Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values

Ce Xu

REVIEW 3 major objections 4 minor 29 references

Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One integral method proves six zeta-value conjectures.

desk verdict Standard-technique proofs of several Borwein–Bradley–Broadhurst conjectures, but (1.13)–(1.14) as printed use divergent first-exponent-1 zeta values and a missing bar appears to be the cause. read the letter →

arxiv 1908.03065 v1 pith:74BER5OG submitted 2019-08-02 math.NT

Ce Xu This is my paper · ORCID

classification math.NT MSC 11M0611M4040B0533E20

keywords multiplepolylogarithmfunctionsiteratedintegralsalternatingzetavaluesunit-exponentharmonicstarsumssixconjecturedidentitiescircled-productlinearrelationsof

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

This paper proves that a broad family of alternating multiple zeta values—zeta-type sums whose summands carry signs attached to the summation labels—can be rewritten using only ordinary multiple zeta values together with the simplest alternating ones, where every exponent after the first is 1. The proof passes through iterated integrals of multiple polylogarithm functions: a change of variables turns each integral into a finite sum over sign choices, and the special point $1/2$ links these integrals to the unit-exponent alternating values. This reduction yields explicit formulas for the six conjectured identities (1.9)--(1.14) stated in the introduction, and more generally expresses every value of the form ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) in terms of unit-exponent alternating values. Section 5 extends the method to circled-product multiple zeta values, and Section 6 derives a general linear-relation identity for alternating multiple zeta values. If the identities are right, a family of special values previously approachable only case-by-case is captured by one uniform evaluation.

What carries the argument

The central machinery is the iterated-integral representation (2.1) of the multiple polylogarithm function—a nested series whose partial sums are multiple harmonic sums—combined with the change-of-variables identity (2.2). That identity rewrites the integral, after setting all parameters equal to $a$, as a sum over $2^{p_k}$ sign choices σ_j ∈ {1,a}, weighted by η(1)=1 and η(a)=−a. When $a=1/2$, equation (2.4) gives a one-to-one correspondence between these polylogarithm values and unit-exponent alternating multiple zeta values, and when $a=-1$, the same identity produces alternating-multiple-zeta-value relations directly. This single substitution rule is what carries every later evaluation in the paper.

What would settle it

Take identity (1.11) with $m=n=0$ and compute both sides as partial sums of their defining multiple harmonic series up to $N=10^6$ terms: the left side is $\zeta(\bar{1},2,2)$ and the right side is an explicit combination of unit-exponent alternating values, so a mismatch beyond truncation error would refute the central identity.

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Extended reading notes

Core claim

The paper's central discovery is that the alternating multiple zeta value ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) is a finite 𝔺-linear combination of ordinary multiple zeta values and unit-exponent alternating multiple zeta values. This is Theorem 3.4 together with the correspondence at $1/2$; in the special case where all $p_i=1$ it gives Theorem 4.1, from which the six identities (1.9)--(1.14) follow as corollaries. The paper further proves that certain circled-product multiple zeta values, such as ζ(({2}ᵃ,3,{2}ᵇ) ⊛ (0,{2}ᵅ)⋆), lie in the ring 𝔺[ζ(2),ζ(3),ζ(4),…] generated by ordinary single zeta values, and it closes with a general linear-relation identity for alternating multiple zeta values obtained from an extended poset integral.

Load-bearing premise

The entire reduction rests on the iterated-integral change-of-variables identity (2.2), whose proof is condensed into a 'direct calculation'; if that step is wrong, every later evaluation of an alternating multiple zeta value inherits the error.

Editorial extensions

If this is right

Every alternating multiple zeta value of the shape ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) can be evaluated in terms of ordinary multiple zeta values and unit-exponent alternating values, so no genuinely new class of constants is needed for these values.
The six conjectured identities (1.9)--(1.14) hold for all nonnegative integers $m,n$; for instance, ζ(¯1, {1}ᵀ, 2, {1}ⁿ) = ζ(¯1, {1}ⁿ, ¯1, ¯1, {1}ᵀ) − ζ(¯1, {1}ᵀ+ⁿ+²).
Several families of circled-product multiple zeta values, including ζ(({2}ᵃ,3,{2}ᵇ) ⊛ (0,{2}ᵅ)⋆), are shown to lie in 𝔺[ζ(2),ζ(3),ζ(4),…], with explicit rational-linear formulas such as the displayed evaluation of ζ((3,2) ⊛ (0,2,2)⋆).
The (p+2)-poset integral picture yields a family of linear relations among alternating multiple zeta values, with explicit small cases exhibited as equation (6.11).
The method proves the six conjectures uniformly, replacing a situation in which one of the identities had previously required a separate, later proof.

