Joint level-weight murmurations: prime averaging and the cubic pointwise range

Julien Cardi

Joint level-weight murmurations: prime averaging and the cubic pointwise range

T0 review · 0 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves an unconditional joint level–weight murmuration law: after averaging over primes, the trace converges to an explicit atomic measure, and fixed primes follow the known density up to the cubic weight range.

desk verdict Unconditional joint level-weight murmuration theorem that looks right in every spot I checked; the one real risk is the 'exhaustive' local Euler-product tables that fix the atomic measure. read the letter →

arxiv 2607.08982 v2 pith:DK67VPUC submitted 2026-07-09 math.NT

Julien Cardi This is my paper · ORCID

classification math.NT MSC 11F1111F3011M2011N13

keywords murmurationsholomorphicnewformsrootnumbersquarefreelevelweightaspectHurwitzclassnumbersatomiclimitingmeasureHeckeeigenvalues

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 murmurations: the correlation between the root number of a holomorphic newform and its Hecke eigenvalue at a prime, when the squarefree level N≍X and the even weight k≍K vary together. It proves two unconditional results. First, after logarithmically averaging over primes, the normalized correlation tends to a completely explicit atomic measure μ_at, supported at points (q/a)², uniformly for every fixed polynomial range of K. Second, for a fixed prime p≍XK², the trace has an asymptotic given by the known weight-aspect murmuration density, valid up to K=X^{3−ε}. The proof avoids unproved hypotheses by combining an exact trace formula, character-sum truncation, and a Poisson-summation decomposition that isolates a 'zero frequency' main term; the atomic measure emerges from a Bessel summation identity.

What carries the argument

The central objects are the explicit atomic measure μ_at of (1.4) in the prime-averaged theorem and the fixed-weight density M_k(y) in the fixed-prime theorem. The identity that carries the argument is a localized Bessel summation identity (Σ_{n odd} n J_n(x) = x/2, smeared) that converts the Bessel series defining M_k into point masses at (d/s)². Around it: an exact trace formula expressing the spectral sum as a finite sum of Hurwitz class numbers; a character-sum truncation of the quadratic L-series; and a Poisson-summation decomposition in the level that groups the Fourier expansion by exact additive conductor, isolating the zero frequency that carries the main term. For prime averaging,

What would settle it

Independently enumerate the local averages a_{ℓ,N,r}(j,t) for small primes ℓ (say 2, 3, 5) and small j,t, and compare with the two tables; any mismatch invalidates the main terms. Alternatively, compute the left side of (1.5) for a test function F concentrated near a single atom (q/a)² and check that the limit equals the asserted mass within the claimed logarithmic error.

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Extended reading notes

Core claim

The central claim is that the main term in both regimes is fully explicit and unconditional. In the prime-averaged regime, the normalized trace equals the integral of the test function against an explicit atomic measure μ_at, with a logarithmic error term; in the fixed-prime regime, the trace equals a weighted convolution of the test function with the fixed-weight murmuration density M_k, plus a power-saving error. The proof traces the spectral sum to Hurwitz class numbers via the Atkin–Lehner trace formula, truncates the quadratic L-series by a character-sum estimate, and applies Poisson summation in the level. The key structural point is that the zero frequency of this decomposition carrie

Load-bearing premise

The proof's main term and the atomic measure inherit their exact coefficients from two local Euler-factor tables that are asserted to be exhaustive (and whose expansion was partly AI-assisted); if any local class at an odd prime or at 2 is missed or misweighted, both theorems fail.

Editorial extensions

If this is right

The joint level–weight murmuration law holds unconditionally in every fixed polynomial range X^{ε} ≤ K ≤ X^{A}, with no unproved hypothesis needed to describe the root-number-weighted trace after prime averaging.
The fixed-prime asymptotic is valid up to K = X^{3−ε}; the cubic endpoint is the limit of the present absolute-value treatment of the nonzero Poisson frequencies, not a proven barrier.
The limiting measure is atomic and explicit, with atoms at (q/a)² and computable masses, so the murmur is a discrete rather than continuous phenomenon in this regime.
Without the root-number weighting, the prime-averaged trace is asymptotically just the average of the test function; the murmur is entirely produced by the root-number sign.
The proof separates two mechanisms: fixed primes require controlling nonzero additive frequencies from Poisson summation in the level, while prime averaging replaces them by the distribution of primes in arithmetic progressions.

