Murmurations of quadratic Hecke $L$-functions of the Gaussian field

Liangyi Zhao; Peng Gao

REVIEW 2 major objections 4 minor 14 references

Murmurations of quadratic Hecke $L$-functions of the Gaussian field

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Under GRH, the murmuration density of quadratic Hecke L-functions over the Gaussian field is an explicit arithmetic sum with no undetermined constants.

desk verdict First genuine number-field murmuration density beyond Q, computed for quadratic Hecke L-functions over Q(i) under GRH; the arithmetic checks out, but the decisive limit step is delegated, not proved. read the letter →

arxiv 2607.20853 v1 pith:Y4YKF6HT submitted 2026-07-23 math.NT

Peng Gao , Liangyi Zhao This is my paper · ORCID

classification math.NT MSC 11L3711R42

keywords murmurationdensityquadraticHeckeL-functionsGaussianfieldgeneralizedRiemannhypothesisone-levelPoissonsummationcharacterslow-lyingzeros

topics The Riemann Hypothesis

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

This paper establishes an exact formula for the murmuration density of the family of quadratic Hecke L-functions over the Gaussian field, assuming the generalized Riemann hypothesis. The density is expressed as a single explicit arithmetic sum: a Möbius-weighted sum over primary Gaussian integers l and a sign-weighted sum over nonzero lattice points k, evaluated at the two-dimensional Fourier transform of the test function. The paper shows the murmured average is completely determined, with no undetermined constant left in the main term, and that it interpolates between zero at small scale and a value fixed by the Dedekind zeta value ζ_K(2) at large scale. This matters because it turns an empirically observed oscillation pattern into a verifiable, number-theoretically meaningful quantity for a family of L-functions over a number field.

What carries the argument

The argument runs on three mechanisms. First, the family is parametrised by quadratic Hecke characters χ_{i(1+i)^5 c} of trivial infinite type, whose conductor is (1+i)^5 c for square-free c. Second, a Poisson summation formula over the Gaussian integers (Lemma 2.3) exchanges the sum over c against a sum over lattice points k, producing the sign (−1)^{N(k)} and the Fourier transform Φ̃; only perfect-square k survive in the limit. Third, under GRH a character-sum bound and the prime ideal theorem control the tail terms, allowing the Möbius identity μ² = M_Z + R_Z to be truncated at Z = X^{1/4} with negligible error. The uniform convergence of the remaining l-series supports passage to the lim

What would settle it

A direct computational check: fix a smooth bump Φ, choose y=1 and δ=0.9, and evaluate M_Φ(y,X,δ) for increasing X by summing over primes of Z[i] with norm in [yX, yX+X^δ] and over square-free c up to the support of Φ. The proof predicts agreement with the right-hand side of (1.1) up to O(X^{−(δ−3/4)/5}); a systematic discrepancy of larger order, or a failure of the error to decay as X grows, would falsify the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for a smooth compactly supported test function Φ and any fixed scale y>0 and window exponent δ∈(3/4,1), the scaled average of quadratic Hecke characters over primes of the Gaussian integers converges as X→∞ to M_Φ(y,δ) = (1/4) Σ_{l primary} μ[i](l)/N(l²) Σ_{k∈O_K, k≠0} (−1)^{N(k)} Φ̃(N(k)√(1/(2y N(l²)))), where 'primary' means congruent to 1 modulo (1+i)^3 in Z[i]. The same limit has the integral representation M_Φ(y,δ)=∫_0^∞ Φ(x) M(y/x) dx with an explicitly displayed kernel M(x), so the murmured signal is a convolution of the test function with a fixed arithmetic density. The boundary behaviours — the limit is 0 as y→0⁺ and −Φ̃(0)/(3ζ_K(2)) as y→∞ — identi

Load-bearing premise

The load-bearing assumption is GRH: it supplies the character-sum estimate and the prime-ideal-theorem count that make the two error terms vanish, and without it the limit in Theorem 1.1 is not supported.

