pith. sign in

arxiv: 2510.13685 · v2 · submitted 2025-10-15 · 🧮 math.NT · math.CO

Congruences for an analogue of Lin's partition function

Russelle Guadalupe This is my paper

Pith reviewed 2026-05-18 06:02 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords partition functionRamanujan congruencesq-seriesmodular functionsrestricted partitionsgenerating functions
0
0 comments X

The pith

An analogue B(n) of Lin's restricted partition function satisfies Ramanujan-type congruences for certain sums modulo 2, 3, 5, 7, and 9.

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

The paper defines B(n) to count partition triples (π1, π2, π3) of n where π1 and π2 have distinct odd parts and π3 consists of parts divisible by 4. It then proves that selected sums involving these values obey congruences modulo 2, 3, 5, 7, and 9. The proofs rely on elementary q-series manipulations and properties of modular functions applied to the generating function of B(n). A reader would care because the results supply fresh examples of arithmetic regularity in a new family of restricted partitions, extending the classical Ramanujan congruences in a concrete and checkable way.

Core claim

We introduce B(n) counting the number of partition triples (π1, π2, π3) of n such that π1 and π2 comprise distinct odd parts and π3 consists of parts divisible by 4. Using elementary q-series techniques and modular functions we establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain sums involving B(n).

What carries the argument

The generating function of B(n), which is shown to admit a form permitting direct application of q-series identities and modular-function properties to produce the claimed congruences.

If this is right

  • Linear combinations of B(n) vanish or satisfy simple divisibility conditions in specified residue classes modulo 2, 3, 5, 7, or 9.
  • The generating function of B(n) obeys modular relations that mirror those satisfied by classical partition generating functions.
  • The same q-series approach yields explicit recurrence or closed-form expressions for the sums that appear in the congruences.
  • B(n) provides a new concrete family of restricted partitions whose values are constrained by modular arithmetic.

Where Pith is reading between the lines

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

  • If the generating function of B(n) can be expressed as an eta-product or similar modular form, the same method may produce congruences for additional moduli or for related counting functions.
  • The existence of these congruences suggests that density or distribution results for B(n) in arithmetic progressions can be obtained by standard circle-method or modular-form techniques.
  • The construction may be varied by altering the divisibility condition on π3 or the odd-part restrictions on π1 and π2, potentially producing further families with analogous arithmetic properties.

Load-bearing premise

The generating function for B(n) can be written in a form that permits direct application of q-series identities and modular-function properties.

What would settle it

Compute the relevant sum of B(n) for a concrete arithmetic progression and modulus (for example n ≡ 1 mod 5) and check whether it violates the stated congruence for any n up to a few hundred.

read the original abstract

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and $\pi_3$ consists of parts divisible by $4$. With the help of elementary $q$-series techniques and modular functions, we establish Ramanujan-type congruences modulo $2,3,5,7$, and $9$ for certain sums involving $B(n)$.

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 defines B(n) as the number of partition triples (π1, π2, π3) of n where π1 and π2 consist of distinct odd parts and π3 consists of parts divisible by 4. It then uses elementary q-series techniques and properties of modular functions to establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain (unspecified in the abstract) sums involving B(n).

Significance. If the derivations hold, the work extends the study of arithmetic properties of restricted partition functions by producing explicit new congruences for this analogue of Lin's function. The reliance on standard q-series and modular-form methods is appropriate for the field and yields falsifiable predictions that can be checked computationally for small n.

minor comments (2)
  1. The abstract and introduction refer to 'certain sums involving B(n)' without an explicit definition or notation for these sums (e.g., S_k(n) or similar); this should be introduced with a clear formula early in the paper.
  2. A short table of small values of B(n) and the relevant sums would aid readability and allow immediate verification of the claimed congruences for small moduli.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The summary accurately reflects the definition of B(n) and the methods employed. We are pleased that the referee views the congruences as extending prior work on restricted partition functions and that the results are computationally verifiable. Since the recommendation is for minor revision and no specific major comments were raised, we will incorporate clarifications to improve readability.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external identities

full rationale

The paper defines the function B(n) combinatorially as the number of partition triples with specified restrictions on parts, yielding a generating function that is a product of standard q-series for distinct odd parts and 4-divisible parts. Congruences are then derived using elementary q-series techniques and properties of modular functions, which are invoked as established external tools rather than derived internally or fitted to the target results. No self-citations are load-bearing for the central claims, no parameters are fitted and renamed as predictions, and no step reduces by construction to its own inputs. The argument chain remains independent of the specific congruences being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of B(n) together with the domain assumption that its generating function is amenable to standard q-series and modular-function manipulations; no free parameters, invented entities, or ad-hoc axioms are indicated.

axioms (1)
  • domain assumption The generating function for B(n) admits a representation that allows application of q-series identities and modular form properties to derive the congruences.
    Invoked throughout the proof strategy described in the abstract.

pith-pipeline@v0.9.0 · 5613 in / 1344 out tokens · 55992 ms · 2026-05-18T06:02:29.591946+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Amdeberhan and M

    T. Amdeberhan and M. Merca, From cranks to congruences,Mediterr. J. Math.22 (2025), Paper no. 190, 22 pp

    work page 2025

  2. [2]

