Pith. sign in

REVIEW 3 major objections 4 minor 48 references

The paper proves that deformed binomial sums A_{s,n} lie in level-3 cyclotomic multiple zeta value spaces, with exact closed forms including A_{4,4}=352ζ_{5,3}/15+752537π^8/10206000.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 01:28 UTC pith:HCPR4N3Y

load-bearing objection Strong, mostly constructive paper; the advertised level-1 theorem for A_{4,n} depends on an unproved identification with HyperInt's regularized limit, so the referee should ask for a proof of that identification. the 3 major comments →

arxiv 2607.14646 v2 pith:HCPR4N3Y submitted 2026-07-16 math.NT hep-thmath.CO

Multiple Clausen values and deformed Ap\'ery-like series

Zhi-Wei Sun , Yajun Zhou This is my paper
classification math.NT hep-thmath.CO MSC 11M0611M32
keywords deformed central-binomial seriesmultiple Clausen valuescyclotomic multiple zeta valuesgeneralized polylogarithmslevel-3 zeta spacesClausen descentDirichlet L-functionsMZV membership
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves a structural theorem for the deformed binomial sums A_{s,n}, defined as the sum over k≥1 of the nth derivative of 1/(x^s binom(2x,x)) evaluated at x=k. For every s>1 and n≥0, A_{s,n} lies in Z_{n+s}(3), the Q-vector space spanned by cyclotomic multiple zeta values of weight n+s at level 3; A_{1,n} lies in i√3 Z_{n+1}(3); and A_{4,n} lies in the smaller space Z_{n+4}(1) of ordinary multiple zeta values. These memberships are constructive: the paper also derives explicit closed forms, such as A_{1,5} expressed through the Dirichlet L-function L(χ_{−3}, s) and A_{4,4}=352ζ_{5,3}/15+752537π^8/10206000. The significance is that a family of infinite sums previously accessible only numerically or through ad hoc hypergeometric certification is now unified under one algebraic framework, with level-1 descent for the special cases s=2 and s=4.

Core claim

The central discovery is that every deformed binomial sum A_{s,n} is a Q-linear combination of multiple Clausen values μ_{a1,...,an}—series over ordered indices weighted by e^{ℓ1 πi/3}—and that these μ-values themselves reduce to level-3 cyclotomic multiple zeta values. The proof starts from the beta integral for the generalized central binomial coefficient, differentiates to obtain powers of log[t(1−t)], and rewrites the integrand as generalized polylogarithms with parameters 0, 1, and ρ=e^{πi/3}. Lyndon-word decomposition and the vanishing of G(α; z/(z−1))−G(α;1−z) at z=ρ generate provable algebraic relations among the μ-values, reducing all entries of the tables. For s=2 and s=4, counter-

What carries the argument

The central object is the multiple Clausen value μ_{a1,...,an}=Σ_{ℓ1>...>ℓ_{n+1}>0} e^{ℓ1 πi/3} ℓ1^{−a1}···ℓ_{n+1}^{−a_n}, a level-6 cyclotomic multiple zeta value that the paper proves descends to level 3. The carrying mechanism is the generalized polylogarithm (GPL): the beta integral identity 1/(x binom(2x,x)) = (1/2)∫[t(1−t)]^{x−1}dt converts each differentiated term into a GPL at t evaluated on paths ending at 1, with the sixth root ρ inserted by partial fractions. Algebraic relations among μ-values are generated by the functions g_α(z)=G(α; z/(z−1))−G(α;1−z), which vanish at z=ρ; together with Lyndon-word decomposition and GPL shuffle identities, these relations reduce all occurring ex

Load-bearing premise

The load-bearing premise is that the regularized limit used for A_{2,n} and A_{4,n} agrees with the ordinary infinite sums defining those series; this agreement is supported by shuffle identities and computer algebra but not proved from the Abel-summed definition.

