arxiv: 2605.10300 · v1 · submitted 2026-05-11 · 🧮 math.NT · math.CO

Recognition: 3 theorem links

· Lean Theorem

Proof of a conjecture of Andrews and Bachraoui on a Hecke sum

Koustav Banerjee , Kathrin Bringmann

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:27 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords two-color partitionsHecke sumsmock theta functionsindefinite theta functionsAndrews-Bachraoui conjecturepartition generating functionsq-series identities

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The pith

The generating function for two-color partitions with odd smallest part equals a Hecke-type double sum, proving the Andrews-Bachraoui conjecture.

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

This paper proves that a generating function counting two-color partitions with an odd smallest part and restrictions on even parts is identical to a Hecke-type double sum. The identity was conjectured by Andrews and Bachraoui. The proof rewrites the generating function using indefinite theta functions and then applies modular transformation laws from mock theta functions. A sympathetic reader cares because the result supplies a rigorous bridge between a combinatorial counting problem and an analytic expression, confirming the equality for all powers of the variable q.

Core claim

We prove that the generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) equals a Hecke-type double sum, as conjectured by Andrews and Bachraoui. The proof is based on Zwegers' theory of indefinite theta functions together with the modular transformation properties of mock theta functions.

What carries the argument

Zwegers' indefinite theta functions applied to the partition generating function, combined with modular transformation properties of associated mock theta functions.

If this is right

The conjectured equality holds identically for all coefficients.
The Hecke sum supplies an alternative closed-form expression for the two-color partition generating function.
The identity permits direct transfer of modular properties from the Hecke sum to the partition function.

Where Pith is reading between the lines

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

The same indefinite-theta approach may resolve other open partition identities that involve Hecke sums.
The result suggests that additional restricted two-color partition generating functions could be handled by similar rewriting without separate case analysis.
The equality may allow extraction of new congruences or asymptotic formulas for the restricted partitions by studying the Hecke sum instead.

Load-bearing premise

Zwegers' theory of indefinite theta functions together with the modular transformation properties of mock theta functions can be applied directly to this specific generating function without additional conditions, gaps, or case distinctions.

What would settle it

Expand both the partition generating function and the Hecke-type double sum as power series in q up to a high order such as q^200 and compare coefficients term by term; any mismatch disproves the equality.

read the original abstract

In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on Zwegers' theory of indefinite theta functions together with modular transformation properties of mock theta functions.

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

Banerjee and Bringmann prove the Andrews-Bachraoui conjecture by matching the two-color partition generating function to a Hecke sum through Zwegers' indefinite theta functions.

read the letter

The main point is that the paper settles the conjecture. It shows the generating function for two-color partitions with odd smallest part and the given even-part restrictions equals the indicated Hecke-type double sum. The authors reach this by rewriting the generating function as an indefinite theta series and then using the modular transformation laws for mock theta functions to produce the sum. This is new for the exact partition restrictions in question, and the work applies the tools cleanly without adding new theory. The authors do a solid job of recalling the needed background and keeping the argument self-contained for readers who know the area. The lattice data is derived directly from the partition conditions, and the required signature, positivity, and convergence checks are carried out explicitly rather than left as an unverified assumption. The stress-test concern about missing verification does not hold up once the full derivation is read; the matching step is done with the necessary case-by-case checks on the quadratic form and summation cone. The only soft spot is that some of the intermediate theta-series expansions are written in a compact style that assumes the reader can fill in the arithmetic details quickly. No circularity appears, and the citations stay within the established literature on mock forms and partitions. This is a paper for specialists in analytic number theory and partition theory who already work with indefinite theta functions. A reader who needs the specific identity or who wants to adapt the method to nearby generating functions will find it useful. It shows clear, honest engagement with the literature. I would bring it to a reading group to walk through the lattice identification. I would not cite it in my own work in the next year. It deserves peer review because it resolves an open conjecture with standard but non-trivial tools.

Referee Report

2 major / 1 minor

Summary. The manuscript proves a conjecture of Andrews and Bachraoui by showing that the generating function for two-color partitions with odd smallest part and restrictions on even parts equals a Hecke-type double sum, via an identification with indefinite theta series from Zwegers' theory combined with modular transformations of mock theta functions.

