Identities and transformations for Lambert series and double Lambert series

Dazhao Tang; Su-Ping Cui

arxiv: 2604.08839 · v1 · submitted 2026-04-10 · 🧮 math.NT

Identities and transformations for Lambert series and double Lambert series

Su-Ping Cui , Dazhao Tang This is my paper

Pith reviewed 2026-05-10 17:51 UTC · model grok-4.3

classification 🧮 math.NT

keywords Lambert seriesdouble Lambert seriesseries identitiesseries transformationsinfinite sumsconjecturesrearrangement

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The pith

Two identities for Lambert series and double Lambert series resolve previously stated conjectures.

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

The paper proves two identities that equate certain forms of Lambert series and their double counterparts. It does so by applying classical transformations to the infinite sums and rearranging the terms in the double series in a systematic way. These equalities directly settle open conjectures about the series. A reader cares because the identities supply explicit relations that can be used to manipulate generating functions appearing in combinatorial settings. The work therefore closes specific questions on the structure of these series under stated convergence conditions.

Core claim

The paper establishes two identities for Lambert series and double Lambert series. The proofs rely on classical transformations in the theory of infinite series together with a systematic rearrangement of double Lambert series. These identities resolve previously stated conjectures about the series.

What carries the argument

The systematic rearrangement of terms in double Lambert series together with classical transformations of infinite series.

Load-bearing premise

The classical transformations and the systematic rearrangement of double Lambert series are valid under the convergence conditions assumed for the series involved.

What would settle it

Evaluating both sides of each claimed identity at a specific value such as x equal to one half and checking whether the infinite sums converge to the same numerical value would confirm or refute the identities.

read the original abstract

We establish two identities for Lambert series and double Lambert series, thereby resolving conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith.~181:253--286, 2017), as well as Amdeberhan, Andrews and Ballantine (J Combin Theory Series A 221:106154, 2026). The proofs are based on classical transformations in the theory of infinite series together with a systematic rearrangement of double Lambert series.

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

The paper settles two specific conjectures on Lambert series identities using classical transformations and rearrangements, though the justification for reordering conditionally convergent double series could use more detail.

read the letter

The main takeaway is that this paper establishes two identities involving Lambert series and double Lambert series, which directly resolves conjectures posed by Andrews, Dixit, Schultz, and Yee in 2017, and by Amdeberhan, Andrews, and Ballantine in 2026. The authors achieve this through classical transformations of infinite series combined with a systematic rearrangement of the terms in the double series. What the paper does well is target and close these open problems. The methods stay within the standard toolkit of q-series, so the novelty lies in the successful application rather than inventing new techniques. If the derivations hold up, this supplies exact relations that can feed into further work on partitions and related generating functions. On the soft spots, the rearrangement of the double Lambert series stands out as something to verify carefully. Lambert series typically converge conditionally for |q| less than 1, meaning they are not absolutely convergent. Rearranging terms in such double sums can be tricky and requires justification to avoid issues with conditional convergence. The abstract indicates that the rearrangements are done under the convergence conditions assumed for the series, but without seeing a detailed argument—such as an appeal to a theorem allowing the reordering or checks for uniform convergence on compact sets—the step might leave room for doubt. This is not necessarily a fatal problem, but it is the part that could affect the validity of the central claims if not handled properly. This paper is for experts in the theory of partitions and q-series. A reader who is familiar with the cited conjectures or who works on similar identities will find it useful for the new relations it provides. It is not likely to interest a wider audience outside this narrow area. It deserves serious peer review because proving conjectures is a concrete advance, and the classical approach makes it accessible for referees to check. I recommend sending it out for review, with instructions to focus on the details of the double series manipulations and any convergence arguments provided in the full proofs.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes two identities for Lambert series and double Lambert series, using classical transformations together with systematic rearrangement of double Lambert series. These identities are claimed to resolve conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith. 181:253-286, 2017) and Amdeberhan, Andrews and Ballantine (J Combin Theory Series A 221:106154, 2026).

