On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series
Pith reviewed 2026-05-21 01:45 UTC · model grok-4.3
The pith
The coefficient of q to the power N times two to the a in this double Lambert series equals the sum of the divisors of N.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors establish that the coefficient of q^{N 2^a} in the expansion of sum_{m,k >=1} q^{mk 2^a} / [(1 + q^{k 2^{a-1}})(1 - q^{2m-1})] equals sigma_1(N). They prove this by converting the double Lambert series into a single sum, which simultaneously supplies a new representation of the quasi-modular form E_2(q).
What carries the argument
The algebraic transformation that rewrites the given double Lambert series as an equivalent single sum and thereby yields a new expression for E_2(q).
If this is right
- The equality holds for every positive integer N and every nonnegative integer a.
- A new representation of the quasi-modular form E_2(q) follows directly from the series transformation.
- The sum-of-divisors function sigma_1(N) appears as the coefficient of q^{N 2^a} in this specific double Lambert series.
Where Pith is reading between the lines
- The same transformation technique might extend to Lambert series built on other integer bases instead of powers of two.
- Analogous coefficient identities could exist for higher-order divisor functions such as sigma_k for k greater than 1.
- The new expression for E_2(q) may simplify certain calculations involving quasi-modular forms in partition theory.
Load-bearing premise
The algebraic transformation that converts the double Lambert series into an equivalent single sum remains valid for all positive integers N and all relevant a without further restrictions.
What would settle it
Extract the coefficient of q^{N 2^a} directly from the original double sum for small concrete values such as a equals 1 and N equals 3, then check whether the result equals 4, which is sigma_1(3).
read the original abstract
In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series. More precisely, they conjectured that \[ \coeff{q^{N2^a}} \sum_{m,k\geq 1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =\sigma_1(N), \] where $\sigma_1(N)$ is the sum of all the positive divisors of $N$. The main idea of the proof is to first transform a double Lambert series on the left-hand side into a single sum. This leads us to derive a new representation of quasi-modular forms $E_2(q)$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves the conjecture of Amdeberhan, Andrews and Ballantine that the coefficient of q^{N 2^a} in the double Lambert series sum_{m,k >=1} q^{mk 2^a} / [(1 + q^{k 2^{a-1}})(1 - q^{2m-1})] equals sigma_1(N). The proof first transforms the double series into a single sum and then obtains a new representation for the quasi-modular form E_2(q).
Significance. If the central identity is established, the result confirms a recent conjecture on Lambert series and supplies a novel expression for E_2(q). The direct algebraic derivation without ad-hoc parameters or self-referential fitting is a clear strength of the manuscript.
major comments (2)
- [Main proof, double-to-single sum transformation] The algebraic transformation of the double Lambert series into a single sum (the first main step of the proof) requires explicit justification of the geometric expansions. The factor 1/(1 + q^{k 2^{a-1}}) produces an alternating series whose signs must cancel exactly against contributions from 1/(1 - q^{2m-1}); without a term-by-term accounting or a formal reordering argument valid for all a >= 1 and all positive integers N (including even N), it is unclear whether the extracted coefficient of q^{N 2^a} is guaranteed to be sigma_1(N).
- [Derivation of E_2(q) representation] The new representation of E_2(q) obtained after the transformation is used to conclude the coefficient identity. This representation should be stated explicitly (with the precise range of summation and any convergence conditions) and its equivalence to the classical definition of E_2(q) verified by direct coefficient comparison or by appeal to a known identity, so that the final step does not rest on an unverified equivalence.
minor comments (2)
- [Abstract] The abstract states that the work leads to a new representation of E_2(q) but does not indicate its form; a one-sentence description would improve readability.
- Ensure consistent numbering of all displayed equations and that the coefficient-extraction operator is defined once at the beginning.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the constructive comments on the proof structure. We address each major comment below and will incorporate the suggested clarifications in the revised version.
read point-by-point responses
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Referee: The algebraic transformation of the double Lambert series into a single sum (the first main step of the proof) requires explicit justification of the geometric expansions. The factor 1/(1 + q^{k 2^{a-1}}) produces an alternating series whose signs must cancel exactly against contributions from 1/(1 - q^{2m-1}); without a term-by-term accounting or a formal reordering argument valid for all a >= 1 and all positive integers N (including even N), it is unclear whether the extracted coefficient of q^{N 2^a} is guaranteed to be sigma_1(N).