Reading between the lines

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

The reduction in Theorem 2.2 is algorithmic: the sign-sum expansion and the $1/2$ correspondence could be implemented as a normal-form procedure that rewrites any alternating multiple zeta value of the stated shape into a basis expression, giving a practical way to compute the 𝔺-span at fixed weight without case-by-case guessing.
Because the key change of variables works for a general real parameter $a$, the same proof scheme should produce analogous reduction identities for colored multiple zeta values at roots of unity, provided convergence conditions are handled; the paper does not itself make that extension.
The poset integral identity of Section 6 is strong enough to suggest a conjectural complete set of linear relations for alternating multiple zeta values, parallel to the conjecture for ordinary multiple zeta values that one integral-series identity generates all relations; the paper stops short of stating that conjecture.

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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

Xu proves a batch of Borwein–Bradley–Broadhurst identities (1.9)–(1.12) and gives general reduction theorems (2.2, 3.4, 4.1, 5.5) for alternating MZVs to unit-exponent ones. The method is standard iterated integrals, and the main body looks credible: Theorem 4.1 plus the corollary at (4.5) really does give (1.9) and (1.11), and the derivations of (1.10) and (1.12) from Corollary 3.6 are plausible. The paper also contains useful identities for Kaneko–Yamamoto zeta values and a linear relation in Section 6. Within the MZV subfield this is a meaningful result, and the novelty claim is honest: (1.10) is explicitly credited to Wang–Liu–Chen.

The soft spot is real and, as printed, load-bearing for two of the six conjectures. Equations (1.13)–(1.14) contain ζ(1,{1}^n,...) with first exponent 1. Under the paper's own convergence definition, a multiple zeta value with first exponent 1 diverges: after summing the inner variables, the tail behaves like a constant divided by n1, and alternating signs in later slots do not rescue a positive first sign. The derivation of (1.13) at (4.9) actually ends with Li{...}(-1,{1}^n,...), which is ζ(\bar1,{1}^n,...), not ζ(1,{1}^n,...). So either a bar is missing in the statement or the identity uses an undefined regularized value. This needs fixing, and it matters because (1.13)–(1.14) are advertised as proven conjectures.

Other soft spots are minor by comparison: Theorem 2.2's key step is 'by a direct calculation,' and Theorems 5.1 and 5.2 are described as 'similar' proofs. The extracted text is garbled enough that full line-by-line checking is impossible, but I found no contradiction in the convergent part. The reliance on the author's own earlier papers for some intermediate identities is not a red flag; those are published results.

Recommendation: send it to a qualified referee if the author supplies a corrected version of (1.13)–(1.14) with the bars restored or a regularization defined. As it stands, the paper deserves referee time—the main theorems are likely right and the standard for this subfield is explicit identities—but it should not be accepted without addressing the divergence.

Referee Report

3 major / 4 minor

Summary. The paper develops iterated-integral identities for multiple polylogarithm functions and applies them to alternating multiple zeta values. Section 2 proves a transformation theorem (Theorem 2.2) and an integration-by-parts identity (Theorem 2.4). Section 3 derives a general reduction (Theorem 3.4) expressing alternating MZVs of the form ζ(\bar1,{1}^{m1-1},p1+1,{1}^{m2-1},...,pk+1,{1}^{mk+1-1}) in terms of ordinary MZVs and infinite sums involving multiple harmonic star sums. Section 4 applies these results to prove the six Borwein-Bradley-Broadhurst conjectures numbered (1.9)-(1.14). Section 5 extends the method to Kaneko-Yamamoto multiple zeta values, and Section 6 gives general linear relations via (p+2)-posets.

