REVIEW 3 major objections 6 minor 1 cited by
Pentagonal number recurrence relations for $p(n)$
T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves Euler's pentagonal-number recurrence for p(n) is the ν=0 case of an infinite family indexed by even weights, with ν=6 yielding a formula for Ramanujan's tau-function.
desk verdict A nice generalization of Euler's recurrence with a removable but real flaw: the multiplier for 1/η is wrong in the unfolding section, and convergence is handwaved, so Theorem 1.4 is not yet proven as printed. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is Pν(τ) := [1/η(τ), η(τ)]ν, the ν-th Rankin–Cohen bracket: a bilinear differential operator that combines derivatives of two modular forms to produce a modular form of weight k+l+2ν. It carries the argument because P0 = 1 reproduces Euler's identity, while knowing Pν as a modular form gives the q-coefficient identity that solves for p(n). The proof then uses Proposition 3.4, expressing 1/η as a weight −1/2 Poincaré series, and Lemma 3.8, which unfolds the Petersson inner product of Pν against a cusp form; the key output is Theorem 3.10, Df = 24ν ⟨[1/η,η]ν, f⟩, converting the inner product into weighted sums of twisted quadratic Dirichlet series.
What would settle it
Take ν=6 and compare the right-hand side of Theorem 1.4 with p(n) for a range of n, evaluating Tr12(n) directly from the definition of the twisted Dirichlet series D(Δ;s) with increasing cutoffs; any mismatch beyond numerical truncation would falsify the explicit trace formula, and divergence as the cutoffs grow would point specifically at the unfolding step in Lemma 3.8.
Extended reading notes
Core claim
For every integer ν≥0 the q-series Pν(τ) := [1/η(τ), η(τ)]ν, the ν-th Rankin–Cohen bracket of the partition generating function 1/η with η, is a holomorphic modular form of weight 2ν on SL2(Z). Expanding Pν with Euler's pentagonal number theorem gives a coefficient identity that solves for p(n); determining Pν as a modular form yields Theorem 1.4: for ν≥6, ν≠7 and n≥1, p(n) equals 1/gν(n,0) times the divisor sum −(4ν/B2ν) binom(2ν−2,ν−2) σ2ν−1(n), the weight-2ν Hecke trace Tr2ν(n), and the pentagonal sum Σ_{k≠0} (−1)^{k+1} gν(n,k) p(n−ω(k)). In the ν=6 case, Tr12(n) = −33108590592/691 τ(n), giving a partition-theoretic formula for Ramanujan's tau-function that implies τ(n) ≡ σ11(n) (mod 691). The proof shows that 1/η is a negative-weight Poincaré series, unfolds the Petersson inner product of Pν against Hecke eigenforms, and evaluates the resulting integrals as infinite weighted sums of twisted quadratic Dirichlet series.
Load-bearing premise
The load-bearing premise is that the Poincaré series for 1/η(τ) can be unfolded against the Petersson inner product despite the convergence issues the proof sets aside; the explicit form of the Hecke trace Tr2ν(n) in Theorem 1.4 depends on this interchange.
Editorial extensions
If this is right
- Euler's classical recurrence is the ν=0 case of the family, and the ν=1 case is the zero modular form, so the classical result is genuinely the first member of the family.
- For ν ∈ {2,3,4,5,7} there are no nontrivial cusp forms, so the recurrence involves only the divisor sum σ2ν−1(n) and the pentagonal sum.
- For ν=6 the trace term is a constant multiple of Ramanujan's tau-function, producing a partition-number formula for τ(n) that immediately implies τ(n) ≡ σ11(n) (mod 691).
- For every ν≥6 with ν≠7, the recurrence expresses p(n) through a divisor function, Hecke eigenvalues, and twisted quadratic Dirichlet series, giving explicit arithmetic formulas at each weight.
- The ν=12 example shows that the weight-24 Hecke trace can be recovered from the first two partition numbers p(1) and p(2), so the recurrences also compute Dirichlet-series data from partition data.
Reading between the lines
- A natural extension the paper leaves implicit is that any weakly holomorphic modular form with arithmetic coefficients and a Poincaré-series expansion should yield an analogous family of recurrences; p(n) is only the first such example.
