Pith. sign in

REVIEW 2 major objections 5 minor 20 references

Multiple Zeta Values

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read These lecture notes propose that ordinary, finite, symmetric, and q-analogue multiple zeta values, together with multiple Eisenstein series, are one quasi-shuffle algebra viewed through different limits and specializations.

desk verdict A solid, useful survey of MZVs and q-analogues; no new theorems, and its central framework leans on an announced proof that isn't in the text, so treat it as expository material with one missing load-bearing brick. read the letter →

arxiv 2608.02230 v1 pith:MKBEKD4U submitted 2026-08-03 math.NT math.CO

Henrik Bachmann This is my paper · ORCID
classification math.NTmath.CO MSC 11M3211F0311F11
keywords multiplezetavaluesfinitesymmetricq-analoguesofEisensteinseriesquasi-shufflealgebrasKaneko-Zagierconjecturemodularforms
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 notes argue that five families—ordinary, finite, symmetric, and q-analogue multiple zeta values, plus multiple Eisenstein series—are not separate objects but realizations of a single quasi-shuffle algebra. The differences between them come from which regularization and which limit is taken: q tending to 1, reduction modulo primes, or symmetrization. The load-bearing bridge is the Kaneko–Zagier conjecture, which says finite and symmetric multiple zeta values satisfy exactly the same rational linear relations. A sympathetic reader should care because, if this picture holds, the algebraic structure can be worked out once in a universal word algebra and then transferred to every family. Much of the common framework is explicitly conjectural, and the notes flag the unproved pieces carefully.

What carries the argument

The central object is the quasi-shuffle algebra: a word algebra whose multiplication recursively interleaves two words and combines their first letters using a commutative product. Choosing the trivial product gives the shuffle product; choosing the product that merges indices gives the stuffle product; choosing a modified rational-coefficient product gives the multiplication for the q-series g. Regularization, expressed as the isomorphism between words and polynomials in a variable T, extends the value map from admissible to all indices and produces the extended double shuffle relations. The q-shuffle product with a formal parameter and the involutive map sigma encode the q-analogue relatio

What would settle it

Compute the full space of Q-linear relations among finite multiple zeta values of the form ζ_A(1,2a,1,2b) in a fixed even weight such as 12 or 16 and compare it with the corresponding space for symmetric multiple zeta values modulo π²; a relation holding in one family but not the other refutes the Kaneko–Zagier isomorphism. For the multiple Eisenstein series part, check the conjectured dimension formula at small weights: if dim_Q E_k deviates from the table given after Conjecture 1.60 for any k, the announced proof cannot be correct as stated.

Watch

Extended reading notes

Core claim

The notes' central claim is that the objects in the title are not a loose collection of analogues but several realizations of the same algebraic structure. In each family, the space of values is the image of an algebra homomorphism from a quasi-shuffle algebra of words; ordinary multiple zeta values and their finite or symmetric counterparts differ only in the chosen regularization and limit. The load-bearing bridge is the Kaneko–Zagier map from finite to symmetric multiple zeta values modulo pi-squared, conjectured to be an isomorphism of Q-algebras and therefore to make finite and symmetric values obey identical relations. Alongside this, multiple Eisenstein series are presented as the ana

Load-bearing premise

The unified picture rests on two unproved pillars: the Kaneko–Zagier isomorphism between finite and symmetric multiple zeta values, and an announced proof of the multiple Eisenstein series structure that is still in preparation; if either fails, the common-framework narrative in the later chapters needs substantial revision.

Editorial extensions

If this is right

  • A relation proved once in the universal quasi-shuffle word algebra descends to ordinary, finite, symmetric, and q-analogue multiple zeta values alike.
  • If the Kaneko–Zagier isomorphism holds, finite and symmetric multiple zeta values have identical linear-relation spaces, and the dimension of the weight-k finite space is predicted to be d_{k-3}.
  • If the announced proof of the multiple Eisenstein series conjectures is correct, multiple Eisenstein series form a weight-graded Q-algebra with an sl2-action, making the dictionary between constant terms, Fourier coefficients, and q-limits an algebra homomorphism rather than a numerical accident.
  • The BTT picture implies that a Q-linear relation among ordinary multiple zeta values with analytic limits gives simultaneous finite and symmetric relations of the same shape, offering an explanatory mechanism for the Kaneko–Zagier conjecture.
  • Explicit depth-two identities such as G6 + 3G4,2 − 6G3,3 = 0 show that modular-form relations are shadows of identities among multiple Eisenstein series and q-analogues, so modular forms enter the framework as a structural feature rather than a coincidence.