Editorial extensions

If this is right

The murmuration density for this family is completely explicit: for any test function and any scale y, the limit value is a finite computation from the Gaussian-integer Möbius function and the Fourier transform Φ̃.
The large-scale limit is determined by ζ_K(2): M_Φ(y,δ) → −Φ̃(0)/(3ζ_K(2)) as y→∞.
The small-scale limit is zero, so the murmured signal dies out as the prime window shrinks, consistent with the phase transition in the one-level density.
The error analysis identifies the exact range of validity: for δ>3/4 the non-square contributions and the truncated Möbius tail decay like X^{−(δ−3/4)/5}.
Because the formula is fully explicit, it can be evaluated numerically and compared with the finite-X average, giving a direct test of the predicted convergence rate.

Reading between the lines

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

If the formula is correct, the same mechanism should yield analogous densities for other imaginary quadratic fields; the only field-dependent data would be the Dedekind zeta value and the units group, so the shape of the kernel M(x) should be universal.
The explicit kernel offers a way to probe the one-level density phase transition without computing zeros: localising or differentiating Φ in M_Φ(y,δ)=∫Φ(x)M(y/x)dx should expose the crossover scale predicted by random-matrix heuristics.
The proof's reliance on GRH is likely only removable at the cost of weaker unconditional character-sum bounds, which would move δ closer to 1; testing whether the formula persists for δ≤3/4 without GRH would separate arithmetic content from analytic engine.
As a proof-theoretic caution, the final interchange of limits is said to follow by arguing as in a lemma from the predecessor Dirichlet-character paper rather than carried out in full here; a self-contained derivation of that interchange would be a natural next step.

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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 the first murmuration density for a number field beyond the rationals, and the object is genuinely new. The authors evaluate the density for quadratic Hecke L-functions over Q(i), with quartic-symbol character structure, and produce an explicit formula (1.1) that is not in the cited literature. The limits as y→0 and y→∞ are plausible and the internal bookkeeping holds up at the points I checked: the residue at s=2 in (3.30) cancels the constant exactly, the Euler product over primary l gives the 3ζ_K(2) denominator, and the y→0+ computation is coherent. No fitted parameters, no imported target answer, no invented entities.

The main caveat is not GRH, which is stated explicitly as a hypothesis. The real soft spot is the limit interchange after (3.10). The paper asserts that the l-series converges uniformly and is continuous in N(ϖ), then says “argue in a way similar to [10, Lemma 2.9]” to get the main formula (3.12). That is the step where the exact main term is born, and it requires a Riemann–Stieltjes estimate with the l and k sums interchanged with the prime average, uniformly in the test function. The paper does not provide that estimate. It may well be fillable by copying the template from [10] with the rapid decay in (2.1), but it is not a routine one-line step for this family. The same delegation reappears in the continuity argument in §3.3.

Two lesser points. The authors lean heavily on their own [2] for the Poisson summation lemma, the character-structure facts, and the transform; that is not circular, but it makes the paper hard to read independently. Also, several displayed formulas are garbled in the version I saw—(1.2) and (1.4) in particular—so I could not verify every normalization factor exactly. That is a presentation defect, not evidence of a mathematical error.

Overall: the central result is likely true conditional on GRH, and the proof structure is sound. The gap is real but local. This is exactly the kind of paper a serious referee should see, with a request to make the limit interchange self-contained and to clean up the displayed integrals.

Referee Report

2 major / 4 minor

Summary. The paper evaluates the murmuration density for the family of quadratic Hecke L-functions over the Gaussian field K=Q(i), under GRH. The main result (Theorem 1.1) gives an explicit limit for the short-interval prime average M_Φ(y,δ), expresses it through a Zubrilina-type kernel, and evaluates the y→0 and y→∞ limits. The proof adapts the authors' one-level density machinery [2] and the Dirichlet-character murmuration framework of Lee–Oliver–Pozdnyakov [10]. The theorem is stated under GRH, which is used in the character-sum bounds and the prime ideal theorem estimates. No free parameters are introduced; the main term is determined by the test function and the arithmetic of Gaussian integers.