- The ν-family suggests higher-weight analogues of the τ(n) ≡ σ11(n) (mod 691) congruence: the denominators in Corollary 1.3, such as 3617 and 43867, are the natural Bernoulli-modulus candidates for congruences between τ2ν(n) and σ2ν−1(n), though the paper does not state these.
- A testable computational extension is to use the recurrences as independent checks on large partition computations: each weight ν gives a separate identity that must hold, so agreement across several weights would provide strong numerical confirmation.
- The method of extracting normalized Petersson norms and Dirichlet-series values from a few partition coefficients, as done for weight 24, could be pushed to higher weights where the cusp-form space has dimension greater than two, yielding new trace computations.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to generalize Euler's pentagonal-number recurrence for the partition function p(n). For each integer ν ≥ 0 it defines P_ν(τ) = [1/η(τ), η(τ)]_ν as a Rankin–Cohen bracket, states that P_ν is a holomorphic modular form of weight 2ν on SL_2(Z), and then derives recurrences for p(n) by identifying the Eisenstein and cuspidal parts of P_ν. For ν ∈ {2,3,4,5,7} the recurrence involves only a divisor sum and the pentagonal-number tail; for ν ∈ {6,8,9,10,11,13} it additionally involves the coefficient τ_{2ν}(n) of a one-dimensional cusp space; and for general ν ≥ 6, ν ≠ 7, Theorem 1.4 expresses p(n) in terms of a divisor sum, a Hecke trace Tr_{2ν}(n) built from twisted quadratic Dirichlet series, and the pentagonal-tail sum. The ν = 6 case is rewritten as a formula for Ramanujan's tau-function.
Significance. If the proof is completed, the paper provides a genuinely new infinite family of explicit recurrences for the partition function, unifying Euler's classical recurrence with modular-form trace formulas. The main structural idea — realizing 1/η as a Poincaré series and using unfolding to compute Petersson inner products of Rankin–Cohen brackets — is attractive and potentially reusable. The paper also gives explicit constants, a clean degeneration to known divisor-sum recurrences for small weights, and a striking partition-theoretic formula for τ(n) that immediately yields Ramanujan's congruence τ(n) ≡ σ_11(n) (mod 691). The detailed lemmas on derivatives of Whittaker functions and hypergeometric evaluation are a useful contribution. However, the proof as printed contains a load-bearing multiplier-system inconsistency and an explicitly deferred convergence interchange, so the central claim is not yet fully supported.
major comments (3)
- [§2.1, Lemma 2.2] Lemma 2.2 assigns the same multiplier ε to both η and 1/η. Taking reciprocals in the first identity forces the multiplier for 1/η to be ε⁻¹; for the translation T these differ, since η(τ+1)=e^{πi/12}η(τ) but 1/η(τ+1)=e^{-πi/12}/η(τ). This is not cosmetic: Lemma 2.3(2), the proof of Theorem 1.1, and Proposition 3.4 all use the same ε for 1/η. If both factors had multiplier ε, then the Rankin–Cohen bracket [1/η,η]_ν would have multiplier ε², not the trivial multiplier used in Lemma 3.8 and Theorem 1.4. The proof should be run with the inverse multiplier for 1/η throughout, and the statements of Lemmas 2.2, 2.3(2), Proposition 3.4, and Lemma 3.8 must be adjusted consistently.
- [§3.4, Lemma 3.8] The proof begins with 'Ignoring the convergence issues' and then interchanges an infinite Poincaré series for 1/η with the Rankin–Cohen bracket and with the Petersson inner product integral. The summands involve derivatives of Whittaker functions and η, and no dominated convergence or truncation argument is supplied. This interchange is precisely what produces the termwise integral in Lemma 3.9 and, through Lemmas 3.11–3.13, the identity D_f = 24^ν ⟨[1/η,η]_ν,f⟩ in Theorem 3.10. Without a justification of this step, the explicit form of Tr_{2ν}(n) in Theorem 1.4 is unsupported. The authors should either provide a convergence proof or replace the interchange by a limiting argument from truncated Poincaré series.
