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REVIEW 3 major objections 5 minor 33 references

On the parity of coefficients of eta powers

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Pith's one-line read The paper establishes a complete structural picture of the parity density of eta powers: D(r) exists, is dyadic rational, vanishes exactly for r dividing or divisible by 32 or 48, and obeys the upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4…

desk verdict A genuinely useful paper on parity densities of eta powers, with a real but fixable gap in the level-9 citation and some omitted verifications. read the letter →

arxiv 2411.17638 v1 pith:AJWYH4E5 submitted 2024-11-26 math.NT

Steven Charlton , Lukas Mauth , Anna Medvedovsky This is my paper · ORCID
classification math.NT MSC 11F3311F0311F2011F8011P8311R45
keywords Dedekindetafunctionmodularformsmodulo2GaloisrepresentationsdensityofFouriercoefficientsdihedralChebotarevtheorempartitionparitylevel9
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

The paper studies, for every positive integer r, the density D(r) of primes ℓ for which the leading coefficient of the Hecke operator U_ℓ applied to the normalized q-series η^r(m_r τ) is odd. It proves three structural theorems: D(r) always exists and is a dyadic rational; D(r)=0 exactly when r is a divisor or a multiple of 32, or a divisor or a multiple of 48; and D(r) obeys the strict bounds D(n)<1, D(2n)<1/2, and D(4n)<1/4 with exactly four exceptions (n=9,15,18,30). It also computes D(r) explicitly for several infinite families of r attached to dihedral and abelian mod-2 modular forms, obtaining values that are powers of 1/2. The motivation is the parity problem for the partition function: the tested subsequence is modelled on the sequence δ_ℓ=$24^{{-1}}$ mod ℓ that supports partition congruences.

What carries the argument

The central object is the sequence δ_{ℓ,r}, defined as the least positive integer n satisfying n≡b_r (mod m_r) and ℓ|n, so that p_r(δ_{ℓ,r}) is the leading coefficient of U_ℓ(P_r); D(r) is the density of primes for which this coefficient is odd. The workhorse identity is Proposition 2.4: P_r(q)≡$Δ^{{b_r}}$ (mod 2) when 3|r, and P_r(q)≡$C^{{b_r}}$ (mod 2) otherwise, with C=η(3τ)^8. This reduces eta-power questions to the prime coefficients of the mod-2 modular forms Δ^s and C^s. Those coefficients are frobenian, meaning that the value at ℓ depends only on the Frobenius conjugacy class in a finite Galois group, so Chebotarev counting computes the relevant densities; for the level-1 basis forms m(a,0) and m(0,a), the density δ(m(a,0))=1/$2^{{u(a)+v(a)+1}}$ follows from counting residue classes modulo $2^{{d(a)+1}}$ in which the coefficient of x^a in a modified Chebyshev polynomial is 1, a count governed by the base-2 digit statistics of a.

What would settle it

Compute, for each i∈{5,7,13,17,19}, the coefficients a_ℓ(α_i) for primes ℓ up to a few thousand; a single prime with a_ℓ(α_i)=1 while ℓ≢i (mod 24), or a_ℓ(α_i)=0 while ℓ≡i (mod 24), would disprove Proposition 6.1 and change the level-9 densities and the Theorem C(iii) exceptions.

Watch

Extended reading notes

Core claim

The paper's central claim is a complete description of the parity density D(r) of eta powers. For the normalized series P_r(q)=η^r(m_r τ), the quantity D(r) measures the proportion of primes ℓ for which the first formal coefficient of U_ℓ(P_r) is odd. The paper proves that D(r) always exists and is a dyadic rational (a fraction whose denominator is a power of 2), that D(r)=0 if and only if 32 or 48 divides r, or r divides 32 or 48, and that D(r) satisfies the strict upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4 with the four exceptions n=9,15,18,30. For the infinite families in Theorem D, where the relevant mod-2 forms are dihedral, the densities are explicit powers of 1/2 determined by the base-2 digits of the exponents. The proofs identify P_r modulo 2 with $Δ^{{b_r}}$ or $C^{{b_r}}$, where C=η(3τ)^8 is the unique normalized cusp form of weight 4 and level 9, then express D(r) as a sum of prime-coefficient densities of the resulting mod-2 forms, computed by counting Frobenius elements in finite Galois extensions of Q.

Load-bearing premise

The level-9 density values rest on Proposition 6.1, which asserts that six explicit forms α_i are abelian with field of determination Q(μ_24) and that a_ℓ(α_i)=1 if and only if ℓ≡i mod 24, but the proof is given in full only for i=11, the other five cases being dismissed as straightforward.