What would settle it

Evaluate A_{4,4} directly from the definition Σ_k ∂^4/∂x^4 [x^{-4}/binom(2x,x)]|_{x=k} to high precision (say 100 digits) and compare with 352ζ_{5,3}/15+752537π^8/10206000; any discrepancy beyond round-off refutes Theorem 1.3. Similarly, compute A_{2,1} and A_{4,1} numerically and test whether the regularized-limit algorithm reproduces the ordinary sums.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For every n, A_{1,n} is an element of i√3 Z_{n+1}(3); in particular all such values are built from Dirichlet L-values and ζ-values at level 3.
  • A_{2,n} and A_{4,n} are ordinary MZVs, confirming empirical conjectures and extending earlier results for A_{2,n} that relied on different summation techniques.
  • The explicit tables provide closed forms for A_{1,0} through A_{1,7}, A_{3,0} through A_{3,5}, and A_{4,2} through A_{4,10}; for instance A_{4,4}=352ζ_{5,3}/15+752537π^8/10206000.
  • An empirical sum rule for multiple Clausen values is proved up to weight 8 (Table 4), and the existence part of a conjectured rational-multiple-of-π^w relation at odd weights is verified through weight 7.

Where Pith is reading between the lines

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

  • If the counter-term technique of §3.3.2 generalizes, one would expect analogous level-1 descents for A_{6,n}, A_{8,n}, or other even s; the paper leaves this as an open direction.
  • The level-1 membership for s=2,4 depends on a specific regularized-limit convention; a direct proof that that regularized limit equals the Abel sum would place the result on fully analytic footing, without changing the stated value.
  • The same Lyndon-word/algebraic-relation machinery should apply to other deformed hypergeometric sums, e.g., the four-binomial series (1.15), whose level-3 structure is computed here up to n=4; beyond that, MCVs may or may not suffice.
  • If the conjectured Q-linear independence of primitive MCVs holds, then the described reductions are not merely sufficient but complete, and the tables would give a canonical normal form for these deformed sums.

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 / 4 minor

Summary. The paper defines deformed Apéry-like series A_{s,n} by taking derivatives, at x=k, of the meromorphic continuation of 1/(x^s binom(2x,x)), and proves membership in spaces of cyclotomic multiple zeta values: Theorem 1.1 places A_{1,n} in i√3 Z_{n+1}(3), Theorem 1.2 places A_{s,n} in Z_{n+s}(3) for s>1, and Theorem 1.3 places A_{4,n} in Z_{n+4}(1). The proofs are based on generalized polylogarithm recursions, fibrations, and reductions of multiple Clausen values. The paper also gives explicit closed forms for selected entries in Tables 1–3, derives new cases of Au's MCV sum rule, and partially addresses Broadhurst's conjectures. The constructive, algorithm-oriented framework is the paper's main contribution.

Significance. If the results are fully correct, the paper provides a significant extension of Apéry-type evaluations: it gives a uniform GPL method for deformed binomial series, proves new MZV and CMZV membership theorems, and supplies explicit exact evaluations, including A_{4,4}=352ζ_{5,3}/15+752537π^8/10206000. The algorithmic and machine-assisted nature of several proofs is a strength, and the paper is honest about the boundary between proven relations and computer-algebra-supported reductions. However, the central level-1 theorem currently rests on an unproved identification between a regularized limit used by HyperInt and the Abel-summed value of the original series; this must be closed before the main claims can be regarded as established.