Significance. If correct, this is the first explicit murmuration density for a family of Hecke L-functions over a number field, extending the results of Zubrilina [14] and Lee–Oliver–Pozdnyakov [10] to a quadratic field setting. The formula is fully explicit and the proof uses machine-checkable? (no) but transparent analytic number theory. The connection to the one-level density phase transition is also of interest. However, the current version has two load-bearing issues: the theorem statement and the proof's conclusion are inconsistent, and the key limit interchange is delegated to an external lemma without verification. These issues must be fixed before the result can be considered established.

major comments (2)

[Theorem 1.1, Eq (1.1); Eq (3.12)] The main theorem states the argument of Φ̃ as N(k)√(2yN(l)), but the proof's conclusion (3.12) gives Φ̃(N(k)√(1/(2yN(l²)))) = Φ̃(N(k)/(N(l)√(2y))). These differ by more than a typo: the first is dimensionally and structurally inconsistent with the Poisson summation formula in Lemma 2.3 and with the derivation around Eq (3.8)–(3.12). Since (1.1) is the paper's central claim, the correct formula must be stated and used consistently throughout.
[Section 3.2, after Eq (3.10)] The step converting the prime average of F_X(V) = Σ_l μ[i](l)/N(l²)(−Φ̃(0)+T(X,N(l),V)) into the pointwise limit (3.12) is delegated to '[10, Lemma 2.9]' without stating the lemma or verifying its hypotheses. This is the exact point where the main formula is born. What is needed is a uniform/Stieltjes estimate showing that the prime average of the l-series converges to the sum of the pointwise limits, with errors controlled uniformly in V. The paper does not provide the required estimate, and it is not a routine rephrasing of the cited lemma. Please either state and prove the relevant lemma or give a self-contained argument.

minor comments (4)

[Eq (3.9)] The bound T(X,N(l),N(ϖ)) ≪ (X/(N(l²)N(ϖ)))^{1/4} is not a direct consequence of Lemma 2.4, because T includes the k=0 term Φ̃(0), which is not small. The estimate applies to the oscillatory error term in Lemma 2.4; the constant part cancels in the subsequent tail bound. Please correct the statement of (3.9) accordingly.
[Eq (3.30)] The residue computation has a typo: the residue of B(s/2) at s=2 is 2 Res_{s=1} B(s) = π/(3ζ_K(2)), not (2π/6)·(π/(3ζ_K(2))). The final result after substitution is correct, but the intermediate display should be fixed.
[Eq (1.2)] The definition of the additive character ẽ(z) = exp(2πi(z/(2i) − ar z/(2i))) appears to give e^{2πi Im z}, which is not oscillatory. The later polar-coordinate formula (1.4) indicates the intended Fourier kernel is e^{−2πi t Im z}; please clarify/repair the definition.
[Title/Abstract] There are typos in the title ('MURMURA TIONS', 'HECKEL-FUNCTIONS') and in the abstract; please proofread.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity; derivation transforms a fixed average via published lemmas; the only soft spot is a delegated GRH-conditional limit interchange, which is a rigor gap not a circular reduction.

full rationale

Walked the derivation chain of Theorem 1.1. M_Φ(y,X,δ) is a fixed arithmetic average with no fitted parameters. The proof uses Möbius inversion, a Poisson summation formula imported from the authors' earlier paper [2, Lemma 2.3], and GRH-conditional character-sum estimates to discard the non-square k terms and the R_{Φ,Z} tail. The final limit is obtained by converting the short interval over primary primes via the prime ideal theorem and a uniform-convergence argument for the l-series. No step identifies the target murmuration density with an input by construction, and no fitted constant is renamed as a prediction. The reliance on [2] is real, published, and independent of the present result; the cited lemmas are standard Fourier/lattice estimates, not the target. The one delicate point is the limit interchange after (3.10): the paper says 'We then argue in a way similar to the proof of [10, Lemma 2.9]' and does not write out the Riemann–Stieltjes estimate for this l-series. This is a gap in exposition/rigor, but it is not circular: [10] is a different family and the cited lemma is not a restatement of (1.1). GRH is an explicit hypothesis, not a hidden input. Score 2 reflects the heavy self-citation and the delegated estimate, not constructional circularity.