- [§3.2, Proposition 3.4] To apply Lemma 3.3, the difference between 1/η and the Poincaré series P_∞(τ,24,−1/2,ε) must have vanishing principal parts at all cusps of SL_2(Z). The proof only checks the principal part at ∞ and states that ε 'clearly vanishes' on Γ(24³). A multiplier system does not vanish, and equality at one cusp does not imply equality at all cusps. This step needs a proof or a reference; otherwise the Poincaré-series representation of 1/η, which is the input to Lemma 3.8, is not established.
minor comments (6)
- [§2.2/§4] The Eisenstein coefficient α_ν = −4ν/B_{2ν} binom(2ν−2,ν−2) in Theorem 1.4 is asserted without derivation; the text says it is 'straightforward to compute the constant terms,' but the computation is not shown. Please include it or provide a precise reference.
- [§1, Theorem 1.4 and examples] The paper should state explicitly that p(m) = 0 for negative integers m, since the recurrences involve p(n−ω(k)) for large |k|.
- [§3.1, proof of Proposition 3.4] The phrase 'ε clearly vanishes on Γ(24³)' should be replaced by 'ε is trivial on Γ(24³)'.
- [§3.3, Lemmas 3.6 and 3.7] In Lemma 3.6, the factor (−3/2)_r on the right-hand side should be (−3/2)_j, and the proof of Lemma 3.7 should state the summation index clearly.
- [§1, ν=6 numerical check] The numerical check Ĥ_∆(100,2000) = −49.608382… is described only as a 'short computer computation'; please include the code or state the numerical precision and the exact truncation, so the check is reproducible.
- [§1, ν=12 example] The ν=12 example uses p(1)=1 and p(2)=2 to solve for D_{f_i}/||f_i||; this is an illustration of Theorem 1.4 rather than a proof of it, and the text should say so explicitly to avoid any appearance of circularity.
Circularity Check
No significant circularity: the recurrences are derived from modular-form decompositions and unfolding, not from the partition function itself.
full rationale
The derivation chain is self-contained. P_nu is defined as the Rankin-Cohen bracket [1/eta, eta]_nu (eq. 1.5), and its q-expansion (1.6) follows from Euler's product identities for eta and 1/eta (eq. 1.3 and the pentagonal number theorem); this expansion contains p(n - omega(k)) by construction, but that is the quantity being re-expressed, not an assumed property of the modular form. Theorem 1.1 (modularity of P_nu) is proved from Ramanujan differential identities and general transformation laws; it does not invoke the target recurrence. Corollaries 1.2 and 1.3 compare coefficients in a finite-dimensional space; the constants beta_nu are coefficients of the fixed modular form P_nu in the basis {E_{2nu}, Delta_{2nu}}, so they are determined by the modular identity, not fitted to the recurrence. For the general case, Theorem 3.10 computes <[1/eta, eta]_nu, f> by unfolding (Lemma 3.8) and evaluates the resulting integrals in terms of the twisted Dirichlet series D(f;s), which is defined independently of p(n) (eqs. 1.10-1.13). The final recurrence (Theorem 1.4) is then obtained by comparing the q-coefficient of the identity P_nu = alpha E_{2nu} + cusp part and solving for g_nu(n,0)p(n); this is an algebraic rearrangement of a proved identity. The nu=12 example uses known p(1)=1 and p(2)=2 to evaluate the Hecke trace constants after the theorem is established; this is an illustration, not part of the proof, and does not feed back into the derivation. The quoted self-citations (e.g., Bringmann-Ono [4] for harmonic Maass Poincare series) are to standard published results, and the key representation of 1/eta as a Poincare series is proved in Proposition 3.4 using Lemma 3.3, which is also proved in the text. The manuscript's own caveat 'Ignoring the convergence issues' in Lemma 3.8 and the multiplier assignment in Lemma 2.2 are potential mathematical gaps, but they are correctness risks, not circular reductions: no step assumes the conclusion that p(n) satisfies the stated recurrence.
