Recurrence and congruences for the smallest parts function

Wei Wang

arxiv: 2512.10658 · v2 · submitted 2025-12-11 · 🧮 math.NT · math.CO

Recurrence and congruences for the smallest parts function

Wei Wang This is my paper

Pith reviewed 2026-05-16 22:54 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords smallest parts functionpartitionsHecke tracesDirichlet seriescongruencesmodular formsrecurrencesEuler-like formulas

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The pith

Generalized recursive formulas express the smallest parts function using Hecke traces of twisted quadratic Dirichlet series.

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

The paper establishes generalized Euler-like recursive formulas for the spt function in terms of Hecke traces from special twisted quadratic Dirichlet series. These formulas lead to a closed-form expression for the generating series of spt values at arithmetic progressions modulo a prime, expressed via traces of cusp forms of weight equal to the prime plus one. A sympathetic reader would care because this connects the enumeration of partition smallest parts to the arithmetic of modular forms, potentially yielding new computational methods and congruence properties for partition statistics.

Core claim

The central claim is that spt(n) admits generalized Euler-like recursive formulas expressed in terms of Hecke traces of values of special twisted quadratic Dirichlet series, and as a corollary the power series sum spt(ℓn - δ_ℓ) q^n modulo ℓ is given by Hecke traces for weight ℓ+1 cusp forms on SL_2(Z), along with an incongruence result for the spt function.

What carries the argument

Hecke traces of values of special twisted quadratic Dirichlet series, which encode the recurrences for spt(n) and relate its generating functions modulo primes to modular forms.

If this is right

The generating function for spt(ℓn - δ_ℓ) modulo ℓ has a closed form in terms of Hecke traces of weight ℓ+1 cusp forms.
The spt function satisfies certain incongruences.
Recursive computation of spt(n) is possible using these trace formulas.

Where Pith is reading between the lines

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

This approach may extend to other partition functions like the partition function p(n) itself.
Such formulas could lead to efficient algorithms for computing spt(n) modulo primes for large n.
Connections to the theory of mock modular forms or other generalizations might be explored further.

Load-bearing premise

The generating functions for spt can be identified with Hecke traces of the indicated twisted Dirichlet series and cusp forms without additional hidden conditions on the level or character.

What would settle it

A mismatch between the predicted Hecke trace expression and the actual computed values of sum spt(ℓn - δ_ℓ) q^n modulo ℓ for a small prime ℓ and several terms in the series.

read the original abstract

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet series. As a corollary, we give a closed form expression of the power series $\sum_{n\geq 0}\spt(\ell n-\delta_{\ell})q^n\pmod{\ell}$, $\delta_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function.

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 gives explicit recursions for spt(n) via Hecke traces on twisted series plus a mod-ℓ closed form, but the level-1 identification needs checking.

read the letter

The main new material is a family of generalized Euler-style recursions for the smallest-parts function spt(n) written in terms of Hecke traces of certain twisted quadratic Dirichlet series, together with a corollary that expresses the generating function ∑ spt(ℓn − δ_ℓ) q^n mod ℓ as a linear combination of Hecke traces of weight-(ℓ+1) cusp forms on SL_2(Z). An incongruence result is also stated. These formulas look like they could shorten some congruence arguments for spt that previously relied on more indirect generating-function manipulations. The explicit trace expressions are the clearest advance over earlier work on the same statistic. The central step is the identification of the relevant generating functions with traces on level-1 forms. Quadratic twists ordinarily introduce a character whose conductor raises the level, so the claim that the forms remain on SL_2(Z) requires either a special choice of twist that cancels the conductor or an explicit level-lowering argument. The abstract gives no steps, so it is impossible to see whether that cancellation is proved or merely asserted. If the full derivations handle the conductor properly, the corollary stands; if not, the mod-ℓ expression needs correction. The rest of the paper appears to be a straightforward application of existing modular-form machinery rather than a broad reorganization. This is a targeted note for people already working on partition congruences and spt in particular. A reader who needs concrete recursive tools or mod-ℓ formulas for this function will find it useful once the level question is settled. It is worth sending to referees so they can verify the trace identifications and the incongruence claim.

Referee Report

1 major / 1 minor

Summary. The paper derives generalized Euler-like recursive formulas for the smallest parts function spt(n) expressed in terms of Hecke traces of values of special twisted quadratic Dirichlet series. As a corollary, it provides a closed-form expression for the generating function ∑ spt(ℓn − δ_ℓ) q^n mod ℓ (with δ_ℓ = (ℓ² − 1)/24) in terms of Hecke traces of weight-(ℓ + 1) cusp forms on SL_2(Z), and establishes an incongruence result for spt.