Reading between the lines

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

  • If the framework is right, a natural next step is to turn the universal algebra into a proof machine: counting the extended double shuffle relations, the paper's Open Problem 2.41, should recover the conjectural dimensions d_k and give an independent route to the known upper bounds.
  • The depth-two appearance of cusp forms suggests a concrete test: the period-polynomial construction for double zeta values should have a finite-multiple-zeta-value analogue in depth four, with one independent relation per cusp form of the relevant weight; the paper points toward this in its later sections.
  • The Q-values at roots of unity, read through the BTT diagram, suggest that q-multiple-zeta identities at p-th roots of unity may package p-adic congruences and real-analytic limits into one statement, potentially proving finite and symmetric relation families simultaneously.
  • Schur multiple zeta values, though introduced separately, appear as constant terms of Schur multiple Eisenstein series; the quasi-shuffle framework suggests they too admit the same regularized double-shuffle treatment, likely through the same word algebra with a different alphabet.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript is a set of lecture notes (version 27) surveying multiple zeta values and their main variants: finite and symmetric MZVs, q-analogues, and multiple Eisenstein series. The stated goal is to exhibit a common quasi-shuffle algebraic framework for these objects. The main structural bridges are the Kaneko–Zagier conjecture (Conjecture 1.39), the algebraic setup of quasi-shuffle algebras and double shuffle/regularization (Chapter 2), and the conjectural sl2-algebra structure of multiple Eisenstein series (Conjectures 1.54 and 1.55). Many statements are classical or are labeled as conjectures or exercises; the notes do not claim to prove the central open conjectures.

Significance. If the announced proof and the conjectures are correct, the notes would be a genuinely useful unified reference: they collect explicit formulas, including the Fourier expansion of multiple Eisenstein series (Theorems 1.51 and 1.58), the quasi-shuffle formalism for q-analogues, the BTT philosophy, and a large set of exercises and pointers to the literature. The expository value is real, and the notes generally distinguish theorems from conjectures. Their main weakness is epistemic rather than mathematical: a central structural assertion is announced as proved in an unpublished paper, and that assertion is load-bearing for the presentation of the multiple Eisenstein series framework.

major comments (2)
  1. [§1.5, Remark 1.56] The text states that the author 'found a proof' of Conjectures 1.54 and 1.55, but the proof is relegated to [Ba13], which is 'currently in preparation'. These conjectures are load-bearing: Conjecture 1.54 asserts gradedness of E, Conjecture 1.55 defines the operators W and δ and asserts an sl2-triple, and Chapter 7 and Conjecture 1.60 rely on this structure. Since the proof is not available to the reader, the claim 'found a proof' is currently uncheckable and cannot support the surrounding structural statements. The notes should either include the proof, make the dependence on [Ba13] explicit at every point where E, W, or δ is used as an established object, or revert these statements to purely conditional form.
  2. [§1.5 / §6–7] The exposition after Remark 1.56 can easily mislead: Theorem 6.55 establishes the derivative D, but the well-definedness of W and δ, the sl2-triple relations, and the gradedness of E remain parts of Conjectures 1.54 and 1.55. Sections 6 and 7 nevertheless present the sl2-structure of the full space E as part of the common framework. Please add a standing caveat at the beginning of Chapters 6 and 7 stating precisely which statements are proved and which depend on the unpublished [Ba13] or on the Kaneko–Zagier conjecture, so that a reader does not mistake conditional constructions for established theorems.
minor comments (5)
  1. [§2.3.2] The letters z_k in the z-alphabet are denoted with the same symbol as the words z_k = x^{k-1}y in H. This overloading is a frequent source of confusion, especially in Example 2.34; a distinct notation (e.g., u_k or a bold z) would improve readability.
  2. [§1.1, Theorem 1.3(ii)] The reference [Fi] is cited as 'Fischler, 2026'. If this is a preprint or forthcoming work, please give the arXiv number or another verifiable pointer, since the reader cannot otherwise locate the claimed improvement.
  3. [§1.6.4, Figure 1.1] The BTT diagram is heuristic and is used to motivate the Kaneko–Zagier conjecture. The arrows are not maps of the same kind; the text already says this, but labeling the diagram with 'analytic limit' and 'algebraic reduction' at the relevant arrows would make the intended meaning clearer.
  4. [§1.5, Conjecture 1.60] The dimension table for E is presented as conjectural, but the derivation of the expected dimensions and relation counts is only indicated by references to [Ok], [BK2], and [Ba3]. Please state more explicitly which ingredients are rigorous, which are numerical, and which depend on the unresolved Conjectures 1.54/1.55.
  5. [§2.7, Proposition 2.67] The resummation duality is stated with a citation to [Tak1]. For a self-contained set of notes, a sketch of the proof or a precise statement of the algebraic mechanism would be helpful, since Proposition 2.69 is used later in the section.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the notes are a survey; central statements are either proved in-text, cited to published work, or explicitly labeled as conjectures.

