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Extending Recent Congruence Results on $(\ell,\mu)$-Regular Overpartitions

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Pith's one-line read The paper proves new infinite families of congruences for counts of overpartitions into parts avoiding two fixed integers, using only elementary q-series manipulations.

desk verdict A clean, elementary extension of known congruences for (ℓ, μ)-regular overpartitions; the new families are real, the proofs are checkable, and only a minor forward-reference blemish keeps it from being fully polished. read the letter →

arxiv 2505.21989 v1 pith:YPQFKTBE submitted 2025-05-28 math.NT math.CO

Bishnu Paudel , James A. Sellers , Haiyang Wang This is my paper
classification math.NTmath.CO MSC 11P8305A17
keywords overpartitionsregularpartitioncongruencesq-seriesdissectionformulasinfinitefamiliesofelementaryproofs
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies $(\ell,\mu)$-regular overpartitions: ways to write an integer as a sum where no part is divisible by either of two fixed coprime integers $\ell$ and $\mu$, and where each part size may be marked once. For the pairs $(2,3)$, $(4,3)$, and $(4,9)$, it establishes new infinite families of congruences for the counts $R_{\ell,\mu}(n)$. Highlights include $R_{2,3}(3^{\beta}(3n+2))\equiv 0\pmod 6$ for all $\beta\ge 1$, $R_{4,9}(4(kn+r))\equiv 0\pmod{24}$ whenever $r$ is a quadratic nonresidue modulo $k\ge 2$, and exact congruences modulo $96$, $48$, and $216$. The proofs are entirely elementary, using classical $q$-series dissections and coefficient extraction instead of modular forms.

What carries the argument

The central object is the generating function $$\sum_{n\ge 0}R_{\ell,\mu}(n)q^n=\frac{f_2 f_{\ell}^2 f_{\mu}^2 f_{2\mu\ell}}{$f_1^{2}$ f_{2\ell} f_{2\mu} f_{\mu\ell}^2},\qquad f_k=(q^k;q^k)_\infty.$$ The proofs reduce this series modulo the relevant primes using the congruence lemma $f_{p^k l}\equiv f_{p^{k-1} lp}\pmod{p^k}$, expand the reduced products through classical dissection identities such as those numbered (18), (25), (27), and (29), extract coefficients whose exponents lie in a prescribed progression, and substitute $q^m\mapsto q$. Lemma 2.6 is a zero-detection device: if a series $G(q)$ is congruent to $q^s G(q^t)$ modulo $k$, then all its coefficients vanish modulo $k$. The mod $8$ and mod $24$ results hinge on rewriting the reduced generating function as a product of the $\theta$ function $\varphi(q)=\sum_n q^{n^2}$ and reading off which square exponents can survive a given progression.

What would settle it

Test the congruences by direct computation of the first few coefficients of the generating function, for example by enumerating $(4,9)$-regular overpartitions of $n$ for $n\le 200$ and checking $R_{4,9}(4(kn+r))\pmod{24}$ with $k=3,r=2$ and $k=5,r=2$; a single nonzero residue would refute Theorem 1.7. Likewise, checking $R_{2,3}(3^{\beta}(3n+2))\pmod 6$ for small $\beta,n$ would test Corollary 1.2.

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Extended reading notes

Core claim

The paper claims that several known congruence phenomena for $(\ell,\mu)$-regular overpartitions belong to infinite families and that each family admits a purely elementary proof. Concretely, it proves $R_{2,3}(27n)\equiv R_{2,3}(3n)\pmod 3$, and therefore $R_{2,3}(3^{\beta}(3n+2))\equiv 0\pmod 6$ for every $\beta\ge 1$; it proves $R_{4,3}(2n)\equiv 0\pmod 4$, with the stronger modulus $8$ whenever $n$ is not an odd square; and it proves $R_{4,9}(4n)\equiv 0\pmod{12}$, $R_{4,9}(3n)\equiv 0\pmod 8$, and $R_{4,9}(4(kn+r))\equiv 0\pmod{24}$ for every $k\ge 2$ and every quadratic nonresidue $r$ modulo $k$. It also proves $R_{4,9}(18n+12)\equiv 0\pmod{96}$, $R_{4,9}(18n+15)\equiv 0\pmod{48}$, and $R_{4,9}(24n+20)\equiv 0\pmod{216}$, and records the exact identity $R_{4,3}(2n)=ppo(n)$, the number of overpartition pairs into odd parts.

Load-bearing premise

The arguments depend on a battery of quoted $q$-series dissections and a congruence lemma taken from earlier papers without proof; if any of those quoted identities is mistranscribed or applied outside its valid range, the derived congruences need not follow.