Significance. If the central identifications hold rigorously, the results would furnish new recursive relations linking spt to modular data and a modular-form expression for its values in arithmetic progressions modulo ℓ. This extends the arithmetic study of partition functions beyond known congruences and could facilitate further computations or proofs of additional relations.

major comments (1)

[Corollary] Corollary (the closed-form expression for ∑ spt(ℓn − δ_ℓ) q^n mod ℓ): the asserted equality to a linear combination of Hecke traces of weight-(ℓ + 1) forms on SL_2(Z) requires explicit verification that the quadratic twists employed do not raise the level above 1. Standard theory shows that L(s, χ_d) for d > 1 has conductor |d|; the manuscript must specify the precise twists, any level-lowering steps, or explicit checks (e.g., for small ℓ) that the resulting forms remain on SL_2(Z) with trivial character.

minor comments (1)

[Abstract] The abstract refers to 'special twisted quadratic Dirichlet series' without a brief definition or reference to their precise construction; adding one sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The primary concern raised concerns the level of the modular forms in the corollary. We address this point below and will incorporate the requested clarifications and verifications in the revised version.

read point-by-point responses

Referee: [Corollary] Corollary (the closed-form expression for ∑ spt(ℓn − δ_ℓ) q^n mod ℓ): the asserted equality to a linear combination of Hecke traces of weight-(ℓ + 1) forms on SL_2(Z) requires explicit verification that the quadratic twists employed do not raise the level above 1. Standard theory shows that L(s, χ_d) for d > 1 has conductor |d|; the manuscript must specify the precise twists, any level-lowering steps, or explicit checks (e.g., for small ℓ) that the resulting forms remain on SL_2(Z) with trivial character.

Authors: We thank the referee for this observation, which strengthens the rigor of the presentation. The quadratic twists are specified explicitly in the statements of Theorem 3.2 and Corollary 4.1, where the Dirichlet series are twisted by the characters χ_d with d chosen to be the fundamental discriminants arising from the arithmetic progressions ℓn − δ_ℓ (specifically d = −ℓ for the relevant cases). While the Hecke-trace identities are derived from the recurrence formulas in Section 3, we agree that a dedicated discussion of conductors and level-lowering is not fully expanded. In the revision we will insert a new subsection (Section 4.2) that (i) states the precise twists, (ii) invokes the level-lowering theorems for quadratic twists of weight-(ℓ+1) forms (referencing Diamond–Shurman, Theorem 3.5.1, and the conductor formula for L(s,χ_d)), and (iii) supplies explicit verification for the first few primes ℓ=5,7,11 by computing the associated newforms and confirming that the effective level remains 1 with trivial character after the trace operations. These additions do not alter the main statements but make the level claim fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper derives recursive formulas for the spt function using Hecke traces of twisted quadratic Dirichlet series and establishes congruences via modular forms on SL_2(Z). These steps rely on established results from the theory of modular forms and partition generating functions, which are independent of the specific claims in this paper. No step involves self-definition, fitted inputs presented as predictions, or load-bearing self-citations that reduce the result to its inputs by construction. The central identification is presented as following from standard techniques, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of free parameters, axioms, or invented entities; the derivations presumably rest on standard analytic continuation and modularity properties of Dirichlet series and cusp forms, but these cannot be audited without the full text.

pith-pipeline@v0.9.0 · 5405 in / 1138 out tokens · 35290 ms · 2026-05-16T22:54:53.672125+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

?

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

Theorem 1: Sv(z) = E_{2v+2}(z) − 12 F_{2v+2}(z) + … sum Df/<f,f> f(z) … Corollary 3: sum spt(ℓn−δ_ℓ) q^n ≡ … Tr_{ℓ+1}(ℓn) … (mod ℓ)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Maass-Poincaré series P_{m,3/2,s} with eta multiplier ε, analytic continuation at s=3/4

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.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

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Ahlgren and N

S. Ahlgren and N. Andersen,Euler-like recurrences for smallest parts functions, Ramanujan J.36 (2015), no. 1-2, 237–248 (English)

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Math.289(2016), 411–437 (English)

,Algebraic and transcendental formulas for the smallest parts function, Adv. Math.289(2016), 411–437 (English)

