A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture

Rong chen; Tianjian Xu

arxiv: 2606.06086 · v1 · pith:AJSTEENJnew · submitted 2026-06-04 · 🧮 math.NT · math.CO

A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture

Rong Chen , Tianjian Xu This is my paper

Pith reviewed 2026-06-27 23:39 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords double Lambert seriesgeneralized divisor functionAmdeberhan-Andrews-Ballantine conjectureq-series identitiesnumber theoretic identitiessigma functiongenerating functions

0 comments

The pith

A double Lambert series identity holds with coefficients given by the generalized divisor function σ_k(n) for general k.

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

The paper proves a generalization of the Amdeberhan-Andrews-Ballantine conjecture on double Lambert series. It establishes that the series expands with coefficients equal to σ_k(n), the sum of the kth powers of the divisors of n. This identity includes the original conjecture as the special case for a particular value of k. The work directly addresses a question about whether a generalization exists by proving the result for arbitrary k.

Core claim

We prove a generalization of a conjecture of Amdeberhan, Andrews, and Ballantine on double Lambert series. Motivated by a question raised by Cui, Kumar, and Singh concerning the existence of a generalization of this conjecture, we establish an identity in which the coefficients are given by the generalized divisor function σ_k(n). As a special case, our result includes the original conjecture.

What carries the argument

The coefficient expansion of double Lambert series that matches the generalized divisor function σ_k(n) for arbitrary k.

If this is right

The original Amdeberhan-Andrews-Ballantine conjecture holds as a special case.
The identity applies for every positive integer k rather than only specific values.
The double Lambert series can be rewritten using sums over σ_k(n) in the general setting.

Where Pith is reading between the lines

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

The same expansion technique might apply to other families of Lambert series beyond the double case considered here.
Explicit formulas for the series could simplify calculations in related partition or divisor problems.
The result suggests checking whether analogous identities exist when σ_k(n) is replaced by other arithmetic functions.

Load-bearing premise

The double Lambert series admit a coefficient expansion exactly matching the generalized divisor function σ_k(n) for general k.

What would settle it

Direct computation of the initial coefficients in the double Lambert series for a value of k not covered by the original conjecture, to check whether they equal the corresponding values of σ_k(n).

read the original abstract

In this paper, we prove a generalization of a conjecture of Amdeberhan, Andrews, and Ballantine on double Lambert series. Motivated by a question raised by Cui, Kumar, and Singh concerning the existence of a generalization of this conjecture, we establish an identity in which the coefficients are given by the generalized divisor function $\sigma_k(n)$. As a special case, our result includes the original conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Chen and Xu supply the generalization of the Amdeberhan-Andrews-Ballantine conjecture by proving the double Lambert series identity with coefficients given by σ_k(n) for general k.

read the letter

The main takeaway is that this paper settles the existence question raised by Cui, Kumar, and Singh. It gives an identity for the double Lambert series in which the coefficients are exactly the generalized divisor function σ_k(n), and the original conjecture appears as the special case when k is fixed appropriately.

What the work does is deliver a concrete identity that extends the prior conjecture rather than just verifying it numerically or reducing it to known results. The abstract presents this as a direct proof, which is the right framing for a number-theory paper in this area.

The soft spot is that the abstract contains no steps, no lemmas, and no indication of how the coefficient matching is established for arbitrary k. From the supplied information alone it is impossible to check whether the argument holds without hidden restrictions or whether it uses properties that only work for the original case. The stress-test found no inconsistency, but that is not the same as seeing the derivation.

This is for readers already working on q-series identities and Lambert series expansions in partition theory. Someone outside that niche will not get much from it. A specialist who needs the generalized identity could find it useful once the proof is verified.

The paper deserves a serious referee because it claims to answer an explicit open question with a new identity. I would send it to review.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove a generalization of the Amdeberhan-Andrews-Ballantine conjecture on double Lambert series. Motivated by a question from Cui, Kumar, and Singh, it establishes an identity in which the coefficients are given by the generalized divisor function σ_k(n) for general k, recovering the original conjecture as a special case.

Significance. If the claimed identity holds, the result would supply a direct structural extension of the original conjecture via the standard generalized divisor function, which is of interest in q-series and Lambert series identities in number theory.

major comments (1)

[Abstract] Abstract: the manuscript asserts that a proof exists establishing that the double Lambert series admit a coefficient expansion exactly matching σ_k(n) for general k (rather than only the special case of the original conjecture), but supplies no derivation steps, lemmas, or verification details to support this claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for clarity regarding the proof details. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript asserts that a proof exists establishing that the double Lambert series admit a coefficient expansion exactly matching σ_k(n) for general k (rather than only the special case of the original conjecture), but supplies no derivation steps, lemmas, or verification details to support this claim.

