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arxiv: 2404.15716 · v3 · pith:B7BOY2PXnew · submitted 2024-04-24 · 🧮 math.CO · math.AC· math.NT

Parity of the coefficients of certain eta-quotients, III: two special classes

William J. Keith , Fabrizio Zanello This is my paper

Pith reviewed 2026-05-24 01:59 UTC · model grok-4.3

classification 🧮 math.CO math.ACmath.NT
keywords eta-quotientscoefficient paritypartition functiondensity theoremsq-seriessingular overpartitionsDedekind eta function
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0 comments X

The pith

New results determine the parity of coefficients for eta-quotients of the form f_t^3/f_1 and certain pure powers f_1^t.

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

The paper continues a series on the parity of coefficients in families of eta-quotients. It examines two specific classes and derives additional results on when the coefficients are even or odd. These results carry implications for the parity of the partition function. They also contribute to a broader conjecture concerning related q-series, with one class connected to singular overpartitions.

Core claim

The paper establishes new results on the parity of the coefficients of eta-quotients of the form f_t^3/f_1, a distinguished case of Andrews' singular overpartitions, and certain pure eta-powers f_1^t, appending to known density theorems.

What carries the argument

The two special classes of eta-quotients f_t^3/f_1 and f_1^t, analyzed via analytic and combinatorial techniques from prior work to determine coefficient parities.

If this is right

  • The parity patterns of the coefficients in these two classes follow from the density theorems established in the series.
  • New information becomes available on the parity of the partition function.
  • Additional support is provided for an overarching conjecture on related q-series.
  • The distinguished case f_t^3/f_1 receives parity results in line with recent attention to singular overpartitions.

Where Pith is reading between the lines

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

  • The same approach might yield parity results for other eta-quotient families beyond the two classes treated here.
  • Explicit small-t verifications could serve as a practical check on the density claims for these forms.
  • The connection to singular overpartitions opens the possibility of translating the parity statements into statements about overpartition functions.

Load-bearing premise

The analytic and combinatorial techniques from the authors' prior papers extend directly to these two classes without new obstructions or case distinctions that would require separate handling.

What would settle it

A concrete computation of coefficients for a small value of t in either class that shows a parity different from the one predicted by the extended density theorems would disprove the results.

read the original abstract

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on two classes. One consists of eta-quotients of the form $f_t^3/f_1$, a distinguished case of Andrews' singular overpartitions that has recently attracted attention among researchers. In addition, we investigate the parity of certain pure eta-powers $f_1^t$, appending new results to known density theorems.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript continues the authors' series on the parity of coefficients of eta-quotients. It focuses on two distinguished classes: quotients of the form f_t^3/f_1 (a case of Andrews' singular overpartitions) and pure eta-powers f_1^t. Parity results are obtained via generating-function and dissection methods from prior papers in the series, with explicit case distinctions introduced for the cubic numerator and pure-power forms; small-t verifications are provided and new density statements are appended to existing theorems, with implications noted for the partition function and an overarching q-series conjecture.

Significance. If the derivations hold, the results extend the density theorems for coefficient parities in these eta-quotient families and reinforce the applicability of the analytic-combinatorial framework across the series. Explicit verification for small t and the documented adjustments for the special forms constitute verifiable strengths that support the broader program on partition parities.

minor comments (2)
  1. §2: the notation f_t is used without an immediate cross-reference to its definition in Paper I of the series; adding one sentence would improve readability for readers entering at this installment.
  2. Theorem 5.2: the density statement appends to a prior result but does not restate the exact form of the known theorem being extended; a one-line reminder would clarify the increment.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper continues prior work in a series but introduces explicit case distinctions and adjustments for the distinguished forms f_t^3/f_1 and pure powers f_1^t, deriving new parity results via generating-function dissections with small-t verification and appending density statements to known theorems. No load-bearing step reduces a claimed prediction or uniqueness result to a self-citation, fitted parameter, or definitional renaming by construction; the central claims retain independent combinatorial content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or ad-hoc axioms; the work relies on standard background from partition theory and q-series.

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Ahlgren:The partition function modulo composite integers M, Math

    S. Ahlgren:The partition function modulo composite integers M, Math. Ann.318(2000), 795–803

    work page 2000

  2. [2]

    The Theory of Partitions,

    G. Andrews: “The Theory of Partitions,” Encyclopedia of Mathematics and its Applications, Vol. II. Addison-Wesley, Reading, Mass.-London-Amsterdam (1976)

    work page 1976

  3. [3]

    Andrews:Singular overpartitions, Int

    G. Andrews:Singular overpartitions, Int. J. Number Theory5(11) (2015), 1523–1533

    work page 2015

  4. [4]

    Aricheta:Congruences for Andrews’(k, i)-singular overpartitions, Ramanujan J.43(2017), 535– 549

    V.M. Aricheta:Congruences for Andrews’(k, i)-singular overpartitions, Ramanujan J.43(2017), 535– 549

    work page 2017

  5. [5]

    Aricheta, J

    V.M. Aricheta, J. Dimabayao, and H.J. Shi:Density results for the parity of(4k, k)-singular overparti- tions, J. Number Theory262(2024), 354–370

    work page 2024

  6. [6]

    Ballantine, M

    C. Ballantine, M. Merca, and C.-S. Radu:Parity of 3-regular partition numbers and Diophantine equa- tions, preprint (arxiv.org/abs/2212.09810)

    work page arXiv

  7. [7]

    Barman and A

    R. Barman and A. Singh:New density results and congruences for Andrews’ singular overpartitions, J. Number Theory229(2021), 328–341

    work page 2021

  8. [8]