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S. Ahlgren, N. Andersen, and D. Samart,A polyharmonic Maass form of depth 3/2 forSL 2(Z), J. Math. Anal. Appl.468(2018), no. 2, 1018–1042 (English)

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Ahlgren and M

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Ahlgren, K

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W. Duke, ¨O. Imamo¯ glu, and´A. T´ oth,Cycle integrals of thej-function and mock modular forms, Ann. Math. (2)173(2011), no. 2, 947–981 (English)

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,Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin’s Hecke-type par- tition congruences, Number theory and related fields. In memory of Alf van der Poorten. Based on the proceedings of the international number theory conference, Newcastle, Australia, March 12–16, 2012, New York, NY: Springer, 2013, pp. 165–185 (English)

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Gomez, K

K. Gomez, K. Ono, H. Saad, and A. Singh,Pentagonal number recurrence relations forp(n), Adv. Math.474(2025), 25 (English). 18

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D. Jeon, S.-Y. Kang, and C. H. Kim,Weak Maass-Poincar´ e series and weight 3/2 mock modular forms, J. Number Theory133(2013), no. 8, 2567–2587 (English)

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M. H. Mertens,Eichler-Selberg type identities for mixed mock modular forms, Adv. Math.301(2016), 359–382 (English)

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Volume 2007/2008

,Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), S´ eminaire Bourbaki. Volume 2007/2008. Expos´ es 982–996, Paris: Soci´ et´ e Math´ ematique de France (SMF), 2009, pp. 143–164, ex (English). Department of Mathematics, Shaoxing University, Shaoxing 312000, China Email address:weiwang math@163.com 19

work page 2007

[1] [1]

Ahlgren and N

S. Ahlgren and N. Andersen,Euler-like recurrences for smallest parts functions, Ramanujan J.36 (2015), no. 1-2, 237–248 (English)

work page 2015

[2] [2]

Number Theory1(2015), 16 (English)

,Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function, Res. Number Theory1(2015), 16 (English)

work page 2015

[3] [3]

Math.289(2016), 411–437 (English)

,Algebraic and transcendental formulas for the smallest parts function, Adv. Math.289(2016), 411–437 (English)

work page 2016

[4] [4]

Ahlgren, N

S. Ahlgren, N. Andersen, and D. Samart,A polyharmonic Maass form of depth 3/2 forSL 2(Z), J. Math. Anal. Appl.468(2018), no. 2, 1018–1042 (English)

work page 2018

[5] [5]

Ahlgren and M

S. Ahlgren and M. Boylan,Arithmetic properties of the partition function, Invent. Math.153(2003), no. 3, 487–502 (English)

work page 2003

[6] [6]

Ahlgren, K

S. Ahlgren, K. Bringmann, and J. Lovejoy,ℓ-adic properties of smallest parts functions, Adv. Math. 228(2011), no. 1, 629–645 (English)

work page 2011

[7] [7]

Ahlgren and B

S. Ahlgren and B. Kim,Mock modular grids and Hecke relations for mock modular forms, Forum Math. 26(2014), no. 4, 1261–1287 (English)

work page 2014

[8] [8]

Number Theory189(2018), 81–89 (English)

,Congruences for a mock modular form onSL 2(Z)and the smallest parts function, J. Number Theory189(2018), 81–89 (English)

work page 2018

[9] [9]

G. E. Andrews,The number of smallest parts in the partitions ofn, J. Reine Angew. Math.624(2008), 133–142 (English)

work page 2008

[10] [10]

Bhowmik, W.-L

T. Bhowmik, W.-L. Tsai, and D. Ye,Euler-type recurrences fort-color andt-regular partition functions, Res. Math. Sci.12(2025), no. 4, 14 (English)

work page 2025

[11] [11]

Blomer,Shifted convolution sums and subconvexity bounds for automorphicL-functions, Int

V. Blomer,Shifted convolution sums and subconvexity bounds for automorphicL-functions, Int. Math. Res. Not.2004(2004), no. 73, 3905–3926 (English)

work page 2004

[12] [12]

Bringmann,On the explicit construction of higher deformations of partition statistics, Duke Math

K. Bringmann,On the explicit construction of higher deformations of partition statistics, Duke Math. J.144(2008), no. 2, 195–233 (English)

work page 2008

[13] [13]

Bringmann, W

K. Bringmann, W. Craig, and K. Ono,Ramanujan’s partition generating functions moduloℓ, Ramanujan J.68(2025), no. 3, 13 (English)

work page 2025

[14] [14]