full rationale

The notes do not derive a target from an input that already contains it. The quasi-shuffle framework in Section 2.3 is set up abstractly (letters, product ⋄) and each family—MZVs, finite MZVs, q-series g, generalized qMZVs—is then verified to be an example (Propositions 1.66, 2.12, 2.20, Lemma 2.18); the framework is not defined in terms of the values it is claimed to organize. The finite/symmetric bridge is stated as Conjecture 1.39, not proved or fitted. The q→1 root-of-unity limits (Theorems 1.77/1.78) are results about limits, not a fit renamed as prediction. The only passage resembling a self-referential support is Remark 1.56, which announces a proof of Conjectures 1.54/1.55 in the author's in-preparation [Ba13]; however the text explicitly 'keep[s] both statements as conjectures for now', and subsequent use of D is based on the published derivative theorem [BKM] while W and δ are presented as conjectural. A promised proof can be a reliability concern, but it is not a circular reduction: the notes' equations do not take [Ba13] as an input and output it as a conclusion. No step reduces to its own definition or to a fitted parameter, so the circularity burden is not met.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The notes introduce many formal algebraic objects (formal multiple zeta values, formal double Eisenstein spaces, combinatorial multiple Eisenstein series, Q-multiple zeta values), but these are explicitly constructed mathematical definitions rather than unexplained postulated entities, and their conjectured properties are clearly flagged. The free-parameter count is zero: no numbers are fitted to data. The main epistemic burden is carried by open conjectures and by external theorems cited as black boxes.

assumptions (6)
  • domain assumption Conjecture 1.18 (Zagier): dim_Q Z_k = d_k with generating series 1/(1-X^2-X^3)
    Used throughout as the expected dimension law for MZVs and as a comparison benchmark for the multiple Eisenstein series dimension conjecture. Only the upper bound dim_Q Z_k ≤ d_k is known via Terasoma and Deligne–Goncharov.
  • domain assumption Conjecture 1.39 (Kaneko–Zagier): φ_KZ : Z_A → Z/π²Z is an isomorphism of Q-algebras
    Core bridge between finite and symmetric multiple zeta values, motivating definitions and conjectures in Sections 1.3, 3.3, and 7.5. It is unproved.
  • domain assumption Conjectures 1.16, 2.40, 2.45: Z is graded by weight; extended double shuffle relations generate all relations; Hoffman relations plus finite double shuffle relations suffice
    These conjectures frame the relation-counting tables and the claim that all linear relations among MZVs have a common algebraic source. They are not proved; Conjecture 2.40 is reported as checked only up to weight 21.
  • ad hoc to paper Conjectures 1.54 and 1.55: E is graded by weight and (D,W,δ) is an sl2-triple, with proof announced in unpublished [Ba13]
    Remark 1.56 explicitly says the proof will appear in a paper in preparation; the notes keep the statements as conjectures. The derivative part is stated to be proved in [BKM], but the full structure is not independently verifiable from this manuscript.
  • standard math External deep theorems: Brown (Theorem 1.22), Terasoma/Deligne–Goncharov (Theorem 1.20), Apéry/Fischler/Zudilin/Lai–Zhou (Theorem 1.3), and the q-series results of Hirose–Maesaka–Watanabe (Theorems 1.75, 1.76)
    Relied on as black boxes for dimension bounds, generation by Hoffman elements, irrationality results, and equality of q-analogue spaces. They are cited but not proved in the notes.
  • domain assumption Conjecture 1.46 (Kaneko–Zagier–Risan): dimension and spanning structure of depth-4 finite MZV spaces
    Presented as a conjecture based on numerical experiments by Kaneko, Zagier, and Risan; used in Section 1.4.3 to predict relations attached to cusp forms. Not proved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multiple Zeta Values." pith.science (2026). https://pith.science/paper/MKBEKD4U