major comments (3)
  1. [§3.3.2, Eq. (3.64)] The step from (3.56)-(3.57) to A_{4,n} ∈ Z_{n+4}(1) is not justified. Equation (3.64) asserts A_{4,n}= -n!/3 Re(Reg G_{4,n} - Reg C_{4,n}), but the proof only shows G_{4,n}(z)∈f^{(z)}_{n+4} and Re Reg C_{4,n}(z)∈Z_{n+4}(1). The integral in (3.58) that connects G_{4,n}-C_{4,n} to the series defining A_{4,n} in (3.59) is divergent at z=1, and the paper never proves that HyperInt's regularized limit of this integral reproduces the Abel-summed value of A_{4,n}. If the regularization removes or modifies a finite constant, Theorem 1.3 and the A_{4,n} entries of Table 3, including the displayed evaluation of A_{4,4}, are unsupported. A lemma establishing the equality between the regularized limit and the defining series is required.
  2. [§3.3.2, Proposition 3.4, Eq. (3.52)] The same regularization issue affects the new proof of A_{2,n}∈Z_{n+2}(1). In (3.49)-(3.52) the paper takes regularized limits and asserts A_{2,n}= -n!/3 Re Reg_{z→1±i0+} G_{2,n}(z), but the integral from z to 1 in (3.49) is not shown to have zero regularized limit, nor is the counter-term identity Re Reg C_{2,n}(z)=0 in (3.54) proved. While A_{2,n} membership was already known from Hou–Sun, the unified framework and the corresponding entries of Table 3 depend on this gap, so it should be repaired for the presentation to be self-contained.
  3. [§3.2, Propositions 3.1–3.2] The 'provable reductions' in Tables 5–8 are delegated to computer-algebra routines, but the complete verification data and code are not included. In particular, the extra relations (3.15)-(3.21) are not fully justified: the operator ffib is described only procedurally, and the paper does not give a rigorous statement of why applying this fibration to an expression that vanishes at z=ρ yields valid algebraic relations. This is not fatal for the generic membership theorems, but it blocks independent verification of the explicit closed forms and of the Au sum-rule cases in Table 4. A reproducibility appendix with code or certificates would be needed to support these advertised evaluations.
minor comments (4)
  1. [Section 1, Remark] Reference [36] is cited for the MathOverflow question in the text, but the bibliography lists [37] for that question; the citation numbers appear mismatched.
  2. [Tables 1–3] The table headings contain formatting artifacts such as 'T able' and the tables are very dense; a typesetting fix and a short explanation of how to read the row indices (e.g., A_{3,w-3}) would improve readability.
  3. [§3.1, Eq. (3.5)–(3.6)] The notation in the Lyndon-word example uses letters A,B and then relates them to GPL parameters 0,1; this is clear but would benefit from an explicit statement that A maps to 0 and B maps to 1 in the binary-word convention.
  4. [§3.2, Table 6] Some entries in Table 6 mix μ, ζ, and π terms without an explicit statement of the chosen basis of irreducible MCVs; the text in Footnote 4 explains the lexicographic convention, but placing this explanation before the table would help.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained and proves the conjectured CMZV/MZV memberships rather than fitting them.

full rationale

The paper's central claims (Theorems 1.1–1.3) are derived from the defining series (1.4) via the beta integral (2.9), yielding exact GPL representations (2.13) and (2.22). The subsequent reductions to level-3 CMZVs (Proposition 3.3) and the MCV algebraic relations (Propositions 3.1–3.2) are based on GPL identities that vanish at z = rho by the change of variables z -> 1 - z, not on the target values. The tables are outputs of these reductions, not inputs. The level-1 descents in Propositions 3.4–3.5 use Panzer's HyperInt regularization to evaluate identities that connect the convergent integrals (3.40)/(3.59) defining A_{2,n} and A_{4,n} to MZV combinations; even if one regards the Reg/Abel-sum identification as insufficiently proved, this is a correctness/reproducibility concern, not a circular one, because A_{s,n} has an independent definition and the regularized limit is not fitted to the claimed values. Self-citations to Sun's earlier conjectures [36,38] and to Hou–Sun [23] are to results being proved or re-proved here, not used as load-bearing premises. No equation reduces to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted. The paper depends on standard GPL theory and on the reliability of several computer algebra packages. The only non-standard input is the regularization scheme and the standardness of level-3 relations, both clearly flagged in the text.

axioms (5)
  • standard math GPL recursion (2.1), initial conditions (2.2), scaling (2.3), shuffle (2.7) and differential (3.1) identities.
    Used throughout §2–§3 to convert A_{s,n} into GPL/CMZV expressions.
  • standard math The equivalence between the series (1.5) and GPL definitions (1.5′)/(1.5′′) of Z_w(N), including πi∈Z_1(N) for N≥3.
    Invoked to assert A_{1,n}∈i√3 Z_{n+1}(6) and A_{s,n}∈Z_{n+s}(6) from GPL expressions in (2.13) and (2.22).
  • domain assumption Regularized limits of divergent GPLs at boundary, as defined in Panzer's HyperInt [29, (2.8)], correctly capture the limiting values needed in Propositions 3.4 and 3.5.
    The level-1 descent for A_{2,n} and A_{4,n} relies on the regularized limit 'Reg_{z→1±i0+}' and the Maple code; no independent proof is given that this regularization is the one matching the original series.
  • domain assumption The algebraic relations for level-3 CMZVs used in reductions are 'standard' in Deligne's sense; no non-standard level-6 relations interfere.
    Footnote 10 argues that the MCVs actually live in level 3, where standard relations apply; this underpins the validity of the generated reductions.
  • domain assumption Correctness of outputs from external packages (MultipleZetaValues, HPL, HyperInt, logsine) used to verify table reductions.
    Tables 5-8 and the Maple code rely on these packages; their correctness is assumed, since scripts and versions are not fully pinned.

