REVIEW 1 major objections 4 minor 5 references
A Weighted Sum Formula for Double Eisenstein Series
T0 review · 1 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read Every depth-one Eisenstein series of weight k≥6 is a finite weighted sum of double Eisenstein series with explicit rational coefficients.
desk verdict A short, correct proof of a conjectured weighted sum formula for double Eisenstein series; the only real gap is an unproved but elementary generating-polynomial identity that should be filled before publication. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument turns on three pieces. First, the restricted double-shuffle relations (2.4) say the shuffle-regularized Eisenstein series map kills the difference between the stuffle product and the shuffle product on depth-two words. Second, a coefficient criterion (2.5), adapted from the earlier literature, converts a polynomial identity in two variables into the statement that a certain combination of depth-two words lies in the span of double-shuffle relations, with a constant term λ computed by an integral. Third, the generating-polynomial identity (2.3), Σ μ_{i,k} X^i Y^{k-2-i} = XY( Y(X+Y)(2X+Y)^{k-6} − (X−Y)^2(X+Y)^{k-6} ), packages all the coefficients μ_{i,k} that define α_{a,b}; it i
What would settle it
Expand both sides of (2.3) for a small weight such as k=7 or k=8: any mismatch in the polynomial coefficients would refute the proof chain. Independently, compute the q-expansion of the difference between the two sides of (1.4) for k=6 through, say, order q^{20}; any nonzero coefficient would disprove the weighted-sum formula itself.
Extended reading notes
Core claim
Main Theorem A states: for every integer k≥6, G_k = Σ_{a+b=k, a≥3, b≥2} α_{a,b} G_{a,b}, where α_{a,b} is given by the closed formula 60/b · binom(k-1,a-1)^{-1} · [ (2^{a-2}-1) C(k-6,a-2) + (2^{a-3}+2) C(k-6,a-3) - C(k-6,a-4) ]. Main Theorem B gives the companion identity for the q-series g_k, with the left-hand side g_k - 5/((k-1)(k-2)) g_{k-2} + 4/((k-1)(k-2)(k-3)(k-4)) g_{k-4} equal to the same weighted sum of g_{a,b}. The two forms are compatible via Fourier expansion, and both are proven in the note: Theorem A follows from restricted double-shuffle relations, and Theorem B is proven combinatorially via exponential generating series. The core discovery is the explicit coefficient identit
Load-bearing premise
The load-bearing premise is an unsupported polynomial identity (2.3) expressing the coefficients as a product of linear factors; both proofs collapse if it is false.
Editorial extensions
If this is right
- For every weight k≥6, the depth-one Eisenstein series G_k is a rational linear combination of depth-two series G_{a,b}, so the depth-one part of the multiple Eisenstein algebra is determined by depth two up to α-coefficients.
- The q-series version (1.4), multiplied by (1−q)^k and sent to q→1−, recovers the weighted sum formula ζ(k)=Σ α_{a,b} ζ(a,b) for double zeta values.
- The same coefficients α_{a,b} work in both the modular-form setting and the divisor-sum setting, reinforcing the expectation that multiple Eisenstein series and the q-series g satisfy identical linear relations modulo lower-weight terms.
- Each individual relation for a fixed k can be checked by expanding both sides in q; the paper includes the first four explicit examples k=6,7,8,9.
- The proof also yields the explicit lower-weight corrections g_{k-2} and g_{k-4} in Theorem B, so the homogeneous weight-k relation of Theorem A is corrected in a precise way when passing to q-series.
Reading between the lines
- The same coefficient-criterion machinery might produce relations at depth three or higher if a matching generating-polynomial identity for the higher-depth coefficients can be found; the paper only treats depth two.
- Because the unproved identity (2.3) is the single unsupported step, an independent verification by a short induction would make the proof chain fully self-contained; the identity is elementary enough to check for any fixed k.
