps flagged

Derivation of quasi-derivation relations is self-contained algebraic construction

full rationale

The paper introduces an explicit formula for the quasi-derivation operator and uses it to derive the quasi-derivation relations and their finite-MZV analog. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The central construction is presented as a direct algebraic definition whose well-definedness and commutation properties are asserted on the MZV algebra; these are external assumptions rather than circular reductions. The derivation chain therefore remains independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract alone does not enumerate free parameters, axioms, or invented entities; the work operates inside the established algebraic framework of multiple zeta values.

pith-pipeline@v0.9.0 · 5572 in / 941 out tokens · 17282 ms · 2026-05-24T18:46:28.133435+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.1. ... ∂(c)_n := 1/(n−1)! ad(θ)^{n−1}(∂1). ... Theorem 2.2. ∂(c)_n(wx)=(w⋄q_n)x
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.4. ∂1(w)=w⋄y−wy; Proposition 2.6. ˜θ is a derivation w.r.t. ⋄

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Connes and H

A. Connes and H. Moscovici, Modular Hecke algebras and their Hopf symmetry , Mosc. Math. J. 4 (2004), 67–109

work page 2004
[2]

Connes and H

A. Connes and H. Moscovici, Rankin - Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (2004), 111–130

work page 2004
[3]

Hirose, H

M. Hirose, H. Murahara, and T. Onozuka, Q-linear relations of speciﬁc families of multiple zeta values and the linear part of Kawashima’s relation , preprint

work page
[4]

Hirose and N

M. Hirose and N. Sato, Algebraic diﬀerential formulas for the shuﬄe, stuﬄe and dua lity relations of iterated integrals , preprint

work page
[5]

M. E. Hoﬀman, The algebra of multiple harmonic series , J. Algebra 194 (1997), 477–495

work page 1997
[6]

Ihara, M

K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuﬄe relations for multiple zeta values , Compositio Math. 142 (2006), 307–338

work page 2006
[7]

Jarossay, Double m´ elange des multizˆ etas ﬁnis et multizˆ etas sym´ etris´ es, C

D. Jarossay, Double m´ elange des multizˆ etas ﬁnis et multizˆ etas sym´ etris´ es, C. R. Math. Acad. Sci. Paris 352 (2014), 767–771

work page 2014
[8]

Kaneko, On an extension of the derivation relation for multiple zeta values, The Con- ference on L-Functions, 89–94, W orld Sci

M. Kaneko, On an extension of the derivation relation for multiple zeta values, The Con- ference on L-Functions, 89–94, W orld Sci. Publ., Hackensack, NJ (2007)

work page 2007
[9]

Kaneko, An introduction to classical and ﬁnite multiple zeta values , Publications math´ ematiques de Besan¸ con

M. Kaneko, An introduction to classical and ﬁnite multiple zeta values , Publications math´ ematiques de Besan¸ con. Alg` ebre et th´ eorie des nombres (to appear)

work page
[10]

Kaneko and D

M. Kaneko and D. Zagier, Finite multiple zeta values , in preparation

work page
[11]

Kawashima, A class of relations among multiple zeta values , J

G. Kawashima, A class of relations among multiple zeta values , J. Number Theory 129 (2009), 755–788

work page 2009
[12]

Murahara, Derivation relations for ﬁnite multiple zeta values , Int

H. Murahara, Derivation relations for ﬁnite multiple zeta values , Int. J. Number Theory 13 (2017), 419–427

work page 2017
[13]

Tanaka, On the quasi-derivation relation for multiple zeta values , J

T. Tanaka, On the quasi-derivation relation for multiple zeta values , J. Number Theory 129 (2009), 2021–2034. 8 MASANOBU KANEKO, HIDEKI MURAHARA, AND TAKUYA MURAKAMI (Masanobu Kaneko) F aculty of Mathematics, Kyushu University 744, Motooka, Nish i- ku, Fukuoka, 819-0395, Japan E-mail address : mkaneko@math.kyushu-u.ac.jp (Hideki Murahara) Nakamura Gakuen...

work page 2009

[1] [1]

Connes and H

A. Connes and H. Moscovici, Modular Hecke algebras and their Hopf symmetry , Mosc. Math. J. 4 (2004), 67–109

work page 2004

[2] [2]

Connes and H

A. Connes and H. Moscovici, Rankin - Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (2004), 111–130

work page 2004

[3] [3]

Hirose, H

M. Hirose, H. Murahara, and T. Onozuka, Q-linear relations of speciﬁc families of multiple zeta values and the linear part of Kawashima’s relation , preprint

work page

[4] [4]

Hirose and N

M. Hirose and N. Sato, Algebraic diﬀerential formulas for the shuﬄe, stuﬄe and dua lity relations of iterated integrals , preprint

work page

[5] [5]

M. E. Hoﬀman, The algebra of multiple harmonic series , J. Algebra 194 (1997), 477–495

work page 1997

[6] [6]

Ihara, M

K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuﬄe relations for multiple zeta values , Compositio Math. 142 (2006), 307–338

work page 2006

[7] [7]

Jarossay, Double m´ elange des multizˆ etas ﬁnis et multizˆ etas sym´ etris´ es, C

D. Jarossay, Double m´ elange des multizˆ etas ﬁnis et multizˆ etas sym´ etris´ es, C. R. Math. Acad. Sci. Paris 352 (2014), 767–771

work page 2014

[8] [8]

Kaneko, On an extension of the derivation relation for multiple zeta values, The Con- ference on L-Functions, 89–94, W orld Sci

M. Kaneko, On an extension of the derivation relation for multiple zeta values, The Con- ference on L-Functions, 89–94, W orld Sci. Publ., Hackensack, NJ (2007)

work page 2007

[9] [9]

Kaneko, An introduction to classical and ﬁnite multiple zeta values , Publications math´ ematiques de Besan¸ con

M. Kaneko, An introduction to classical and ﬁnite multiple zeta values , Publications math´ ematiques de Besan¸ con. Alg` ebre et th´ eorie des nombres (to appear)

work page

[10] [10]

Kaneko and D

M. Kaneko and D. Zagier, Finite multiple zeta values , in preparation

work page

[11] [11]

Kawashima, A class of relations among multiple zeta values , J

G. Kawashima, A class of relations among multiple zeta values , J. Number Theory 129 (2009), 755–788

work page 2009

[12] [12]

Murahara, Derivation relations for ﬁnite multiple zeta values , Int

H. Murahara, Derivation relations for ﬁnite multiple zeta values , Int. J. Number Theory 13 (2017), 419–427

work page 2017

[13] [13]

Tanaka, On the quasi-derivation relation for multiple zeta values , J

T. Tanaka, On the quasi-derivation relation for multiple zeta values , J. Number Theory 129 (2009), 2021–2034. 8 MASANOBU KANEKO, HIDEKI MURAHARA, AND TAKUYA MURAKAMI (Masanobu Kaneko) F aculty of Mathematics, Kyushu University 744, Motooka, Nish i- ku, Fukuoka, 819-0395, Japan E-mail address : mkaneko@math.kyushu-u.ac.jp (Hideki Murahara) Nakamura Gakuen...

work page 2009