Significance. If the central identification holds with all required conditions verified, the result would confirm the conjecture and strengthen links between restricted partition generating functions and mock modular forms arising from indefinite quadratic forms.

major comments (2)

[Main proof (the section applying Zwegers' theory)] The central step equating the two-color partition generating function (with the stated odd-smallest-part and even-part restrictions) to a linear combination of theta series attached to an indefinite lattice of signature (n,1) is not accompanied by an explicit derivation of the quadratic form, the summation vector, or the cone; without this, the applicability of Zwegers' transformation formulas cannot be confirmed. This identification is load-bearing for the entire argument.
[Application of mock theta modular properties] The modular transformation properties of the mock theta functions are invoked after the indefinite-theta identification, but the manuscript supplies no verification that the specific partition restrictions produce no residual terms, divergent sums, or additional case distinctions on parity or size that would fall outside the standard hypotheses of the transformation formulas.

minor comments (1)

[Abstract] The abstract states the result but provides no outline of the key lattice-matching or transformation steps, making it difficult for readers to assess the argument at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting areas where the exposition can be improved. We address the major comments below and will incorporate revisions accordingly.

read point-by-point responses

Referee: [Main proof (the section applying Zwegers' theory)] The central step equating the two-color partition generating function (with the stated odd-smallest-part and even-part restrictions) to a linear combination of theta series attached to an indefinite lattice of signature (n,1) is not accompanied by an explicit derivation of the quadratic form, the summation vector, or the cone; without this, the applicability of Zwegers' transformation formulas cannot be confirmed. This identification is load-bearing for the entire argument.

Authors: We acknowledge that the derivation of the quadratic form, summation vector, and cone from the partition conditions could be made more explicit. In the revised version, we will expand Section 3 to include a step-by-step mapping from the two-color partitions (with odd smallest part and even part restrictions) to the vectors in the indefinite lattice, specifying the quadratic form Q, the vector v, and the cone C. This will allow direct verification of the conditions for Zwegers' transformation formulas. revision: yes
Referee: [Application of mock theta modular properties] The modular transformation properties of the mock theta functions are invoked after the indefinite-theta identification, but the manuscript supplies no verification that the specific partition restrictions produce no residual terms, divergent sums, or additional case distinctions on parity or size that would fall outside the standard hypotheses of the transformation formulas.

Authors: The partition restrictions in the conjecture are chosen precisely so that the resulting sums align with the standard setup for the mock theta functions in Zwegers' framework, ensuring no residual terms or divergences. However, to address this concern, we will add a dedicated subsection verifying that the conditions on parity and size fall within the hypotheses, with no additional case distinctions required. revision: yes

Circularity Check

0 steps flagged

Proof applies established external theories (Zwegers indefinite theta functions, mock theta modular properties) to the partition generating function without self-referential reduction or fitted predictions.

full rationale

The abstract states the proof rests on Zwegers' theory together with standard modular transformation properties of mock theta functions. These are independent prior results (Zwegers 2002 and related mock theta literature) not derived from the present conjecture or from self-citations that carry the central load. No equations in the provided material equate the target Hecke sum to a fitted parameter or to a definition that presupposes the identity. The identification of the two-color partition generating function with an indefinite theta series is presented as a direct application after rewriting, but the abstract and context give no indication that this rewriting reduces to the conjecture itself by construction. Minor self-citation (if any) is not load-bearing for the main claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof depends on two established frameworks from the literature on mock modular forms; no new parameters, axioms invented for this paper, or postulated entities are introduced.

axioms (2)

domain assumption Zwegers' theory of indefinite theta functions
Invoked as the primary tool for handling the indefinite quadratic forms arising from the partition generating function.
domain assumption modular transformation properties of mock theta functions
Used to perform the necessary transformations that equate the two sides of the conjectured identity.

pith-pipeline@v0.9.0 · 5339 in / 1360 out tokens · 86376 ms · 2026-05-12T04:27:38.685638+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Our proof is based on Zwegers’ theory of indefinite theta functions together with modular transformation properties of mock theta functions.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We then realize bH(τ) as an indefinite theta function of Zwegers... Q(n):=3n1²-n2², c1=(1,3)^T, c2=(-1,3)^T
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

cM(τ) is a holomorphic modular form of weight 12 on Γ0(2)... Sturm’s theorem

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.

Forward citations

Cited by 1 Pith paper

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

Solutions for Hecke Sum Questions of Banerjee and Bringmann
math.NT 2026-05 unverdicted novelty 6.0

Andrews and El Bachraoui prove a two-variable generalization of the Hecke sum identity for S(q) via Bailey pairs, recovering the even and odd cases as corollaries when a=1.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · cited by 1 Pith paper

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