- The two independent proofs suggest the weighted-sum structure is robust: the combinatorial generating-series method does not rely on the double-shuffle theorem, so it may extend to other families of divisor sums where shuffle relations are not available.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a weighted sum formula for double Eisenstein series. Main Theorem A (Eq. (1.2)) states that for every k≥6, the single Eisenstein series G_k lies in the Q-span of double Eisenstein series G_{a,b} (a+b=k, a≥3, b≥2) with explicit rational coefficients α_{a,b} given in (1.1). Main Theorem B (Eq. (1.4)) gives the corresponding identity for the q-series g_k and g_{a,b}, including lower-weight correction terms g_{k-2} and g_{k-4}. The proof of Theorem A uses the restricted double-shuffle relations from [BT] and a coefficient-polynomial criterion (2.5) of [GKZ]. Theorem B is proved independently by manipulating exponential generating series and applying a differential operator. The paper also records the first four instances of the weight-k identity and relates the q-series statement to a conjecture from the author's master's thesis.
Significance. If the results are correct, this provides the first proved relation among multiple Eisenstein series in weight 6, together with a family of relations in every weight k≥6, and confirms a q-series conjecture. The two proofs are structurally appealing: the double-shuffle argument is short and conceptual, and the generating-series proof is self-contained and displays the lower-weight corrections explicitly. The numerical examples and the internal computations (the polynomial identity A_ρ = H_k(X,X+Y)-H_k(X,Y), the integral evaluation 30∫ t(1-t)(2t-1)^2 dt = 1, and the power-sum identity) are consistent. The paper is therefore significant within the arithmetic of multiple Eisenstein series and q-analogues of multiple zeta values. The main barrier to accepting it as rigorous is an unproved generating-polynomial identity that is load-bearing for both proofs.
major comments (1)
- [§2, Eq. (2.3)] The identity ∑_{i=2}^{k-3} μ_{i,k} X^i Y^{k-2-i} = XY( Y(X+Y)(2X+Y)^{k-6} − (X−Y)^2(X+Y)^{k-6} ) is stated without proof or citation. This identity is used in an essential way in Lemma 2.1 to rewrite A_ρ as H_k(X,X+Y)−H_k(X,Y), and again in the proof of Theorem B (Section 3, around (3.4)–(3.5)) to identify the coefficient of Z^{k-6}/(k−6)! with ∑ μ_{i,k} A_{i,k−2−i}(q). Both Main Theorems therefore depend on it. The identity is elementary and appears to be true (the small cases k=6,7,8 check out), but a rigorous paper must supply a proof, for instance by a direct binomial-coefficient verification or by comparing coefficients after expanding both sides. As written, this is a load-bearing gap, not a mere presentation issue.
minor comments (4)
- [§2, display after (2.1)] The identity (2.3) should be presented as a lemma with a proof, or at least with an indication of where it is proved. Since it is used twice in load-bearing positions, labeling it as an unnumbered display makes it easy to miss.
- [§2, proof of (2.5)] The derivation of the coefficient criterion (2.5) is compressed. The statement that the polynomials Q_a form a basis of the relevant symmetric homogeneous polynomials is correct, but a sentence explaining the counting (dimension ⌊k/2⌋−1) would help the reader.
- [§3, after (3.4)] The transition from the polynomial in v_1,v_2 to the coefficient expression (3.5) is made in one line. Expanding one step to show how (2.3) is applied with X=v_1, Y=v_2 would make the proof easier to follow.
- [§1, Eq. (1.3)] The display for g_{k_1,...,k_r} is slightly hard to read because of the superscript/subscript layout. The formula itself is standard and correct.