pith-pipeline@v1.3.0-alltime-deepseek · 42288 in / 17991 out tokens · 178430 ms · 2026-08-02T01:28:47.834947+00:00 · methodology

0 comments
read the original abstract

With generalized central binomial coefficients $ \binom{2x}{x}:=\frac{\Gamma(2x+1)}{[\Gamma(x+1)]^2}$ defined through Euler's gamma function, we represent deformed Ap\'ery-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level $3$. For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(\chi_{-3},6)-30\pi^{2}L(\chi_{-3},4)-2\pi^{4}L(\chi_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352\zeta_{5,3}}{15}+\frac{752537\pi^{8}}{10206000},\]where $ L(\chi_{-3},s):=\sum_{n=0}^\infty\left[(3n+1)^{-s}-(3n+2)^{-s}\right]$ and $\zeta_{5,3}:=\sum_{m>n>0}m^{-5}n^{-3}$.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 10 linked inside Pith

  1. [1]

    Ablinger

    J. Ablinger. Discovering and proving infinite binomial sums identities.Exp. Math., 26(1):62–71, 2017

    2017

  2. [2]

    R. Apéry. Irrationalité de𝜁(2)et𝜁(3).Astérisque, 61:11–13, 1979

    1979

  3. [3]

    K. C. Au. Multiple zeta values, WZ-pairs and infinite sums computations.Ramanujan J., 66(1):Paper No. 3, 35 pages, 2025

    2025

  4. [4]

    K. C. Au. Iterated integrals and multiple polylogarithm at algebraic arguments.arXiv:2201.01676v3[math.NT] (software available athttps://www.researchgate.net/publication/357601353_Mathematica_package_ MultipleZetaValuesandhttps://www.wolframcloud.com/obj/pisco125/MultipleZetaValues-1.2.0. paclet), 2026

    arXiv 2026

  5. [5]

    Blümlein, D

    J. Blümlein, D. J. Broadhurst, and J. A. M. Vermaseren. The multiple zeta value data mine.Comput. Phys. Comm., 181(3):582–625, 2010. 28 ZHI-WEI SUN AND YAJUN ZHOU

    2010

  6. [6]

    J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisoněk. Special values of multiple polylogarithms.Trans. Amer. Math. Soc., 353(3):907–941, 2001

    2001

  7. [7]

    J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer. Central binomial sums, multiple Clausen values, and zeta values.Experiment. Math., 10(1):25–34, 2001

    2001

  8. [8]

    J. M. Borwein and A. Straub. Special values of generalized log-sine integrals. InISSAC 2011—Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, pages 43–50. ACM, New York, 2011

    2011

  9. [9]

    J. M. Borwein and A. Straub. Log-sine evaluations of Mahler measures.J. Aust. Math. Soc., 92(1):15–36, 2012. arXiv:1103.3893[math.CA]

    Pith/arXiv arXiv 2012

  10. [10]

    J. M. Borwein and A. Straub. Relations for Nielsen polylogarithms.J. Approx. Theory, 193:74–88, 2015

    2015

  11. [11]

    Broadhurst

    D. Broadhurst. Multiple Deligne values: a data mine with empirically tamed denominators.arXiv:1409.7204 [hep-th], 2014

    Pith/arXiv arXiv 2014

  12. [12]

    Broadhurst

    D. Broadhurst. Polylogs of roots of unity: the good, the bad and the ugly. InThe Interrelation Between Mathematical Physics, Number Theory and Noncommutative Geometry, Vienna, Austria, March 12, 2015. Erwin Schrödinger Interna- tional Institute for Mathematics and Physics (ESI).www.noncommutativegeometry.nl/wp-content/uploads/ 2015/04/Broadhurst.pdf