Authors: We agree that additional detail is warranted to make the geometric expansions fully rigorous. In the revised manuscript we will insert a dedicated subsection that carries out the expansions explicitly: we expand 1/(1 + q^{k 2^{a-1}}) as an alternating geometric series and 1/(1 - q^{2m-1}) as the usual geometric series, then collect like powers of q. We will provide a term-by-term accounting that exhibits the precise cancellation of the alternating signs and supply a formal reordering argument (justified by absolute convergence inside the unit disk) that holds uniformly for every integer a ≥ 1 and every positive integer N, including even N. Small explicit examples for even and odd N will be included to illustrate the cancellation before the general argument. revision: yes
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Referee: The new representation of E_2(q) obtained after the transformation is used to conclude the coefficient identity. This representation should be stated explicitly (with the precise range of summation and any convergence conditions) and its equivalence to the classical definition of E_2(q) verified by direct coefficient comparison or by appeal to a known identity, so that the final step does not rest on an unverified equivalence.
Authors: We accept the referee’s observation. The revised manuscript will state the new representation of E_2(q) in full, specifying the exact range of the summation indices and the region of convergence (the open unit disk). Equivalence to the classical Eisenstein series E_2(q) will be established by direct comparison of Fourier coefficients: we will show that the coefficient of q^n on both sides equals -24 times the sum of the divisors of n (with the usual convention for the constant term). This comparison will be carried out explicitly for the first several powers and then extended by the general formula for the divisor sum. If space permits we will also note the relation to known identities in the literature on quasi-modular forms. revision: yes
Circularity Check
Algebraic series transformation yields coefficient identity without reducing to fitted inputs or self-citations
full rationale
The derivation begins with the double Lambert series and applies direct algebraic manipulations (partial fractions and geometric series expansions) to convert it into an equivalent single sum. The coefficient of q^{N 2^a} is then extracted term-by-term and shown to equal sigma_1(N) by matching divisors, while the same manipulation produces a new generating-function identity for E_2(q). These steps rely on standard q-series identities and the definition of sigma_1(N) as the divisor sum; no parameters are fitted to data, no result is renamed as a prediction, and no load-bearing premise is justified solely by prior work of the same authors. The chain is therefore self-contained and externally verifiable by direct expansion.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Algebraic identities for manipulating double Lambert series into single sums hold in the ring of formal power series.
- domain assumption E2(q) admits representations derived from generating functions involving divisor sums.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 2: X_{m,k≥1} q^{2mk}/((1+q^k)(1-q^{2m-1})) = X_{n≥1} q^{2n}/(1-q^{2n})^2 = X_{N≥1} σ₁(N) q^{2N}
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IndisputableMonolith/Foundation/ArithmeticFromLogic.leanLogicNat recovery unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Proof of Theorem 1 reduces the general-a case to the a=1 identity via substitution x = q^{2^{a-1}} and congruence arguments on ℓ
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
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[1]
T. Amdeberhan, G. E. Andrews and C. Ballantine,Lambert series and double Lambert series, J. Combin. Theory Ser. A221(2026), Paper No. 106154, 22 pp
work page 2026
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[2]
G. E. Andrews, A. Dixit, D. Schultz and A. J. Yee,Overpartitions related to the mock theta functionω(q), Acta Arith.181(2017), no. 3, 253–286
work page 2017
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[3]
S. P. Cui and D. Tang,Identities and transformations for Lambert series and double Lambert series, preprint, https://arxiv.org/pdf/2604.08839, 2026
work page internal anchor Pith review Pith/arXiv arXiv 2026
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[4]
On the Double Lambert Series Conjecture of Andrews-Dixit--Schultz-Yee
Q. Fang,On the double Lambert series conjecture of Andrews-Dixit-Schultz-Yee, preprint, https://arxiv.org/pdf/2604.06242, 2026
work page internal anchor Pith review Pith/arXiv arXiv 2026
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[5]
J. H. Lambert,Anlage zur Architectonic, oder Theorie des ersten und des ein- fachen in der philosophischen und mathematischen Erkenntnis, Vol. 2, Johann Friedrich Hartenoch, Riga, 1771, Philosophische Schriften, vol. 4, Georg Olm, Hildesheim, 1965. Department of Mathematics, Indian Institute of Technology, Roorkee- 247667, Uttarakhand, India Email address...
work page 1965
discussion (0)
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