Editorial extensions

If this is right

  • For every r≥1, the density D(r) is a dyadic rational, so the parity of eta-power coefficients in the tested subsequence has a well-defined, computable limiting frequency.
  • If the theorem holds, the vanishing classification is exact: the only powers of η whose tested coefficient is odd on a set of primes of density zero are those with r dividing or divisible by 32 or 48.
  • The upper bounds imply that for r of the form 2n or 4n, the order of infinity of U_ℓ(η^r) fails to be maximal for a positive proportion of primes—more than half of all primes for D(2n), and more than three-quarters for D(4n) except at the four listed n.
  • For the infinite families in Theorem D, the densities are explicit powers of 1/2, for example D(12·z_n)=2^{-(n+1)} and D(3·w_n)=3·2^{-(n+1)} for n≥2, making the values computable directly from the binary expansions of the exponents.
  • Conditional on the density expectation stated in Remark 7.4, most odd n should satisfy D(3n)=1/2 and D(6n)=1/4, so the explicitly computed small densities in Theorem D would be the atypical cases rather than the generic ones.

Reading between the lines

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

  • A structural reading of the proof of Theorem B is that every zero-density case is a pure congruence/support obstruction: P_r modulo 2 and the tested index δ_{ℓ,r} never align modulo a suitable power of 2 for almost all ℓ. If that reading is right, the zero locus of such densities for other moduli and other q-series would also be characterized by support conditions rather than by finer Galois struc
  • The formula δ(m(a,0))=1/2^{u(a)+v(a)+1} ties these densities to binary digit statistics, suggesting that other families of eta powers with dihedral reduction will have densities expressible as powers of 1/2 determined by the base-2 digits of their exponents.
  • Because the paper's Remark 7.3 shows the same frobenian argument applies to any integral modular form supported on an arithmetic progression, the framework should compute parity densities for other subsequences of coefficients of modular forms, not only powers of eta.
  • The paper does not settle the partition-function analogue D(-1); extending the method to η^{-1} would require a mod-2 model for that nonholomorphic case, and any such extension would directly inform the Partition Parity Conjecture.
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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

3 major / 5 minor

Summary. The paper introduces a notion D(r) measuring, for each normalized eta-power eta^r, the natural density of primes ell for which the leading coefficient of U_ell(eta^r) modulo 2 is nonzero (equivalently, for which the order at infinity of U_ell(eta^r) modulo 2 is minimal). It proves four main results: Theorem A states that D(r) always exists and is a dyadic rational; Theorem B classifies exactly when D(r)=0, namely when r divides or is a multiple of 32 or 48; Theorem C gives unconditional upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4 with four explicit exceptions; Theorem D computes D(r) exactly for several infinite families of eta-powers whose reductions are dihedral mod-2 forms, giving densities such as 2^{-n} and 3*2^{-(n+2)}. The proofs proceed by reducing eta-powers modulo 2 to powers of Delta or of the level-9 form C (Proposition 2.4), expressing D(r) as a finite sum of Bellaiche densities delta(f) (Proposition 7.1 and Corollary 7.2), and then computing delta(f) for dihedral and abelian forms. The paper also supplies proofs of two unpublished Bellaiche results, including a combinatorial formula for the density of dihedral basis elements m(a,0), m(0,a) in terms of the base-2 digits of a (Theorem 4.6).

Significance. If the central claims are correct, the paper gives a complete structural classification of the vanishing of D(r) and near-optimal upper bounds, together with the first exact computations for infinite families of eta-powers; the reduction to Bellaiche density and the combinatorial density formula are elegant and parameter-free. The paper is also valuable for communicating, with proof, two of Bellaiche's unpublished results and for extending some level-1 techniques to level 9. The main theorems are falsifiable and are checked against extensive numerics, and no fitted parameters appear anywhere in the argument. The significance is somewhat reduced, but not destroyed, by the fact that two load-bearing points are not fully proved inside the manuscript: the level-9 nonvanishing theorem is imported by a citation whose exact statement and hypotheses are not given, and five of the six cases of Proposition 6.1 are dismissed as straightforward without proof.