@misc{pith2026260802230,
  author       = {Pith},
  title        = {Pith review of: Multiple Zeta Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKBEKD4U}},
  note         = {Machine review of arXiv:2608.02230}
}
read the original abstract

These lecture notes are based on three courses given at Nagoya University. Their purpose is to give a beginner-friendly introduction to multiple zeta values and several of their variants, such as finite and symmetric multiple zeta values, q-analogues of multiple zeta values, and multiple Eisenstein series. These notes will be updated in the future.

Figures

Figures reproduced from arXiv: 2608.02230 by the authors.

Figure 1.1
Figure 1.1. Overview of the BTT philosophy Proposition 1.79. For all n, k ≥ 1 we have Hn−1(k; e 2πi n ) = − bk (n) k!  n(1 − e 2πi n ) k . Proof. This is Exercise 1.8. For an index set k = (k1, . . . , kr) we define its reverse by k = (kr, . . . , k1). Theorem 1.80. For all n ≥ 1 and all non-empty indices k we have for any primitive n-th root of unity ζn H⋆ n−1 (k; ζn) = (−1)wt(k)+1H⋆ n−1 (k∨; ζn). Theorem 1.41 (ζ ⋆ A(k) = −ζ… view at source ↗
Figure 4.1
Figure 4.1. Overview of some relations among multiple zeta values. [PITH_FULL_IMAGE:figures/full_fig_p118_4_1.png] view at source ↗
Figure 5.1
Figure 5.1. Relations and bases for the formal double zeta space in small weights. [PITH_FULL_IMAGE:figures/full_fig_p133_5_1.png] view at source ↗
Figures from the paper (4 more)
Figure 6.1
Figure 6.1. Figure 6.1: A summand of GRURRU k1,...,k5 (τ ) for τ = i. Proof of Theorem 6.4: For k1, . . . , kr ≥ 2 the Fourier expansion of the multiple Eisenstein series Gk1,...,kr can be computed in the following way (i) Split up the summation into 2r distinct parts Gw k1,...,kr where w a…
Figure 6.2
Figure 6.2. Figure 6.2: One diagram for the calculation of ∆G (I(a0; a1, . . . , a8; a9)). It gives the term I(a0; a1, a4, a7; a9) ⊗ I(a0; a1)I(a1; a2, a3; a4)I(a4; a5, a6; a7)I(a7; a8; a9). For our purpose it will be important to consider the quotient space2 I 1 = I/I(1; 0; 0)I . Let us de…
Figure 6.3
Figure 6.3. Figure 6.3: Overview of the behavior of the operator [PITH_FULL_IMAGE:figures/full_fig_p184_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: The conjugation of the partition 15 = 4 + 4 + 3 + 2 + 1 + 1 is given by [PITH_FULL_IMAGE:figures/full_fig_p184_6_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 4 canonical work pages

  1. [1]

    Journ´ ees Arithm´ etiques de Luminy

    [AK] T. Arakawa, M. Kaneko,多重ゼータ値入門(Introduction to multiple zeta values), Kyushu Uni- versity MI Lecture Note Series23(2010, corrected and revised January 2016), Japanese. (avail- able athttps://www2.math.kyushu-u.ac.jp/ ~mkaneko/papers/MZV_LectureNotes.pdf) [Ap] R. Ap´ ery:Irrationalit´ e deζ(2)etζ(3), in “Journ´ ees Arithm´ etiques de Luminy”, Ast´ eri...