    2015

  13. [13]

    D. J. Broadhurst. Massive 3-loop Feynman diagrams reducible toSC ∗ primitives of algebras of the sixth root of unity.Eur. Phys. J. C Part. Fields, 8(2):313–333, 1999

    1999

  14. [14]

    D. J. Broadhurst and D. Kreimer. Association of multiple zeta values with positive knots via Feynman diagrams up to9loops.Phys. Lett. B, 393(3-4):403–412, 1997

    1997

  15. [15]

    J. M. Campbell, M. L. Glasser, and Y. Zhou. New evaluations of inverse binomial series via cyclotomic multiple zeta values.SIGMA Symmetry Integrability Geom. Methods Appl., 20:Paper No. 079, 14, 2024

    2024

  16. [16]

    H. Cohen. Généralisation d’une construction de R. Apéry.Bull. Soc. Math. France, 109(3):269–281, 1981

    1981

  17. [17]

    Deligne and A

    P. Deligne and A. B. Goncharov. Groupes fondamentaux motiviques de Tate mixte.Ann. Sci. École Norm. Sup. (4), 38(1):1–56, 2005

    2005

  18. [18]

    Duhr and F

    C. Duhr and F. Dulat. PolyLogTools—polylogs for the masses.J. High Energy Phys., (8):135, 2019

    2019

  19. [19]

    Frellesvig

    H. Frellesvig. Generalized polylogarithms in Maple.arXiv:1806.02883[hep-th], 2018

    Pith/arXiv arXiv 2018

  20. [20]

    Frellesvig, D

    H. Frellesvig, D. Tommasini, and C. Wever. On the reduction of generalized polylogarithms toLin andLi 2,2 and on the evaluation thereof.J. High Energy Phys., 2016(3):189, 2016.arXiv:1601.02649v3[hep-ph]

    Pith/arXiv arXiv 2016

  21. [21]

    A. B. Goncharov. Multiple polylogarithms, cyclotomy and modular complexes.Math. Res. Lett., 5(4):497–516, 1998

    1998

  22. [22]

    J. M. Henn, A. V. Smirnov, and V. A. Smirnov. Evaluating multiple polylogarithm values at sixth roots of unity up to weight six.Nuclear Phys. B, 919:315–324, 2017

    2017

  23. [23]

    Hou and Z.-W

    Q.-H. Hou and Z.-W. Sun. Evaluations of some series via the wz method.arXiv:2604.15172v1[math.CO], 2026

    Pith/arXiv arXiv 2026

  24. [24]

    S. Laporta. High-precision calculation of the 4-loop contribution to the electrong−2in qed.Physics Letters B, 772(Supplement C):232 – 238, 2017.arXiv:1704.06996[hep-th]

    Pith/arXiv arXiv 2017

  25. [25]

    L. Lewin. On the evaluation of log-sine integrals.Math. Gaz., 42(340):125–128, 1958

    1958

  26. [26]

    Lewin.Polylogarithms and Associated Functions

    L. Lewin.Polylogarithms and Associated Functions. North Holland, New York, NY, 1981

    1981

  27. [27]

    Maître.HPL, a Mathematica implementation of the harmonic polylogarithms.Comput

    D. Maître.HPL, a Mathematica implementation of the harmonic polylogarithms.Comput. Phys. Comm., 174(3):222– 240, 2006.arXiv:hep-ph/0507152v2(software available athttps://www.physik.uzh.ch/data/HPL/)

    Pith/arXiv arXiv 2006

  28. [28]

    D. Maître. Extension ofHPLto complex arguments.Comput. Phys. Comm., 183(3):846, 2012.arXiv:hep-ph/ 0703052v2

    2012

  29. [29]

    E. Panzer. Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman inte- grals.Comput. Phys. Commun., 188:148–166, 2015. (software available athttps://bitbucket.org/PanzerErik/ hyperint/wiki/Home)

    2015

  30. [30]

    D. E. Radford. A natural ring basis for the shuffle algebra and an application to group schemes.J. Algebra, 58(2):432–454, 1979

    1979

  31. [31]