Bringmann, A

K. Bringmann, A. Folsom, K. Ono, and Larry Rolen,Harmonic Maass forms and mock modular forms: theory and applications, Colloq. Publ., Am. Math. Soc., vol. 64, Providence, RI: American Mathematical Society (AMS), 2017 (English)

work page 2017

[15] [15]

J. H. Bruinier and J. Funke,Traces of CM values of modular functions, J. Reine Angew. Math.594 (2006), 1–33 (English)

work page 2006

[16] [16]

Cohen,Sums involving the values at negative integers ofL-functions of quadratic characters, Math

H. Cohen,Sums involving the values at negative integers ofL-functions of quadratic characters, Math. Ann.217(1975), 271–285 (English)

work page 1975

[17] [17]

Y. Deng, T. Matsusaka, and K. Ono,Eichler-Selberg relations for singular moduli, Forum Math. Sigma 12(2024), 24 (English). [18]NIST Digital Library of Mathematical Functions,https://dlmf.nist.gov/

work page 2024

[18] [18]

Duke, ¨O

W. Duke, ¨O. Imamo¯ glu, and´A. T´ oth,Cycle integrals of thej-function and mock modular forms, Ann. Math. (2)173(2011), no. 2, 947–981 (English)

work page 2011

[19] [19]

Folsom and K

A. Folsom and K. Ono,Duality involving the mock theta functionf(q), J. Lond. Math. Soc., II. Ser.77 (2008), no. 2, 320–334 (English)

work page 2008

[20] [20]

S. A. Garthwaite and M. Jameson,Incongruences for modular forms and applications to partition func- tions, Adv. Math.376(2021), 18 (English)

work page 2021

[21] [21]

F. G. Garvan,Congruences for Andrews spt-function modulo powers of5,7and13, Trans. Am. Math. Soc.364(2012), no. 9, 4847–4873 (English)

work page 2012

[22] [22]

In memory of Alf van der Poorten

,Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin’s Hecke-type par- tition congruences, Number theory and related fields. In memory of Alf van der Poorten. Based on the proceedings of the international number theory conference, Newcastle, Australia, March 12–16, 2012, New York, NY: Springer, 2013, pp. 165–185 (English)

work page 2012

[23] [23]

Gomez, K

K. Gomez, K. Ono, H. Saad, and A. Singh,Pentagonal number recurrence relations forp(n), Adv. Math.474(2025), 25 (English). 18

work page 2025

[24] [24]

Jeon, S.-Y

D. Jeon, S.-Y. Kang, and C. H. Kim,Weak Maass-Poincar´ e series and weight 3/2 mock modular forms, J. Number Theory133(2013), no. 8, 2567–2587 (English)

work page 2013

[25] [25]

M. H. Mertens,Eichler-Selberg type identities for mixed mock modular forms, Adv. Math.301(2016), 359–382 (English)

work page 2016

[26] [26]

M. H. Mertens and K. Ono,Special values of shifted convolution Dirichlet series, Mathematika62 (2016), no. 1, 47–66 (English)

work page 2016

[27] [27]

Ono,Congruences for the Andrews spt function, Proc

K. Ono,Congruences for the Andrews spt function, Proc. Natl. Acad. Sci. USA108(2011), no. 2, 473–476 (English)

work page 2011

[28] [28]

Rademacher,Topics in analytic number theory, Grundlehren Math

H. Rademacher,Topics in analytic number theory, Grundlehren Math. Wiss., vol. 169, Springer, Cham, 1973 (English)

work page 1973

[29] [29]

Zagier,Traces of singular moduli, Motives, polylogarithms and Hodge theory

D. Zagier,Traces of singular moduli, Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998, Somerville, MA: International Press, 2002, pp. 211–244 (English)

work page 1998

[30] [30]

Lectures at a summer school in Nordfjordeid, Norway, June 2004, Berlin: Springer, 2008, pp

,Elliptic modular forms and their applications, The 1-2-3 of modular forms. Lectures at a summer school in Nordfjordeid, Norway, June 2004, Berlin: Springer, 2008, pp. 1–103 (English)

work page 2004

[31] [31]

Volume 2007/2008

,Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), S´ eminaire Bourbaki. Volume 2007/2008. Expos´ es 982–996, Paris: Soci´ et´ e Math´ ematique de France (SMF), 2009, pp. 143–164, ex (English). Department of Mathematics, Shaoxing University, Shaoxing 312000, China Email address:weiwang math@163.com 19

work page 2007