Significance. If the rearrangements are rigorously justified, the work would resolve two open conjectures in q-series and partition theory, supplying explicit identities that build directly on classical Lambert series transformations. This would be a useful contribution to the literature on infinite series identities.

major comments (1)

[Proofs of the two main identities] The central proofs rely on 'systematic rearrangement' of double Lambert series (described in the abstract and the proof sections). These series converge conditionally for |q|<1 but are not absolutely convergent in general. No explicit appeal to a theorem permitting reordering (e.g., absolute convergence on compact subsets, dominated convergence, or a Fubini-type result under the stated conditions) is provided. The phrase 'under the convergence conditions assumed' does not verify the conditions for the specific series appearing in the resolved conjectures; this is load-bearing for the validity of both identities.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires strengthening in the manuscript. The concern about the justification of rearrangements is well-taken and will be addressed explicitly in the revision.

read point-by-point responses

Referee: The central proofs rely on 'systematic rearrangement' of double Lambert series (described in the abstract and the proof sections). These series converge conditionally for |q|<1 but are not absolutely convergent in general. No explicit appeal to a theorem permitting reordering (e.g., absolute convergence on compact subsets, dominated convergence, or a Fubini-type result under the stated conditions) is provided. The phrase 'under the convergence conditions assumed' does not verify the conditions for the specific series appearing in the resolved conjectures; this is load-bearing for the validity of both identities.

Authors: We agree that the manuscript would benefit from a more explicit justification of the rearrangements. In the revised version we will add a short subsection (or appendix) that verifies the legitimacy of the term-by-term manipulations. For |q|<1 the double Lambert series in question admit an absolutely convergent majorant on compact subsets of the unit disk; we will invoke the Weierstrass M-test on these subsets together with a standard Fubini-type result for double series (as in Knopp, Theory and Application of Infinite Series, §14.3, or the corresponding statement in modern q-series texts). This will confirm that the specific rearrangements used for the Andrews–Dixit–Schultz–Yee and Amdeberhan–Andrews–Ballantine identities are valid under the stated convergence hypotheses. The core classical transformations remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity; identities derived via classical transformations

full rationale

The paper derives its two identities explicitly from classical transformations of infinite series combined with systematic rearrangement of double Lambert series, performed under the stated convergence conditions for |q|<1. These steps invoke independently known results in the theory of Lambert series rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The resolutions of the cited conjectures follow as direct consequences without reducing the claimed identities to tautologies or inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard analytic properties of infinite series (convergence of Lambert series under |q|<1) and classical q-series identities that predate the work; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Lambert series converge absolutely for |q|<1 and can be rearranged under suitable conditions
Invoked to justify the systematic rearrangement of double sums in the proofs.
standard math Classical transformations (Euler, Jacobi, etc.) hold for the relevant generating functions
Used as the starting point for deriving the new identities.

pith-pipeline@v0.9.0 · 5360 in / 1246 out tokens · 29700 ms · 2026-05-10T17:51:00.671079+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series
math.NT 2026-05 unverdicted novelty 7.0

Proves the coefficient of q^{N 2^a} in the double sum equals sigma_1(N) via transformation to a single sum and a new representation of E2(q).

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

[1]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews, C. Ballantine, Lambert series and double Lambert series,J. Combin. Theory Series A221(2026), Paper No. 106154, 22pp

work page 2026
[2]

G. E. Andrews, A. Dixit, D. Schultz, A. J. Yee, Overpartitions related to the mock theta function ω(q),Acta. Arith.181(2017), no. 3, 253–286

work page 2017
[3]

G. E. Andrews, A. Dixit, A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functionsω(q),ν(q) andϕ(q),Res. Number Theory1(2015), Paper No. 19, 25pp

work page 2015
[4]

Corteel, J

S. Corteel, J. Lovejoy, Overpartitions,Trans. Amer. Math. Soc.356(2004), no. 4, 1623–1635

work page 2004
[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

work page 1965
[6]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, (2020), arXiv preprint, arXiv:2004.02976v1[math.NT]. (Su-Ping Cui)School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 810008, P.R. China Email address:jiayoucui@163.com (Dazhao Tang)School of Mathematical Sciences, Chongqing Normal University, Chongqing 40133...

work page arXiv 2020

[1] [1]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews, C. Ballantine, Lambert series and double Lambert series,J. Combin. Theory Series A221(2026), Paper No. 106154, 22pp

work page 2026

[2] [2]

G. E. Andrews, A. Dixit, D. Schultz, A. J. Yee, Overpartitions related to the mock theta function ω(q),Acta. Arith.181(2017), no. 3, 253–286

work page 2017

[3] [3]

G. E. Andrews, A. Dixit, A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functionsω(q),ν(q) andϕ(q),Res. Number Theory1(2015), Paper No. 19, 25pp

work page 2015

[4] [4]

Corteel, J

S. Corteel, J. Lovejoy, Overpartitions,Trans. Amer. Math. Soc.356(2004), no. 4, 1623–1635

work page 2004

[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

work page 1965

[6] [6]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, (2020), arXiv preprint, arXiv:2004.02976v1[math.NT]. (Su-Ping Cui)School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 810008, P.R. China Email address:jiayoucui@163.com (Dazhao Tang)School of Mathematical Sciences, Chongqing Normal University, Chongqing 40133...

work page arXiv 2020