major comments (3)
  1. [Section 6.3, Theorem 6.2(ii)] The proof of Theorem 6.2(ii) is only the sentence 'See Bellaiche [B.Im, Theorem I]'. The only nonvanishing theorem for mod-2 forms actually stated in this paper is Theorem 4.3, which is explicitly for the level-1 space K. The level-9 space K(9) is constructed later in Section 6.1, so Theorem 6.2(ii) is not an immediate corollary of Theorem 4.3 unless [B.Im, Theorem I] is itself a level-N theorem. This is load-bearing: the converse half of Theorem B uses Corollary 6.3 to conclude that delta(C^{b_r})=0 only for the stated b_r, and Theorem 6.2(iii) uses delta(g)>0 to exclude all but six forms; without a valid level-9 nonvanishing statement, the claimed classification of r with D(r)=0 and the strictness in Theorem C(iii) are unsupported. Please quote the precise statement of [B.Im, Theorem I], verify that its hypotheses cover the space K(9), or supply a direct proof for this level-9 case.
  2. [Section 6.2, Proposition 6.1] Proposition 6.1 asserts that the six forms alpha_5, alpha_7, alpha_11, alpha_13, alpha_17, alpha_19 are abelian with field of determination Q(mu_24) and satisfy the coefficient criterion a_ell(alpha_i)=1 if and only if ell ≡ i mod 24. The proof gives detailed arguments only for i=11; the other five cases are disposed of with the sentence 'The other cases are straightforward.' These omitted checks are load-bearing because Corollary 6.3, Theorem 6.2(iii), the strictness statement in Theorem C(iii), and the level-9 entries in Remark 1.2 all depend on the coefficient criterion and on the exceptional status of exactly these six forms. An error in any omitted case would change the exceptional set in Theorem C(iii) and the claimed densities. Please provide complete verifications for all five remaining cases, including the representation/uniqueness arguments for the quadratic forms listed in the table.
  3. [Section 7.3, proof of Theorem C(ii)] In the subcase r=2s with s prime to 6, the text says that T_5 C^s = 0 'immediately gives D(r) ≤ 3/8 < 1/4', but the target inequality in Theorem C(ii) is D(2n)<1/2, and 3/8 is not less than 1/4. This appears to be a typographical slip rather than a mathematical error, since the displayed computation gives D(r) ≤ 3/8 < 1/2. Please correct the inequality.
minor comments (5)
  1. [Abstract and throughout] The names 'Nicholas-Serre' and 'Nicolas-Serre' are used inconsistently; the correct spelling in the references is Nicolas-Serre.
  2. [Section 6.1(viii)] The notation 'C^i ∈ K(9)_i for every i relatively prime to 24' would be clearer if the condition that i is taken modulo 24 were repeated, since C^i is supported on exponents congruent to i modulo 24.
  3. [Section 7.2] In the proof of Theorem B, the phrase 'r = 16n with n > 1 odd' should say 'with n odd and prime to 3', because the argument uses the level-9 reduction and Corollary 7.2 for r prime to 6.
  4. [Section 4.2, Lemma 4.12] In the proof of Lemma 4.12, the sentence 'we establish that c, cg, or g^{2^d} are in the kernel of t_f' should read 'are not in the kernel'; otherwise the displayed Cases contradict the conclusion.
  5. [Section 6.2, table in Proposition 6.1] The table lists the field K for alpha_7 and alpha_13 as Q(sqrt(-3)), yet the proposition states that the field of determination is Q(mu_24); please clarify the relation between the quadratic field used for representation by the quadratic form and the full abelian field of determination.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: D(r) is not defined in terms of the densities it predicts, the main formulas are proved from external theorems, and the self-citations are not load-bearing.

full rationale

The paper's core derivation is not circular. Definition 2.1 defines D(r) directly in terms of the coefficients pr(delta_{ell,r}); Proposition 7.1 and Lemma 2.5 then prove an exact identity D(r) = sum delta(T_{u_c}f), rather than assuming it. No fitted parameter is renamed a prediction: the table of D(r) values is labeled 'expected' with proved entries computed in Sections 7.2--7.5, and the proofs use the Bellaiche density delta(f), not the values D(r) themselves. The vanishing classification Theorem B rests on Corollaries 4.5 and 6.3. Corollary 4.5 follows from Theorems 4.3--4.4; the paper supplies a full proof of Theorem 4.4 and cites Bellaiche's published [B.Im, Theorem 1] for Theorem 4.3. Corollary 6.3 uses Proposition 6.1 and Theorem 6.2(ii). The self-citations to [DM, Theorem 3.4] and [Me] are to published independent work by the third author and do not reduce any argument to the present paper's own claims. Two verification gaps are flagged for correctness, not circularity: Section 6.2 gives full details only for alpha_11 and dismisses the other five alpha_i as 'straightforward', and Section 6.3 proves Theorem 6.2(ii) by the bare citation 'See Bellaiche [B.Im, Theorem I]' without stating whether that theorem covers level Gamma_0(9). If [B.Im, Theorem I] does not cover level 9, then Theorem B's converse and Theorem C(iii) lose support; but this is an omitted-proof or unsupported-citation issue, not a reduction of a target equation to its own input. The derivation chain contains no step where an output quantity is defined in terms of itself or where a fitted parameter is presented as a prediction. Score 2 reflects minor self-citations that are not load-bearing; the central claims have independent mathematical content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or fitted constants appear; the input is the fixed sequence of eta powers. The main external input is the published and unpublished Bellaiche-Serre theory of mod-2 modular forms, plus one paper-internal assertion in Proposition 6.1 whose five cases are not shown.