Editorial extensions

If this is right

  • For the pair $(2,3)$, every arithmetic progression $3^{\beta}(3n+2)$ has $R_{2,3}$ divisible by $6$, so the previously known single congruence $R_{2,3}(9n+6)\equiv 0\pmod 6$ becomes the base case of an infinite chain.
  • For the pair $(4,9)$, the theorem $R_{4,9}(4(kn+r))\equiv 0\pmod{24}$ produces infinitely many progressions at once, one for every quadratic nonresidue $r$ modulo every $k\ge 2$; for example $k=4,r=2$ gives $R_{4,9}(16n+8)\equiv 0\pmod{24}$.
  • The identity $R_{4,3}(2n)=ppo(n)$ transfers all known congruence properties of overpartition pairs into odd parts to the sequence $R_{4,3}(2n)$, including divisibility by powers of $2$ in the stated progressions.
  • Since all proofs are elementary $q$-series manipulation, these congruences do not depend on modular forms machinery and can be verified term-by-term by direct computation.
  • The paper thereby supplies elementary proofs for several congruences that were previously obtained through modular forms, fulfilling the request for such proofs in the source papers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A structural pattern visible in the proof but not stated as a theorem: modulo $8$, $R_{4,9}(4n)$ is supported only on exponents that are squares or small sums of squares, so any arithmetic progression avoiding those exponents automatically inherits divisibility by $8$; the quadratic-nonresidue condition is one systematic way to produce such progressions.
  • The zero-detection lemma (if $G(q)\equiv q^s G(q^t)$ then $G\equiv 0$) is a general certification tactic; it could be applied automatically to other $q$-series to discover new vanishing families rather than verifying them by hand.
  • The exact identity connecting $R_{4,3}$ to overpartition pairs into odd parts suggests looking for other pairs $(\ell,\mu)$ whose even dissections collapse to known theta products; any such collapse would immediately import a stock of known congruences for free.
  • One testable extension is to ask whether the mod $24$ family $R_{4,9}(4(kn+r))\equiv 0$ extends to mod $48$ or mod $96$ when $r$ is restricted by a stronger condition, using the same theta-product expansion evaluated modulo higher powers of $2$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This paper studies the arithmetic of the function R_{\ell,\mu}(n), which counts overpartitions of n with no part divisible by \ell or \mu. Using elementary q-series dissections, the authors prove several infinite families of congruences for the pairs (\ell,\mu)=(2,3),(4,3),(4,9). The main new results include Corollary 1.2, stating R_{2,3}(3^\beta(3n+2))\equiv0 (mod 6) for all \beta\ge1; Theorem 1.3, giving congruences for R_{4,3}(2n) modulo 4 and 8; Theorem 1.4, giving R_{4,9}(4n)\equiv0 (mod 12) for n\ge1; Theorem 1.6, giving R_{4,9}(3n)\equiv0 (mod 8); Theorem 1.7, giving the infinite family R_{4,9}(4(kn+r))\equiv0 (mod 24) for quadratic nonresidues r modulo k, together with a companion congruence modulo 4; and Theorem 1.8, giving congruences modulo 96, 48, and 216. The proofs are largely dissection-based, with a battery of quoted q-series identities. The central claims are plausible and the derivations are detailed, but the paper has a load-bearing problem in its statement of the main congruence lemma, as detailed below.

Significance. If the results are correct, the paper makes a solid contribution to the elementary study of biregular overpartition congruences. It extends prior modular-form results of Alanazi-Munagi-Saikia and provides elementary proofs for several previously non-elementary congruences. A particular strength is that the proofs are fully explicit and the main congruences are stated in testable form: for instance, Theorem 1.7 gives a uniform infinite family depending only on a quadratic-residue condition. The paper is transparent about its reliance on quoted dissection identities, most of which are standard. However, the correctness of the exposition depends on the congruence lemma stated in Lemma 2.4, and that lemma is not stated correctly; this must be repaired before the paper can be accepted.

major comments (1)
  1. [Section 2, Lemma 2.4, Eq. (30)] The congruence (30) as printed is not usable: if the right-hand side is read literally as f_{p^{k-1}lp} (or f_{p^k l}), the statement is a tautology; if it is read as the natural intended statement f_{p^k l} \equiv f_{p^{k-1}l}^{p} \pmod{p^k}, it is false. For example, with p=2, k=2, l=1 it would assert f_4 \equiv f_2^2 \pmod4, but f_2^2 has expansion 1 - 2q^2 + O(q^3), while f_4 has no q^2 term. This is load-bearing: the proof of Theorem 1.6 invokes (30) to pass from f_2^4f_3^8/(f_1^8f_6^4) to 1 modulo 8, and comparable reductions are used in the proof of Theorem 1.8. Those reductions are in fact true and can be justified by the standard iterated congruence f_m^{p^a} \equiv f_{pm}^{p^{a-1}} \pmod{p^a}, but they are not consequences of Eq. (30) as stated. The authors should replace Lemma 2.4 with a correct statement and then re-verify or re-derive the affected lines in the proofs of Theorems 1.6 and 1.8.
minor comments (4)
  1. [Section 2, Lemma 2.5] In the proof of Lemma 2.5, the displayed generating function is written as \sum_{n\ge0} R_{\ell,\mu}(n) = ...; it should be \sum_{n\ge0} R_{\ell,\mu}(n)q^n.
  2. [Section 5, proof of Theorem 1.4] The proof of Theorem 1.4 invokes Eq. (6) for the modulo 4 part before Eq. (6) is proved later in Theorem 1.7. The dependency chain is acyclic, but the forward reference should be flagged explicitly, or the proof of (6) should be moved before Theorem 1.4 to avoid any appearance of circularity.
  3. [Section 6, proof of Theorem 1.7, after Eq. (43)] When extracting the terms with exponents of the form 4n from (43), the constant term 1 is omitted from the displayed congruence for \sum R_{4,9}(4n)q^{4n}. As written, the congruence is false at exponent 0. The missing constant does not affect the subsequent extraction of coefficients with exponents 4(kn+r), r\ge1, but the displayed formula should be corrected.
  4. [Section 2, Lemma 2.7] In the statement of Lemma 2.7, the first congruence begins 'For all n \ge 0 and k \ge 4, we have ppo(2^{k+1}n)\equiv0 ...' but then adds 'n \ge 1'. This is confusing; the quantifier on n should be stated consistently, most likely with n\ge1 in the first family only.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the target congruences are derived from independent q-series dissections, the Theorem 1.4-to-(6) forward reference is acyclic, and Sellers-coauthored cited lemmas are published results that do not encode the targets.