    Chen:Congruences for the number ofk-tuple partitions with distinct even parts, Discrete Math

    S.-C. Chen:Congruences for the number ofk-tuple partitions with distinct even parts, Discrete Math. 313(2013), 1565–1568

    work page 2013

  9. [9]

    Chen:On the density of the odd values of the partition function and thet-multipartition function, J

    S.-C. Chen:On the density of the odd values of the partition function and thet-multipartition function, J. Number Theory225(2021), 198–213

    work page 2021

  10. [10]

    Chen:Partition congruences and the vanishing coefficients of products of theta functions, Ra- manujan J.62(2023), 1125–1144

    S.-C. Chen:Partition congruences and the vanishing coefficients of products of theta functions, Ra- manujan J.62(2023), 1125–1144

    work page 2023

  11. [11]

    Cotron, A

    T. Cotron, A. Michaelsen, E. Stamm, and W. Zhu:Lacunary Eta-quotients Modulo Powers of Primes, Ramanujan J.53(2020), 269–284

    work page 2020

  12. [12]

    Gordon and K

    B. Gordon and K. Ono:Divisibility of certain partition functions by powers of primes, Ramanujan J.1 (1997) 25–34

    work page 1997

  13. [13]

    The power of q,

    M. Hirschhorn: “The power of q,” Developments in Mathematics49, Springer (2017)

    work page 2017

  14. [14]

    Hirschhorn:Parity results for certain partition functions, Ramanujan J.4(2000), 129–135

    M. Hirschhorn:Parity results for certain partition functions, Ramanujan J.4(2000), 129–135

    work page 2000

  15. [15]

    Judge, W.J

    S. Judge, W.J. Keith, and F. Zanello:On the density of the odd values of the partition function, Ann. Comb.22(2018), no. 3, 583–600

    work page 2018

  16. [16]

    Judge and F

    S. Judge and F. Zanello:On the density of the odd values of the partition function, II: An infinite conjectural framework, J. Number Theory188(2018), 357–370

    work page 2018

  17. [17]

    Keith and F

    W.J. Keith and F. Zanello:Parity of the coefficients of certain eta-quotients, J. Number Theory235 (2022), 275–304

    work page 2022

  18. [18]

    Keith and F

    W.J. Keith and F. Zanello:Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions, J. Number Theory251(2023), 84–101. PARITY OF THE COEFFICIENTS OF CERTAIN ETA-QUOTIENTS, III 21

    work page 2023

  19. [19]

    Landau: ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzen erforderlichen Quadrate, Arch

    E. Landau: ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzen erforderlichen Quadrate, Arch. Math. Phys. (3)13(1908), 305–312

    work page 1908

  20. [20]

    Li and O.X.M

    X. Li and O.X.M. Yao:New infinite families of congruences for Andrews (K, I)-singular overpartitions, Quaest. Math.41(2018), 1005–1019

    work page 2018

  21. [21]

    Naika and D.S

    M.S.M. Naika and D.S. Gireesh:Congruences for Andrews’ singular overpartitions, J. Number Theory 165(2016), 109–130

    work page 2016

  22. [22]

    Ono:On the parity of the partition function in arithmetic progressions, J

    K. Ono:On the parity of the partition function in arithmetic progressions, J. Reine Angew. Math.472 (1996), 1–15

    work page 1996

  23. [23]

    The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series,

    K. Ono: “The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series,” CBMS Regional Conference Series in Mathematics, Amer. Math. Soc.102, Providence, RI (2004)

    work page 2004

  24. [24]

    Parkin and D

    T.R. Parkin and D. Shanks:On the distribution of parity in the partition function, Math. Comp.21 (1967), 466–480

    work page 1967

  25. [25]

    Pore and S.N

    U. Pore and S.N. Fathima:Some congruences modulo 2, 8 and 12 for Andrews’ singular overpartitions, Note Mat.38(2018), 101–114

    work page 2018

  26. [26]

    Radu:A proof of Subbarao’s conjecture, J

    C.-S. Radu:A proof of Subbarao’s conjecture, J. Reine Angew. Math.672(2012), 161–175

    work page 2012

  27. [27]

    Robbins:Ont-core partitions, Fibonacci Quarterly38(2000), 39–48

    N. Robbins:Ont-core partitions, Fibonacci Quarterly38(2000), 39–48

    work page 2000

  28. [28]

    Elementary Number Theory and its Applications,

    K. Rosen: “Elementary Number Theory and its Applications,” Fourth Ed., Addison-Wesley, Reading, MA (2000), xviii+638 pp

    work page 2000

  29. [29]

    Serre:Divisibilit´ e des coefficients des formes modulaires de poids entier, C.R

    J.-P. Serre:Divisibilit´ e des coefficients des formes modulaires de poids entier, C.R. Acad. Sci. Paris A 279(1974), 679–682

    work page 1974

  30. [30]

    Xia:New congruences modulo powers of 2 for broken 3-diamond partitions and 7-core partitions, J

    E.X.W. Xia:New congruences modulo powers of 2 for broken 3-diamond partitions and 7-core partitions, J. Number Theory141(2014), 119–135

    work page 2014

  31. [31]

    Zanello:Deducing the positive odd density ofp(n)from that of a multipartition function: An uncon- ditional proof, J

    F. Zanello:Deducing the positive odd density ofp(n)from that of a multipartition function: An uncon- ditional proof, J. Number Theory229(2021), 277–281. Department of Mathematical Sciences, Michigan Tech, Houghton, MI 49931-1295 Email address:wjkeith@mtu.edu, zanello@mtu.edu

    work page 2021