Assumptions & free parameters
assumptions (6)
- standard math E4 and E6 generate the graded ring of holomorphic modular forms on SL2(Z), and dim S_{2ν} is 0 or 1 for the listed ν
- standard math Ramanujan's differential identities for E2, E4, E6 (Lemma 2.1)
- standard math Harmonic Maass Poincaré series are absolutely convergent and determined by principal parts (Lemmas 3.1 and 3.3)
- standard math DLMF integral and derivative identities for Whittaker functions (13.23.1, 13.15.15)
- standard math Pfaff-Saalschütz summation identity for 3F2 hypergeometric series
- standard math Hecke eigenforms on SL2(Z) have real Fourier coefficients and are orthogonal for the Petersson inner product
Cite this review
Pith. "Pith review of Pentagonal number recurrence relations for $p(n)$." pith.science (2026). https://pith.science/paper/IX3DVMTZ
@misc{pith2026241116968,
author = {Pith},
title = {Pith review of: Pentagonal number recurrence relations for $p(n)$},
year = {2026},
howpublished = {\url{https://pith.science/paper/IX3DVMTZ}},
note = {Machine review of arXiv:2411.16968}
}
abstract
We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-\omega(k)), $$ where $\omega(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $\nu=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $\nu\geq 0,$ we prove for positive $n$ that $$ p(n)=\frac{1}{g_{\nu}(n,0)}\left(\alpha_{\nu}\cdot \sigma_{2\nu-1}(n)+ \mathrm{Tr}_{2\nu}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_{\nu}(n,k)\cdot p(n-\omega(k))\right), $$ where $\sigma_{2\nu-1}(n)$ is a divisor function, $\mathrm{Tr}_{2\nu}(n)$ is the $n$th weight $2\nu$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_{\nu}(n,k)$ is a polynomial in $n$ and $k.$ The $\nu=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot \tau(n). $$
Forward citations
Cited by 1 Pith paper
-
Euler-type recurrences for $t$-color and $t$-regular partition functions
An infinite family of triangular-number recurrences for the 3-colored partition function, plus Euler-type recurrences for 2-colored and all t-regular partition functions, derived by Rankin-Cohen bracket methods.
Reference graph
Works this paper leans on
-
[1]
G. E. Andrews, The theory of partitions , Cambridge University Press, Cambridge, 1984
work page 1984
-
[2]
G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999
work page 1999
-
[3]
K. Bringmann, A. Folsom, K. Ono, and L. Rolen, Harmonic Maass forms and mock modular forms: Theory and appl ications, Amer. Math. Soc. Colloq. 64, Amer. Math. Soc., Providence 2017
work page 2017
-
[4]
K. Bringmann and K. Ono, Coefficients of harmonic Maass forms , Proceedings of the 2008 University of Florida Conference o n Partitions, q-series and Modular Forms. Dev. Math., vol. 23 (2012), 23–28
work page 2012
-
[5]
Cohen, Sums involving the values at negative integers of L-functions of quadratic characters , Math
H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters , Math. Ann. 217 (1977) 81–94
work page 1977
-
[6]
NIST, Digital Library of Mathematical Functions . https://dlmf.nist.gov
-
[7]
Ono, The web of modularity: Arithmetic of the coefficients of modul ar forms and q-series, Conf
K. Ono, The web of modularity: Arithmetic of the coefficients of modul ar forms and q-series, Conf. Board Math. Sciences no. 102, Amer. Math. Soc., Providence, 2004
work page 2004
-
[8]
Rademacher, On the expansion of the partition function in a series , Ann
H. Rademacher, On the expansion of the partition function in a series , Ann. of Math. 44, (1943), 416-422
work page 1943
Show all 13 references
-
[9]
Ramanujan, On certain arithmetical functions , Trans
S. Ramanujan, On certain arithmetical functions , Trans. Camb. Phil. Soc., 22 (1916), 159–184
1916
-
[10]
R. A. Rankin, The construction of automorphic forms from the derivatives of a given form , J. Indian Math. Soc. 20 (1956) 103–116
1956
-
[11]
R. A. Rankin, Modular forms and functions , Cambridge Univ. Press, Cambridge, 1977
1977
-
[12]
Serre, A course in arithmetic , Graduate Texts in Mathematics, No
J.-P. Serre, A course in arithmetic , Graduate Texts in Mathematics, No. 7, Springer-Verlag, Ne w York, 1973
1973
-
[13]
Indian Acad
Don Zagier, Modular forms and differential operators , Proc. Indian Acad. Sci. Math. Sci., 104 (1) (1994), 57–75. Dept. of Mathematics, University of Virginia, Charlottesvil le, V A 22904 Email address : vhe4ht@virginia.edu Email address : ken.ono691@virginia.edu Dept. of Mathe...
1994
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.