    2010

  2. [4]

    [H1] M. E. Hoffman:Multiple harmonic series, Pacific J. Math.152(1992), no. 2, 275–290, doi:10.2140/pjm.1992.152.275. [H2] M. E. Hoffman:The algebra of multiple harmonic series, J. Algebra194(1997), 477–495. [H3] M. E. Hoffman:Quasi-shuffle products, J. Algebraic Combin.11(2000), 49–68. [H4] M. E. Hoffman:Quasi-symmetric functions and mod p multiple harmo...

    arXiv 1992

  3. [5]

    Software for Algebraic Geometry

    [K2] M. Kaneko:有限多重ゼータ値(Finite multiple zeta values), RIMS Kˆ okyˆ uroku BessatsuB68 (2017), 175–190 (Japanese). [K3] M. Kaneko:An introduction to classical and finite multiple zeta values, Publications Math´ ematiques de Besan¸ con. Alg` ebre et th´ eorie des nombres, 2019/1 (2019), 103–129, doi:10.5802/pmb.31. [K4] M. Kaneko:多重ゼータ値入門(Introduction to Mul...

    work page doi:10.5802/pmb.31 2017

  4. [6]

    ζ240 ζ [BoBr] J. M. Borwein and D. M. Bradley,Thirty-two Goldbach variations, Intl. J. Number Theory2 (2006), 65–103. [BF] J. I. Burgos Gil, J. Fres´ an:Multiple zeta values: From numbers to motives, with contributions by U. K¨ uhn, Clay Mathematics Proceedings, to appear. (available athttp://javier.fresan. perso.math.cnrs.fr/mzv.pdf) [Bu1] A. Burmester:A...

    arXiv 2006

  5. [7]

    Analytic Number Theory

    ζ241 ζ [Eu] L. Euler:De summis serierum reciprocarum, Comment. Acad. Sci. Petropolitanae7(1740), 123–134. Written in 1735.https://scholarlycommons.pacific.edu/euler-works/41/ [F1] H. Furusho,The multiple zeta value algebra and the stable derivation algebra, Publ. Res. Inst. Math. Sci.,39(4) (2003), 695–720. [F2] H. Furusho,Double shuffle relation for asso...

    arXiv 2003

  6. [8]

    Bachmann:Modular forms andq-analogues of modified double zeta values, Abh

    [Ba9] H. Bachmann:Modular forms andq-analogues of modified double zeta values, Abh. Math. Semin. Univ. Hambg.90(2020), no. 2, 201–213, doi:10.1007/s12188-020-00227-7. [Ba10] H. Bachmann:Stuffle regularized multiple Eisenstein series revisited, RIMS Kˆ okyˆ uroku2238 (2023), 73–86, arXiv:2212.10700. [Ba11] H. Bachmann:MacMahon ’s sums-of-divisors and their...

    arXiv 2020

  7. [9]

    Bachmann and K

    [BT] H. Bachmann and K. Tasaka,The double shuffle relations for multiple Eisenstein series, Nagoya Math. J.230(2018), 180–212, doi:10.1017/nmj.2017.9. [BTs] H. Bachmann, H. Tsumura:On multiple series of Eisenstein type, Ramanujan J.42(2017), no. 2, 479–489, doi:10.1007/s11139-015-9738-0. [BTan1] H. Bachmann, T. Tanaka:Rooted tree maps and the derivation r...

    arXiv 2018

  8. [10]

    The Conference on L-Functions

    ζ242 ζ [HS] M. Hirose, N. Sato:Iterated integrals onP 1\{0,1,∞, z}and a class of relations among multiple zeta values, Adv. Math.348(2019), 163–182, doi:10.1016/j.aim.2019.03.005. [HSS] M. Hirose, N. Sato and S. Seki:The connector for the double Ohno relation, Acta Arith.201 (2021), no. 2, 109–118, doi:10.4064/aa200621-18-5. [HST] M. Hirose, N. Sato and K...

    work page doi:10.1016/j.aim.2019.03.005 2019

Show all 20 references
  1. [11]

    Modular Forms and Multiple Zeta Values

    [KY] M. Kaneko, S. Yamamoto,A new integral-series identity of multiple zeta values and regulariza- tions, Selecta Math. (N.S.)24(2018), no. 3, 2499–2521, doi:10.1007/s00029-018-0400-8. [KZ] M. Kaneko and D. Zagier:Finite multiple zeta values, preprint, 50 pp., to appear in the...

    2018 doi

  2. [15]

    (available athttps://www.risan.io/and at drive.google.com) [Rac] G

    ζ244 ζ [R] Risan,Formal finite multiple zeta values and modular forms, Master’s thesis, Graduate School of Mathematics, Nagoya University (2025), 61 pp. (available athttps://www.risan.io/and at drive.google.com) [Rac] G. Racinet:Doubles m´ elanges des polylogarithmes multiples...