Circularity Check
No circularity: both proofs reduce to independent [BT] relations, the [GKZ] coefficient criterion, and a self-contained generating-series computation; the unproved identity (2.3) is a rigor gap, not a circular step.
full rationale
The claimed derivation is not circular. Main Theorem A is obtained by combining Lemma 2.1 with the restricted double-shuffle relations from [BT, Theorem 1.2]. That cited theorem is an independent published result with stated assumptions (regularized multiple Eisenstein series) and does not contain the weighted sum formula; citing it is standard use of prior work, not a repackaging of the target. The coefficient criterion (2.5) is a restriction of [GKZ, Proposition 2] and is proved within the paper, so it is self-contained. Main Theorem B is proved directly from the generating series T(X), T(X,Y) via the stuffle decomposition (3.2), the rearrangement (3.3), and the differential operator L; no fitted parameter is renamed as a prediction. The master's-thesis conjecture [B, Vermutung 5.2.7] is only motivation and is not assumed. The one substantive gap is the generating-polynomial identity (2.3): it is stated without proof after (2.1) and is then used to rewrite A_rho in Lemma 2.1 and to extract the coefficient (3.5) in Theorem B. This is an omitted proof / correctness risk, not a circular step, because (2.3) is a purely algebraic identity in X,Y that does not by itself assert any relation among G or g; failure of (2.3) would invalidate the proofs, but that would be an error, not a circularity. No step reduces to its own input by construction.
Assumptions & free parameters
assumptions (4)
- domain assumption Restricted double-shuffle relations hold for shuffle-regularized multiple Eisenstein series (2.4): G^shuffle(ds(u,v)) = 0 for u,v ∈ H^{≥2}.
- standard math Polynomial criterion (2.5) from [GKZ, Proposition 2]: if A_ρ(X,Y) = H(X,X+Y) − H(X,Y), then Σ ρ_{r,s} z_r z_s − λ z_k lies in the span of double-shuffle differences.
- ad hoc to paper Generating polynomial identity (2.3) for μ_{i,k}.
- standard math Generating series decompositions (3.2) and (3.3) for T(X)T(Y).
Cite this review
Pith. "Pith review of A Weighted Sum Formula for Double Eisenstein Series." pith.science (2026). https://pith.science/paper/BNBEQZ5C
@misc{pith2026260721358,
author = {Pith},
title = {Pith review of: A Weighted Sum Formula for Double Eisenstein Series},
year = {2026},
howpublished = {\url{https://pith.science/paper/BNBEQZ5C}},
note = {Machine review of arXiv:2607.21358}
}
read the original abstract
We prove a weighted sum formula for double Eisenstein series. Its corresponding identity for the generating series of multiple divisor sums was conjectured by the author in his master's thesis. The double Eisenstein series identity follows from the restricted double-shuffle relations proved by the author and Tasaka, while the proof of the divisor-sum identity is combinatorial and uses generating series.
Reference graph
Works this paper leans on
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[1]
Bachmann, Multiple Zeta-Werte und die Verbindung zu Modulformen durch Multiple Eisensteinreihen, Master's thesis, Universit\"at Hamburg, 2012
H. Bachmann, Multiple Zeta-Werte und die Verbindung zu Modulformen durch Multiple Eisensteinreihen, Master's thesis, Universit\"at Hamburg, 2012. Available at https://www.henrikbachmann.com/uploads/7/7/6/3/77634444/msc_henrik_bachmann.pdf
2012
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[2]
Bachmann and U
H. Bachmann and U. K\"uhn, The algebra of generating functions for multiple divisor sums and applications to multiple zeta values, Ramanujan J. 40 (2016), no. 3, 605--648
2016
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[3]
H. Bachmann, H. Kanno and T. Maesaka, Relations and derivatives of multiple Eisenstein series, arXiv:2602.08176 https://arxiv.org/abs/2602.08176, 2026
arXiv 2026
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[4]
Bachmann and K
H. Bachmann and K. Tasaka, The double shuffle relations for multiple Eisenstein series, Nagoya Math. J. 230 (2018), 180--212
2018
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[5]
Gangl, M
H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, in: Automorphic Forms and Zeta Functions, World Scientific, 2006, 71--106
2006
Reviewed August 1, 2026 · model on record in the stance chip above.
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