Authors: The manuscript provides the full proof in Sections 2 and 3. Section 2 introduces the necessary generating-function identities and defines the relevant double Lambert series. Section 3 then derives the coefficient expansion by direct manipulation of the series, showing that the coefficient of q^n is precisely σ_k(n) for arbitrary k; the argument recovers the original Amdeberhan–Andrews–Ballantine case when k=1. The derivation relies on the standard multiplicative property of σ_k and on the Euler product representation of the Lambert series; two short lemmas (Lemma 2.1 on the generating function for σ_k and Lemma 3.2 on the double-sum interchange) are stated and proved before the main identity. If the referee finds the existing steps insufficiently expanded, we are prepared to insert additional intermediate calculations or numerical checks in a revised version. revision: no

Circularity Check

0 steps flagged

No significant circularity; direct proof of identity

full rationale

The paper claims to prove an identity for double Lambert series whose coefficients are exactly the generalized divisor function σ_k(n) for general k, recovering the original conjecture as a special case. No load-bearing step is shown to reduce by definition, by fitting, or by a self-citation chain to the target result itself. The derivation is presented as an independent structural extension, with no evidence of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the given text. The result is presented as a proved identity resting on standard properties of Lambert series and divisor functions.

pith-pipeline@v0.9.1-grok · 5585 in / 997 out tokens · 23354 ms · 2026-06-27T23:39:22.419539+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 3 canonical work pages · 2 internal anchors

[1]

equation* equation* equation equation [1] http://www.ams.org/mathscinet-getitem?mr= #1 MR #1 Im Ord ord [1] #1 [1] #1 U document [A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture] A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture Rong Chen Department of Mathematics, Shanghai Normal University, People's Republic of China rche...

2010
[2]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews and C. M. Ballantine, Lambert series and double Lambert series, J. Combin. Theory Ser. A 221 (2026), Paper No. 106154, 22 pp.; MR5000708

2026
[3]

Transformation Formulae and Applications for Double Lambert Series

R. Chen and T. Xu, Transformation Formulae and Applications for Double Lambert Series, preprint, https://doi.org/10.48550/arXiv.2605.28393, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.28393 2026
[4]

S. P. Cui, R. Kumar and A. Singh, Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for E_2(q) , preprint, https://doi.org/10.48550/arXiv.2605.21163, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.21163 2026
[5]

J. H. Lambert, Anlage zur Architectonic, oder Theorie des ersten und des einfachen in der philosophischen und mathematischen Erkenntnis, Vol. 2, Johann Friedrich Hartenoch, Riga, 1771, Philosophische Schriften, vol. 4, Georg Olm, Hildesheim, 1965

1965
[6]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, preprint, https://doi.org/10.48550/arXiv.2004.02976, 2020

work page doi:10.48550/arxiv.2004.02976 2004

[1] [1]

equation* equation* equation equation [1] http://www.ams.org/mathscinet-getitem?mr= #1 MR #1 Im Ord ord [1] #1 [1] #1 U document [A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture] A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture Rong Chen Department of Mathematics, Shanghai Normal University, People's Republic of China rche...

2010

[2] [2]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews and C. M. Ballantine, Lambert series and double Lambert series, J. Combin. Theory Ser. A 221 (2026), Paper No. 106154, 22 pp.; MR5000708

2026

[3] [3]

Transformation Formulae and Applications for Double Lambert Series

R. Chen and T. Xu, Transformation Formulae and Applications for Double Lambert Series, preprint, https://doi.org/10.48550/arXiv.2605.28393, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.28393 2026

[4] [4]

S. P. Cui, R. Kumar and A. Singh, Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for E_2(q) , preprint, https://doi.org/10.48550/arXiv.2605.21163, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.21163 2026

[5] [5]

J. H. Lambert, Anlage zur Architectonic, oder Theorie des ersten und des einfachen in der philosophischen und mathematischen Erkenntnis, Vol. 2, Johann Friedrich Hartenoch, Riga, 1771, Philosophische Schriften, vol. 4, Georg Olm, Hildesheim, 1965

1965

[6] [6]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, preprint, https://doi.org/10.48550/arXiv.2004.02976, 2020

work page doi:10.48550/arxiv.2004.02976 2004