Significance. If the statements are corrected as indicated below, the paper offers explicit, checkable identities and a uniform method that proves several long-standing conjectures. The approach is not circular: the identities are derived from the iterated integral representation, and the intermediate known identities such as (4.2) and (2.4) are cited from published sources. The paper is systematic and gives many concrete examples and corollaries. However, the displayed BBB identities (1.13)-(1.14) are not well-defined as written, and this directly affects the central claim that those conjectures are proved.

major comments (3)

[Eqs. (1.13)-(1.14) and Section 4 (Eqs. (4.9), (4.14))] As printed, equations (1.13) and (1.14) are not identities among convergent multiple zeta values under the paper's own convergence criterion ℜ(s1+...+sj)>j quoted for (1.3). In (1.13), every summand ζ(1,{1}^n,S_k) has first exponent +1, so its defining series over n1 is a sum of a term that does not tend to zero divided by n1, which diverges; the alternating signs in S_k do not affect the first summation variable. Similarly, the left-hand side of (1.14) is ζ(1,m+1,{1}^n), which also has first exponent +1 and diverges. The derivation in Section 4 indicates that barred exponents are intended: Eq. (4.9) is obtained from Li_{m+1,{1}^n}(-1), which is ζ(\bar{m+1},{1}^n), and its right-hand side is a sum of Li(-1,{1}^n,...)=ζ(\bar1,{1}^n,...), not ζ(1,{1}^n,...). Thus either a bar (and in (4.10) also a sign) has been lost in transcription, or the identities are asserted for a regularized value that is never defined. Since Theorem 3.4 and Theorem 4.1 are advertised as proofs of (1.13)-(1.14), this must be corrected and the corrected statements verified before the central claim can be accepted.
[Theorem 2.2, Eq. (2.2)] The proof of Theorem 2.2, which is the foundation for Theorem 4.1 and for equations (4.9) and (4.14), is compressed into the sentence "by a direct calculation." The summation over σ_j∈{1,a} with η(1)=1 and η(a)=-a, the product ∏ η(σ_j)/σ_j, and the bookkeeping of the Cat blocks are exactly the steps that produce the exponent sequences in (2.2). Please provide the full expansion, including the degenerate cases m1=0 and p_i=0, because the one-to-one correspondence in (2.4) and the later reductions depend on the precise placement of the arguments a/σ_i and σ_i/σ_{i-1}.
[Theorem 3.4, Eq. (3.9)] The proof of Theorem 3.4 says "Continuing this process k times," and the notation E_i, F_i with the □ operation is intricate. The step in which products of harmonic sums are expanded via the stuffle product is asserted in a paragraph rather than shown. Since Theorem 3.4 is the main reduction result underlying Theorem 4.1 and the BBB identities, please supply the induction in detail and state exactly how the ζ(E_i,{1}^{p_i-j}) factors and the infinite harmonic-star sums in (3.9) are obtained, especially in degenerate cases where some p_i=0 or m_i=1.

minor comments (4)

[Proposition 2.1] The convention for m1=0 is explained only after the statement of Proposition 2.1, but the symbol {1}^{-1} appears inside the displayed formula before that convention is introduced. Please state the convention before the proposition.
[References] Reference [18] is incomplete: it gives authors and a title but no journal, year, arXiv number, or DOI. Please supply the full publication data.
[Theorems 5.1 and 5.2] The proofs of Theorems 5.1 and 5.2 are described as "similar" to earlier proofs. Because these theorems are used for Corollary 5.6, please include enough detail (or a supplement) to make the derivations verifiable.
[Throughout] The manuscript contains numerous typographical and OCR artifacts, such as "pol ylogarithm" in the abstract, malformed diagrams in Section 6, and inconsistent spacing in several displayed equations. A careful proofread is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction of the central claims; only minor, non-load-bearing self-citations.

full rationale

Walking the derivation chain, the central reduction theorems are not circular: Theorem 2.2 is obtained by a change of variables in the iterated-integral representation (2.1), Theorem 3.4 follows from (2.1) plus Lemma 3.3, and Theorem 4.1 follows from (2.2). The Borwein-Bradley-Broadhurst identities (1.9)-(1.14) are external targets, and their right-hand sides are not fitted to the left-hand sides; the unit-exponent alternating MZVs on the right are produced by the integral transformations, not assumed. The one-to-one dictionary (2.4) between Li(1/2) values and unit-exponent alternating MZVs is quoted from Borwein et al. and Zlobin, not from the author's own work, and it is used as a stated input rather than as the conclusion being proved. The author's self-citations are peripheral: (4.2) is also derived in the paper from (2.8), and the cited identity Li_{1^j}(1/2) = -ζ(\bar{j}) is a parameter-free special value used in one intermediate step of the proof of (1.14), not the target identity. I do flag, outside the circularity rubric, that Theorem 2.2's proof is compressed as 'by a direct calculation' and that the apparent sign/convergence issues around (1.13)-(1.14) are correctness or rigor risks rather than evidence that the claims reduce to their inputs. Because the paper contains minor self-citations that are not load-bearing, I set the score at the low end of the normal range; no circular step was identified.