Reading between the lines

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

If the local-coefficient tables (5.33) and (5.39) are correct, the atomic masses are fully determined; a direct numerical check of the left side of (1.5) near one atom (q/a)² would test the tables independently of the proof.
The fixed-prime range may extend beyond ρ=3 if distinct nonzero Fourier frequencies cancel rather than being estimated in absolute value; whether a secondary transition occurs at the cubic point is a concrete open question.
The same structural split — zero frequency versus nonzero frequencies, pointwise versus averaged — might transfer to non-squarefree levels or to families with different spectral weights, giving a template for joint level–weight murmurations there.
Because the prime-averaged theorem is unconditional, it provides a benchmark against which conditional (GRH-dependent) results with harmonic weights can be compared; differences between the resulting atomic laws would reveal how the spectral weight shapes the murmur.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuine advance, and I think the main theorems are probably right, but the load-bearing part is a pair of 'exhaustive' local tables that nobody has machine-checked. That's where I'd send a referee first.

What's new: Theorem 1.1 gives an unconditional prime-averaged joint level-weight murmuration law with an explicit atomic measure, valid in every fixed polynomial weight range. Theorem 1.2 pushes the fixed-prime asymptotic up to the cubic point K=X^3, and the paper is honest that the cubic exponent is the limit of the absolute-value treatment, not a claimed barrier. The fixed-prime argument via exact additive conductor plus Burgess truncation, and the prime-averaged argument via smooth Barban-Davenport-Halberstam, are different mechanisms and that division is convincing. The trace formula is used before harmonic weights, so the density is in the natural family.

Credit where due: the bookkeeping is careful. I checked the Burgess truncation exponent, the 8D^2T=o(P) condition, the cubic-endpoint exponent in (8.1), the final error exponents -3ε*/64 and -21ε*/128, and the normalizer constant. All internally consistent. The constants αcφ/12 = B/(πζ(2)) and βcφ/12 = A/(πζ(2)) are matched against Zubrilina's published values, which is an independent cross-check rather than an input. I found no circularity or post-hoc fitting.

The soft spot is exactly the one the stress test flags. Proposition 5.6's local Euler factors, tables (5.33) and (5.39), plus Lemma 7.2's 2-adic classification, determine Bν(r)/ζ(2) and A/(2ζ(2)). Those constants feed the main terms, then W0, then the atomic measure. If one local class is missed or misweighted, both theorems fail even if every other step is fine. The tables are asserted exhaustive, and the disclosure that ChatGPT expanded proof details raises the prior for a silent slip in exactly this kind of finite computation. This is not a demonstrated error; the spot sums I did (e.g., ℓ=2 rows summing to 2/3 and 4/3) are consistent. But a finite table of this complexity is where a hidden mistake would live.

One smaller thing: Lemma 3.3 relies on Assaf's Corollary 5.14 from a preprint. Probably fine, but a published or more thoroughly checked source would be better.

Who this is for: analytic number theorists working on murmurations or trace formulas. It deserves a serious referee; desk rejection would be wrong. I'd ask the referee to spend the time on Section 5 and Lemma 7.2, and on the AI-expanded proof details, and I'd expect the paper to survive unless a local constant is wrong. My own verdict is conditional in the same way the reader's is.

Referee Report

0 major / 5 minor

Summary. The paper proves two joint level-weight murmuration theorems for squarefree levels N as X and even weights k as K. Theorem 1.1 gives, after logarithmic averaging over primes and for every fixed polynomial range X^{eps0} <= K <= X^{A0}, an unconditional asymptotic for the root-number-weighted Hecke trace ratio N_{X,K}(F)/C_{X,K}, with an explicit atomic limiting measure mu_at. Theorem 1.2 gives a fixed-prime asymptotic, uniform for X^{eps0} <= K <= X^{3-eps0}, whose main term is expressed through Zubrilina's density M_k(y). The method uses the exact Atkin-Lehner trace formula, Hurwitz class numbers, Burgess truncation, a smooth Barban-Davenport-Halberstam estimate for the prime-averaged case, and Poisson/Neumann summation for the pointwise case. The proof is organized around a small number of explicitly isolated arithmetic inputs: the local Euler-product constants in Section 5, the 2-adic level classification in Section 7, and the exact lifting identity for the zero Fourier frequency.