full rationale

The paper is not circular. Each target congruence (Theorems 1.1, 1.3, 1.4, 1.6, 1.7, and 1.8) is obtained by reducing the generating function of R_{l,mu} through quoted q-series dissections to an explicit finite expansion, e.g., (35), (39), and (43), whose coefficient shape (square exponents, quadratic residues, theta-series form) is then read off directly. No fitted parameter appears, and no target congruence is assumed in the course of its own proof. The one internal forward dependency, Theorem 1.4's use of (6) from Theorem 1.7, is acyclic: (6) is proved directly from the mod-4 reduction of (43), and the mod-3 half of Theorem 1.4 is proved independently, so there is no cycle linking (4), (5), and (6). The proofs do rely on a battery of quoted dissection identities and on Lemma 2.4 (congruence (30)) taken from [8, Lemma 3], and several of these sources ([5], [8], [13], [14], [21]) are coauthored by Sellers; however, each is a published, externally verifiable general q-series result that does not encode the divisibility statements being proved, so these citations are real evidence rather than assumed conclusions. Acyclic internal reuse also occurs when Theorem 1.8(9) reuses the already-proved G(q) = 0 (mod 3) from the proof of Theorem 1.4. The residual score of 1 reflects only the density of self-citations in the toolbox; no derivation step reduces to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new fitting parameters, particles, or entities. It rests on standard q-series identities and the definition of (ℓ, μ)-regular overpartitions.

assumptions (5)
  • domain assumption Generating function for R_{ℓ,μ}(n): f2 f_ℓ^2 f_μ^2 f_{2μℓ} / (f1^2 f_{2ℓ} f_{2μ} f_{μℓ}^2)
    Follows by removing factors for parts divisible by ℓ and μ from the overpartition product; not proved in the paper.
  • standard math Lemma 2.4 congruence (30): f_{pk}^l ≡ f_{p^{k-1}lp} (mod pk)
    Quoted from [8, Lemma 3] and used repeatedly to reduce series modulo primes.
  • standard math Lemma 2.1 theta identities (11)-(14)
    Quoted from [12]; used in Sections 4 and 6 for mod 8 expansions.
  • standard math Lemma 2.2 and 2.3 dissection identities (15)-(29)
    Quoted from [5,8,9,12,13,14,21,25,26]; backbone of all proofs.
  • standard math Lemma 2.7 congruences for ppo(n) from [1, Theorem 1.2]
    Used in Remark 4.1 to transfer congruences to R_{4,3}(2n); not part of main proofs.

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Cite this review

Pith. "Pith review of Extending Recent Congruence Results on $(\ell,\mu)$-Regular Overpartitions." pith.science (2026). https://pith.science/paper/YPQFKTBE

@misc{pith2026250521989,
  author       = {Pith},
  title        = {Pith review of: Extending Recent Congruence Results on $(\ell,\mu)$-Regular Overpartitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPQFKTBE}},
  note         = {Machine review of arXiv:2505.21989}
}
abstract

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,\mu}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $\mu$, for various integer pairs $(\ell, \mu)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.

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Forward citations

Cited by 2 Pith papers

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

  1. On Some New Congruences For Biregular Overpartitions

    math.NT 2025-07 conditional novelty 5.0 of 10

    B_{2^alpha,3^beta}(n), the number of overpartitions whose parts avoid multiples of 2^alpha and 3^beta, satisfies new congruence families modulo 4, 8, 6, and 12.

  2. New Congruences on Biregular Overpartitions

    math.NT 2025-07 reject novelty 4.0 of 10

    The paper derives infinite congruence families for several biregular overpartition functions, but the headline family (4,3^t) with t=1 is invalid because \bar{B}_{4,3}(3)=6.

Reference graph

Works this paper leans on

26 extracted references · 23 canonical work pages · cited by 2 Pith papers

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