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

No parameters are fitted and no new physical entities are postulated. The paper rests on standard analytic convergence assumptions, harmonic product algebra, and several external results from the literature, including the one-to-one polylog/alternating-MZV correspondence and evaluations of specific MZV families.

assumptions (5)

domain assumption Convergence and admissibility of iterated integrals and multiple polylogarithm series on the specified domains (a ∈ [-1,0)∪(0,1) etc.)
Invoked throughout Section 2 to justify changes of variables and the integral-series identity (2.1); standard convergence conditions are not proved here.
standard math Stuffle (harmonic) product relations for multiple harmonic sums and star sums
Used in Section 3 to expand products ζn(k1)ζ⋆n(k2,k3) into sums of harmonic numbers, a step in Theorem 3.4 and its corollaries.
domain assumption Correspondence between multiple polylogarithms at 1/2 and unit-exponent alternating MZVs, Eq. (2.4)
Cited from Borwein et al. Eq. (6.8) and Zlobin Corollary 5; this correspondence is the bridge that converts the polylog identities into identities for unit-exponent alternating MZVs.
domain assumption Evaluations of ζ({2}a,3,{2}b) and ζ({2}a,1,{2}b) as rational linear combinations of products of Riemann zeta values
Relied on in Corollary 5.6 and cited from [17,26,28]; without these evaluations the Kaneko-Yamamoto results would not reduce to products of single zeta values.
domain assumption The (p+2)-poset integral identity (6.10) and Proposition 6.1 hold when the poset is admissible
The final section generalizes Yamamoto's 2-poset to (p+2)-posets and asserts the integral equals a Kaneko-Yamamoto-type series; the convergence is encoded in admissibility and the proof is sketched via repeated integration.

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Pith. "Pith review of Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values." pith.science (2026). https://pith.science/paper/74BER5OG

@misc{pith2026190803065,
  author       = {Pith},
  title        = {Pith review of: Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/74BER5OG}},
  note         = {Machine review of arXiv:1908.03065}
}

read the original abstract

In this paper we consider iterated integrals of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst \cite{BBBL1997}, and give some general results. Furthermore, we discuss Kaneko-Yamamoto multiple zeta values, and establish some relations between it and multiple zeta values. Finally, we establish a linear relation identity of alternating multiple zeta values.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

Arakawa, M

T. Arakawa, M. Kaneko, Multiple zeta values, poly-Bernoulli numb ers, and related zeta functions, Nagoya Math. J., 153(1999), 189-209

work page 1999

[2] [2]

Bayad, Y

A. Bayad, Y. Hamahata, Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials, J. Number Theory, 131(2011), 1020-1036

work page 2011

[3] [3]

Borwein, D.M

J.M. Borwein, D.M. Bradley and D.J. Broadhurst. Evaluations of k-fold Euler/Zagier sums: a com- pendium of results for arbitrary k . Electron. J. Combin., 1997, 4(2): 1-21. 25

work page 1997

[4] [4]

Borwein, D.M

J.M. Borwein, D.M. Bradley, D.J. Broadhurst and Petr. Lison˘ ek. Special values of multiple polylog- arithms. Trans. Amer. Math. Soc., 2001, 353(3): 907-941

work page 2001

[5] [5]

Coppo, B

M.A. Coppo, B. Candelpergher, The Arakawa-Kaneko zeta func tion, Ramanujan J., 22(2010), 153- 162

work page 2010

[6] [6]

Coppo, B

M.A. Coppo, B. Candelpergher, Inverse binomial series and value s of Arakawa-Kaneko zeta functions, J. Number Theory, 150(2015), 98-119

work page 2015

[7] [7]

Broadhurst

D.J. Broadhurst. Exploiting the 1,440-fold symmetry of the master two-loop diagram . Z. Phys. C Part. Fields, 1986, 32: 249-253

work page 1986

[8] [8]

Hirose, N

M. Hirose, N. Sato. Iterated integrals on P1 \ {0, 1, ∞, z } and a class of relations among multiple zeta values. Adv. Math., 2019, 348: 163-182

work page 2019