Significance. If the main theorems are correct, this is a substantial advance: it removes the GRH assumption for the prime-averaged weight-aspect murmuration law in the squarefree-level family, and it gives a fixed-prime asymptotic with an explicit main term all the way to the cubic weight range. The trace-formula-first approach is natural, and the independent match of the derived constants with Zubrilina's externally published alpha and beta is a strong internal consistency check. The paper also gives a clean explanation of why averaging over primes removes the obstruction that limits the fixed-prime range. The analytic mechanisms are used in a coherent way, and the displayed numerical exponents in Section 8 are internally consistent.

minor comments (5)

[Section 9, Proposition 9.2] In the reduction of the prime sum after Neumann summation there appears to be a missing factor 1/t. With Y_{t,d,s} = t K^2 d^2 / (16 pi^2 s^2), one has p/t = u K^2 d^2 / (16 pi^2 s^2), so the sum over p of (log p)(p/t) V(...) should be (Y_{t,d,s}/t) times the summed psi_K(p/Y_{t,d,s}), not Y_{t,d,s} times it. The final main term and the substitution of the integral are consistent with the corrected version, so this is a typographical or derivation typo, but it should be fixed.
[Section 6, proof of Proposition 6.4] When Lemma 6.2 is applied with 2A in place of A, the second term from the P^2 (log P)^{-2A} bound should be P / (f (log P)^A), i.e. a negative power of log P. The displayed text appears to show a positive exponent. The claimed (log P)^{-B} error in (6.10) depends on this sign, so the exponent should be corrected and a sentence added explaining that the N-sum does not destroy the power saving.
[Sections 5.2 and 7.2, Tables (5.33)/(5.39) and Lemma 7.2] The constants in (5.30) and (5.32) are load-bearing, and the tables are asserted to be exhaustive. I checked the odd-prime rows against (5.34), the 2-adic rows against (5.40)-(5.42), and the row sums (5.35)-(5.38); all entries I checked are internally consistent, and the 2-adic classification in Lemma 7.2 also checks out. Given the centrality of these finite local computations and the paper's disclosure of LLM-assisted expansion of proof details, I recommend that the authors supply a small machine-checkable verification, e.g. a PARI/Sage script, of the two tables and of Lemma 7.2. This is a robustness request, not a correction of a demonstrated error.
[Section 5.2, Table (5.33)] The row for l not dividing Nr, j=0, t >= 2 even should read (l-2)/(l-1) * l^{-t}. In the plain-text rendering this can easily be misread as l^{-2}/(l-1)*l^{-t}; please ensure the typeset formula is unambiguous.
[Sections 6-8] The parameter choices in Section 8 are coherent, but a reader would benefit from a short table summarizing the final values of eta, vartheta, beta_0, sigma, xi and the resulting error exponents. This is purely organizational and would make the proof easier to audit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main terms are derived from the trace formula and external published identities, not fitted or self-cited into the conclusions.

full rationale

I walked the derivation chain. Theorem 3.4 (Atkin–Lehner trace) converts the spectral average into class-number sums; Section 5 reduces those sums to truncated Euler products; Sections 6 and 7 complete the frequencies and identify the zero-frequency main term with the continuous integral of Zubrilina's density M_k; Section 9 applies smoothed Neumann summation to turn the Bessel series into point masses; Lemma 10.1 identifies the resulting atomic measure. At no point is a target average used as an input. The constants Bν(r)/ζ(2) and A/(2ζ(2)) are computed from the local tables (5.33) and (5.39) and only afterward matched to Zubrilina's α,β through the local identities in Proposition 6.5 and Proposition 7.5; the matching is a cross-check, not a construction. Proposition 1.3 is cited from Zubrilina [17], an external published theorem, not from the present author's own prior work, and it is used to re-express a main term already obtained from the trace formula rather than to supply that main term by assumption. There are no self-citations and no fitted parameters renamed as predictions. The asserted exhaustiveness of the local tables and the 'Use of generative artificial intelligence' disclosure are legitimate verification risks: a missed local class would change the constants and break both theorems. But that is a correctness risk, not circular reasoning. No reduction of the target asymptotic to its own input is present in the paper's equations.

Assumptions & free parameters 4 free parameters · 8 assumptions · 1 invented entities

No data are fitted anywhere. The listed free parameters are standard analytic-number-theory proof exponents whose final values are chosen in §8 only to make error terms fixed negative powers of X; they do not enter the main terms. The constants α, β, γ, B, A, W₀, ν(r) are defined objects (convergent Euler products or computed local averages), not fitted values, and they are cross-checked against Zubrilina's published constants. The axioms are standard external theorems plus the locally proved lemmas; the only non-peer-reviewed citation is Assaf [1]. The single invented object, μ_at, is a theorem output with explicitly checkable masses.

free parameters (4)