    2025 arXiv

  3. [17]

    First European Congress of Mathematics

    ζ245 ζ [W] M. Waldschmidt:Lectures on multiple zeta values, preliminary draft of eight lectures at IMSc, Chennai, April 2011 (updated 21 April 2011). (available athttps://webusers.imj-prg.fr/ ~michel.waldschmidt/articles/pdf/MZV2011IMSc.pdf) [Yam] S. Yamamoto:Multivariable Hof...

    2011 arXiv

  4. [19]

    Periods in Quantum Field Theory and Arithmetic

    [Za4] D. Zagier:Evaluation of the multiple zeta valuesζ(2, . . . ,2,3,2, . . . ,2), Ann. of Math. (2)175 (2012), no. 2, 977–1000, doi:10.4007/annals.2012.175.2.11. [Zh1] J. Zhao:Multiple q-zeta functions and multiple q-polylogarithms, Ramanujan J.14(2007), no. 2, 189–221, doi:...

    2012 doi

  5. [20]

    Zudilin:One of the numbersζ(5),ζ(7),ζ(9),ζ(11)is irrational, Russian Math

    [Zu1] W. Zudilin:One of the numbersζ(5),ζ(7),ζ(9),ζ(11)is irrational, Russian Math. Surveys 56(2001), no. 4, 774–776, doi:10.1070/RM2001v056n04ABEH000427. [Zu2] W. Zudilin:Diophantine problems for q-zeta values, Mat. Zametki72(2002), no. 6, 936–940 (Russian). Translation in Ma...

    2001 doi

  6. [23]

    The 1-2-3 of modular forms

    [Za3] D. Zagier:Elliptic modular forms and their applications, in “The 1-2-3 of modular forms”, Universitext, Springer-Verlag, Berlin, 2008, 1–103, doi:10.1007/978-3-540-74119-0

    2008 doi

  7. [1984]

    Li,Regularized double shuffle and Ohno-Zagier relations of multiple zeta values, J

    [L] Z. Li,Regularized double shuffle and Ohno-Zagier relations of multiple zeta values, J. Number Theory133(2013), no. 2, 596–610, doi:10.1016/j.jnt.2012.08.005. [La] S. Lang:Introduction to modular forms, Grundlehren der mathematischen Wissenschaften, No. 222, Springer-Verlag...

    2013 arXiv

  8. [2014]

    Okuda and Y

    (available athttps://hdl.handle.net/2237/20845) [OT] J. Okuda and Y. Takeyama,On relations for the multiple q-zeta values, Ramanujan J.14 (2007), no. 3, 379–387, doi:10.1007/s11139-007-9053-5. [OZ] Y. Ohno and D. Zagier,Multiple zeta values of fixed weight, depth, and height, ...

    2007 arXiv

  9. [2016]

    Maesaka, S

    [MSW] T. Maesaka, S. Seki and T. Watanabe:Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730 (2024). [MJOP] H. N. Minh, G. Jacob, N. E. Oussous and M. Petitot:Aspects combinatoires des polyloga- rithmes et d...

    2024 arXiv

  10. [2017]

    Periods in Quantum Field Theory and Arithmetic

    (available athttps://www.henrikbachmann.com/uploads/7/7/6/3/77634444/ qderiv_bachmann_osaka.pdf) [Ba3] H. Bachmann:Double shuffle relations for q-analogues of multiple zeta values, their derivatives and the connection to multiple Eisenstein series, RIMS Kˆ okyˆ uroku No.2015(2...

    2015 doi

  11. [2024]

    Seki:A proof of the extended double shuffle relation without using integrals, Kyushu J

    [Se3] S. Seki:A proof of the extended double shuffle relation without using integrals, Kyushu J. Math. 79(2025), no. 1, 191–198, doi:10.2206/kyushujm.79.191. [SY] S. Seki and S. Yamamoto,A new proof of the duality of multiple zeta values and its generaliza- tions, Intern. J. N...

    2025 arXiv

  12. [2026]

    Periods in Quantum Field Theory and Arithmetic

    ζ239 ζ [BR] K. Ball, T. Rivoal:Irrationalit´ e d’une infinit´ e de valeurs de la fonction zˆ eta aux entiers impairs, Invent. Math.146(2001), no. 1, 193–207, doi:10.1007/s002220100168. [BS] S. Baumard, L. Schneps:Period polynomial relations between double zeta values, Ramanuja...

    2001

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.