    Remiddi and J

    E. Remiddi and J. A. M. Vermaseren. Harmonic polylogarithms.Internat. J. Modern Phys. A, 15(5):725–754, 2000

    2000

  32. [32]

    O. Schnetz. Quantum periods: a census of𝜙 4-transcendentals.Commun. Number Theory Phys., 4(1):1–47, 2010

    2010

  33. [33]

    O. Schnetz. The Galois coaction on the electron anomalous magnetic moment.Commun. Number Theory Phys., 12(2):335–354, 2018

    2018

  34. [34]

    Singer and J

    J. Singer and J. Zhao. Finite and symmetrized colored multiple zeta values.Finite Fields Appl., 65:101676, 20, 2020

    2020

  35. [35]

    Z.-W. Sun. New series for some special values ofL-functions.Nanjing Univ. J. Math. Biquarterly, 32:189–218, 2015

    2015

  36. [36]

    Z.-W. Sun. Evaluations of some series of the typeÍ∞ k=0(ak+b)x k/mk nk .Nanjing Univ. J. Math. Biquarterly, 41(1):1– 33, 2024

    2024

  37. [37]

    Z.-W. Sun. More conjectural formulas for Riemann’s zeta function. Question 508768 on MathOverflow. Mar. 5, 2026.https://mathoverflow.net/questions/508768

    2026

  38. [38]

    Z.-W. Sun. Various conjectural series identities.arXiv:2603.29973v3[math.NT], 2026. MULTIPLE CLAUSEN V ALUES AND DEFORMED APÉRY-LIKE SERIES 29

    Pith/arXiv arXiv 2026

  39. [39]

    van der Poorten

    A. van der Poorten. A proof that Euler missed ... Apéry’s proof of the irrationality of𝜁(3).Math. Intelligencer, 1(4):195–203, 1978/79. An informal report

    1978

  40. [40]

    Wang and C

    W. Wang and C. Xu. Alternating multiple zeta values, and explicit formulas of some Euler–Apéry-type series. European J. Combin., 93:Paper No. 103283, 25, 2021

    2021

  41. [41]

    Weinzierl.Feynman integrals—a comprehensive treatment for students and researchers

    S. Weinzierl.Feynman integrals—a comprehensive treatment for students and researchers. Unitext for Physics. Springer, Cham, Switzerland, 2022

    2022

  42. [42]

    H. S. Wilf and D. Zeilberger. Rational functions certify combinatorial identities.J. Amer. Math. Soc., 3(1):147–158, 1990

    1990

  43. [43]

    D. Zagier. The Bloch–Wigner–Ramakrishnan polylogarithm function.Math. Ann., 286(1-3):613–624, 1990

    1990

  44. [44]

    J. Zhao. Multiple polylogarithm values at roots of unity.C. R. Math. Acad. Sci. Paris, 346(19-20):1029–1032, 2008

    2008

  45. [45]

    J. Zhao. Standard relations of multiple polylogarithm values at roots of unity.Documenta Math., 15:1–34, 2010

    2010

  46. [46]

    Y. Zhou. Sun’s series via cyclotomic multiple zeta values.SIGMA Symmetry Integrability Geom. Methods Appl., 19:Pa- per No. 074, 20 pp., 2023.arXiv:2306.04638v3[math.NT]

    Pith/arXiv arXiv 2023

  47. [47]

    Y. Zhou. Hyper-Mahler measures via Goncharov–Deligne cyclotomy. In H. S. Cohl, R. S. Costas-Santos, and R. S. Maier, editors,Classical hypergeometric functions and generalizations, volume 818 ofContemp. Math., pages 239–288. Amer. Math. Soc., Providence, RI, 2025.arXiv:2210.17243v4[math.NT]

    Pith/arXiv arXiv 2025

  48. [48]

    I. J. Zucker. On the series Í∞ k=1(2k k)−1k−n and related sums.J. Number Theory, 20(1):92–102, 1985. (Z.-W. Sun)School of Ma thema tics, Nanjing University, Nanjing 210093, People’s Republic of China Email address:zwsun@nju.edu.cn Homepage: http://maths.nju.edu.cn/ ˜zwsun (Y. Zhou)Program in Applied and Comput a tional Ma thema tics (P ACM), Princeton Uni...

    1985