η (Burgess truncation excess) = ε*/256, ε* = min(ε₀,1)
Sets T = (XK)^{1/2+η} in (5.6); chosen in §8 small enough that Burgess savings and the cubic-endpoint exponents are fixed negative powers of X. Hand-chosen proof parameter, not data-fitted.
ϑ (square-divisor and elliptic-index truncation exponent) = ε*/512
Sets D = R = X^ϑ in (5.6); balances outer tails (Lemma 5.2) against the BDH modulus condition 8D²T = o(P) (Prop 6.4).
β₀ (squarefree-sieve cutoff) = ε*/16
Sets Z = X^{β₀}; balances the sieve-tail error (XK)^ξ Z^{-1} against the nonzero-frequency term ZT^{1/2}/X in Prop 7.4.
σ, ξ (auxiliary error exponents) = σ = ε*/64, ξ = σ/4
Chosen so (XK)^ξ ≤ X^σ and the final error exponents (−3ε*/64, −21ε*/128) are fixed negative powers of X.

assumptions (8)

domain assumption Popa's exact Atkin–Lehner trace formula [14, Theorem 4], specialized to Q=N, n=p
Theorem 3.4 rests on this external published theorem; the specialization dictionary is given in Remark 3.1.
domain assumption Zubrilina's level-aspect theorems and identity [17, Theorems 1 and 3] (Proposition 1.3 here)
Supplies the main-term density M_k and identity (1.13); published in Invent. Math. 2025 by an independent author.
standard math Burgess's character-sum bound [4]
Used in Lemma 5.1 to truncate L(1,χ_d) at T = Y^{1/2+η}; standard external theorem.
standard math Hooley's Barban–Davenport–Halberstam theorem [8, Theorem B]
Used in Lemma 6.2 for the smooth prime-averaging estimate; standard external theorem.
domain assumption Martin's dimension formula for newform spaces [13]
Used in Proposition 2.3 to compute the normalizer C_{X,K}.
domain assumption Assaf's newform trace identity [1, Corollary 5.14]
Used in Lemma 3.3: Tr(T_p W_N | S_new) = Tr(T_p W_N | S_k) for p∤N and squarefree N. Cited from an arXiv preprint rather than a peer-reviewed source.
standard math Dirichlet class-number formula in the H₁ normalization (5.2)
Used throughout §5 to expand Hurwitz class numbers into truncated L-series.
standard math Jacobi–Anger identity Σ_{n odd} n J_n(x) = x/2 and Landau's uniform Bessel bounds [12]
Used in Propositions 9.1 and 9.2 to convert the Bessel series into atomic masses.

invented entities (1)

Atomic limiting measure μ_at (1.4) independent evidence
purpose: The limit law for prime-averaged joint level–weight murmurations: point masses at (q/a)² with masses W₀(q/a)⁴∏_{ℓ|q} ℓ/(ℓ²−1)².
This is a derived output, not a postulated input: it is computed from the trace formula, the local Euler products, and the smoothed Neumann summation (Prop 6.5, Prop 9.2, Lemma 10.1). Its masses are explicit numbers any referee can evaluate and compare against numerics, so it carries its own falsifiable handle.

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Cite this review

Pith. "Pith review of Joint level-weight murmurations: prime averaging and the cubic pointwise range." pith.science (2026). https://pith.science/paper/DK67VPUC

@misc{pith2026260708982,
  author       = {Pith},
  title        = {Pith review of: Joint level-weight murmurations: prime averaging and the cubic pointwise range},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DK67VPUC}},
  note         = {Machine review of arXiv:2607.08982}
}

read the original abstract

Let (N) range over squarefree levels (N \asymp X), and let the even weight (k) vary in a smooth window (k \asymp K). We study natural root-number-weighted traces of Hecke eigenvalues at primes (p \asymp XK^2). First, uniformly for (X^{\varepsilon_0} \leq K \leq X^{3-\varepsilon_0}), we prove a fixed-prime asymptotic whose main term is expressed through Zubrilina's murmuration density. The exponent (3) is the endpoint of our absolute treatment of the nonzero Poisson frequencies. Second, after averaging the primes with logarithmic weight, we obtain unconditionally, throughout every fixed polynomial range (X^{\varepsilon_0} \leq K \leq X^{A_0}), an explicit atomic limiting measure. The fixed-prime argument groups the local Fourier expansion by exact additive conductor and uses Burgess truncation, whereas the prime-averaged argument applies a smooth Barban-Davenport-Halberstam estimate.

Figures

Figures reproduced from arXiv: 2607.08982 by the authors.

**Figure 1.** Ranges of the weight parameter. Here A0 is arbitrary but fixed before X → ∞. Proposition 1.3 (Zubrilina’s identity). For every even k ≥ 2 and y > 0, the function in (1.8) satisfies Mk(y) = α(−1)k/2−1 k − 1 X 1≤r<2 √y ν(r) q 4y − r 2 Uk−2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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