    Ballantine and M

    C. Ballantine and M. Merca, Parity of sums of partition numbers and squares in arithmetic progressions,Ramanujan J.44(2017), 617–630

    work page 2017

  3. [3]

    N. D. Baruah, Divisibility of sums of partition numbers by multiples of 2 and 3,Bull. Aust. Math. Soc.110(2024), 271–279

    work page 2024

  4. [4]

    Biswas and N

    S. Biswas and N. Saikia, Divisibility of sums of some restricted partition numbers by 2, 3 and 4,Bol. Soc. Mat. Mex. (3)31(2024), Paper no. 54, 19 pp

    work page 2024

  5. [5]

    Biswas and N

    S. Biswas and N. Saikia, Divisibility properties of sums of 24-regular partition numbers, Indian J. Pure Appl. Math(2025), doi:10.1007/s13226-025-00799-1

    work page doi:10.1007/s13226-025-00799-1 2025

  6. [6]

    J. M. Borwein, P. B. Borwein, and F. G. Garvan, Some cubic modular identities of Ramanujan,Trans. Amer. Math. Soc.343(1994), 35–47

    work page 1994

  7. [7]

    Chan, Ramanujan’s cubic continued fraction and an analog of his ‘most beautiful identity’,Int

    H.-C. Chan, Ramanujan’s cubic continued fraction and an analog of his ‘most beautiful identity’,Int. J. Number Theory6(2010), 673–680

    work page 2010

  8. [8]

    Cooper,Ramanujan’s Theta Functions(Springer, Cham, 2017)

    S. Cooper,Ramanujan’s Theta Functions(Springer, Cham, 2017)

    work page 2017

  9. [9]

    Cooper, M

    S. Cooper, M. D. Hirschhorn, and R. Lewis, Powers of Euler’s product and related identities,Ramanujan J.4(2000), 137–155

    work page 2000

  10. [10]

    Guadalupe, Remarks on a certain partition function of Lin, preprint (2025), arXiv:2503.23996

    R. Guadalupe, Remarks on a certain partition function of Lin, preprint (2025), arXiv:2503.23996

    work page arXiv 2025

  11. [11]

    Guadalupe, A note on congruences for the difference between even cranks and odd cranks,Bol

    R. Guadalupe, A note on congruences for the difference between even cranks and odd cranks,Bol. Soc. Mat. Mex. (3)31(2025), Paper no. 127, 11 pp

    work page 2025

  12. [12]

    G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson,Collected Papers of Srinivasa Ramanujan(Cambridge University Press, 1927)

    work page 1927

  13. [13]

    M. D. Hirschhorn, A note on overcubic partitions,New Zealand J. Math.42(2012), 229–234

    work page 2012

  14. [14]

    M. D. Hirschhorn,The Power ofq, A Personal Journey, Developments of Mathematics, vol. 49 (Springer, Cham, 2017)

    work page 2017

  15. [15]

    W. Hu, O. X. M. Yao, and T. Zhao, New parity results of sums of partitions and squares in arithmetic progressions,Contrib. Discrete Math.14(2019), 117–129

    work page 2019

  16. [16]

    Huber, J

    T. Huber, J. McLaughlin, and D. Ye, Dissections of lacunary eta quotients and iden- tically vanishing coefficients,Adv. Appl. Math.168(2025), Paper no. 102902, 50 pp. 16 RUSSELLE GUADALUPE

    work page 2025

  17. [17]

    Kim, The overcubic partition function mod 3, inRamanujan Rediscovered

    B. Kim, The overcubic partition function mod 3, inRamanujan Rediscovered. Pro- ceedings of a conference on elliptic functions, partitions, andq-series in memory of K. Venkatachaliengar, Bangalore, India, June 1–5, 2009, RMS Lecture Note Series, vol. 14 (Ramanujan Mathematical Society, 2010), pp. 157–163

    work page 2009

  18. [18]

    B. L. S. Lin, The restricted 3-colored partition function mod 3,Int. J. Number Theory 9(2013), 1789–1799

    work page 2013

  19. [19]

    Ono,The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series, CBMS Regional Conference Series in Mathematics, vol

    K. Ono,The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series, CBMS Regional Conference Series in Mathematics, vol. 102 (American Mathematical Society, Rhode Island, 2004)

    work page 2004

  20. [20]

    Paule and C.-S

    P. Paule and C.-S. Radu, A unified algorithmic framework for Ramanu- jan’s congruences modulo powers of 5, 7, and 11 (2018), available at http://www3.risc.jku.at/publications/download/risc 5760/PP submission.pdf

    work page 2018

  21. [21]

    Paule and C.-S

    P. Paule and C.-S. Radu, A proof of the Weierstraß gap theorem not using the Rie- mann–Roch formula,Ann. Comb.23(2019), 963–1007

    work page 2019

  22. [22]

    Radu, An algorithmic approach to Ramanujan–Kolberg identities,J

    C.-S. Radu, An algorithmic approach to Ramanujan–Kolberg identities,J. Symbolic Comput.68(2015), 225–253

    work page 2015

  23. [23]

    N. A. Smoot, On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: an implementation of Radu’s algorithm,J. Symbolic Comput.104(2021), 276–311. Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines Email address:rguadalupe@math.upd.edu.ph

    work page 2021