assumptions (6)
  • domain assumption Bellaiche's generalized eigenform and density theory, including the unpublished Theorem 4.4 and Theorem 4.6, is correct and applies to level 9.
    The authors reproduce proofs for the two unpublished theorems, but cite [B.Im, Theorem I] for the positivity of delta(f) in level 9 in Theorem 6.2(ii).
  • domain assumption The mod-2 Hecke algebra A(9) of level 9 is local with a unique maximal ideal and carries a Galois pseudorepresentation t9 with t9(Frob_l) = T_l.
    Section 6.1, items (vii) through (ix), establishes this by adapting [B.Rep] and [B.Im] to level 9; it is not fully proved within the preprint.
  • standard math Serre reciprocity and the Nicolas-Serre structure theorem A = F2[[T3, T5]] for level-1 mod-2 modular forms.
    Used for uniqueness of the Hecke eigensystem and for the level-1 structure in Sections 3 and 6.
  • standard math Markshaitis-Serre description G = <g, c | c^2 = 1> of the Galois group of the maximal pro-2 extension unramified outside 2.
    Used in Section 3.4 and cited from [B.Sp, Section 2].
  • standard math Kummer's theorem on p-adic valuations of binomial coefficients as stated in Lemma 5.3.
    Used in the proof of the combinatorial lemma in Section 5.
  • ad hoc to paper The five unproved cases of Proposition 6.1, asserting that alpha_i is abelian with field Q(mu_24) and the coefficient criterion a_l(alpha_i) = 1 iff l = i mod 24, are correct.
    The preprint says the remaining cases are straightforward but does not display the computations; these cases are used in the level-9 density results.

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Pith. "Pith review of On the parity of coefficients of eta powers." pith.science (2026). https://pith.science/paper/AJWYH4E5

@misc{pith2026241117638,
  author       = {Pith},
  title        = {Pith review of: On the parity of coefficients of eta powers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJWYH4E5}},
  note         = {Machine review of arXiv:2411.17638}
}
abstract

We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $\eta$-function, analogous to the sequence $\delta_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density $D(r)$ for a normalized eta-power $\eta^r$ measuring the proportion of primes $\ell$ for which the order at infinity of $U_\ell (\eta^r)$ modulo 2 is maximal. We relate $D(r)$ to a notion of density measuring nonzero prime Fourier coefficients introduced by Bella\"iche, and use this to completely classify the vanishing of and establish upper bounds for $D(r)$. Furthermore, for several infinite families of $\eta$ powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bella\"iche, we explicitly compute the densities $D$. We rely on Galois-theoretic techniques developed by Bella\"iche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bella\"iche's unpublished results on densities of mod-$2$ modular forms.

Figures

Figures reproduced from arXiv: 2411.17638 by the authors.

Figure 1
Figure 1. Random walk obtained from parity of the partition function, moving up for p(i) even, and down for p(i) odd. More precisely, the plot shows (n,P i≤n (−1)p(i) ), for 1 ≤ n ≤ 2 · 108 . The blue curve is 0 ± √ n, and the orange curve is 0 ± 2 √ n, indicating the regions 1 and 2 standard deviations away from the mean, with probabilities ∼ 66% and ∼ 95% respectively [PITH_FULL_IMAGE:figures/full_fig_p038_1.png] view at source ↗
Figure 2
Figure 2. Random walk obtained from parity of the partition function, moving up for p(δπi ) even, and down for p(δπi ) odd. More precisely, the plot shows (n,P i≤n (−1)p(δπi ) ), for 1 ≤ n ≤ 2 · 108 . The blue curve is 0 ± √ n, and the orange curve is 0 ± 2 √ n, indicating the regions 1 and 2 standard deviations away from the mean, with probabilities ∼ 66% and ∼ 95% respectively [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.