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

The paper adds no fitted constants and no invented objects. Everything the central claim needs beyond routine analysis is: the explicit hypothesis GRH; lemmas imported from the authors' own [2] (Poisson summation, the −Φ̃(0) identity, primitivity of the characters); a routine Dirichlet-series identity for B(s); and standard zero-free-region/convexity bounds for ζ_K(s). The load is concentrated in the first two items: if the GRH bounds or the Lemma 2.3 transform failed in the needed uniformity, formula (1.1) would not follow.

assumptions (6)

domain assumption GRH (explicitly assumed in Theorem 1.1)
Every error term — the Möbius-tail bound (3.6), the non-square k contribution, and S′ — uses the GRH character-sum bound S(x, χ) ≪ x^{1/2} log(N(c)x) at (3.5), and the ϖ-count uses the GRH prime ideal theorem at (3.11). The condition is stated openly.
domain assumption Lemma 2.3: Poisson summation for the quadratic symbol (imported from the authors' [2], Canadian J. Math 2020)
The whole evaluation of M_{Φ,Z} in §3.2 hinges on this transform relating the a-sum with the quadratic symbol to the dual k-sum with the (−1)^{N(k)} factor and Φ̃; the paper restates it without proof.
domain assumption Primitivity: χ_{i(1+i)^5 c} is a primitive Hecke character of trivial infinite type for square-free c (from [2, Section 2.1])
If these characters were imprimitive or principal for some c in the family, the character sums (3.4)–(3.5) would carry main terms and the error analysis in §3.1–3.2 would break.
domain assumption Lemma 2.4: Σ_{k≠0} (−1)^{N(k)} Φ̃(N(k)/A) = −Φ̃(0) + O(A^{−1/2}) (from [2, Lemma 3.1])
Used to extract the −Φ̃(0) main term and the T-term at (3.9)–(3.10) and again for the y→∞ limit; imported from prior published work.
standard math Euler product identity (3.24): B(s) = (1−2^{−s})/(1−2^{−s−1}) · ζ_K(s)/ζ_K(s+1), with simple pole at s = 1 of residue π/(6ζ_K(2))
Stated as "similar to [10, (5.4)]"; a routine computation from the definition of B(s) over primary n; the 2-primary factor is forced by the primariness restriction. Needed for the contour shift in (3.30).
standard math Zero-free region, convexity, and inversion bounds for ζ_K(s): (3.27), (3.28) from [13], [6], [11]
Used to bound B(s) by (1+|s|)^2 on ℜ(s) ≥ 1/2 and to justify the contour shift; standard analytic number theory background.

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Cite this review

Pith. "Pith review of Murmurations of quadratic Hecke $L$-functions of the Gaussian field." pith.science (2026). https://pith.science/paper/Y4YKF6HT

@misc{pith2026260720853,
  author       = {Pith},
  title        = {Pith review of: Murmurations of quadratic Hecke $L$-functions of the Gaussian field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4YKF6HT}},
  note         = {Machine review of arXiv:2607.20853}
}

abstract

In this paper, we evaluate the murmuration density for the family of quadratic Hecke $L$-functions of the Gaussian field under the generalized Riemann hypothesis.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references

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P. Gao and L. Zhao,One level density of low-lying zeros of quadratic and quartic HeckeL-functions, Canad. J. Math.72(2020), no. 2, 427–454

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [10]

K.-H. Lee, T. Oliver, and A. Pozdnyakov,Murmurations of Dirichlet characters, Int. Math. Res. Not. IMRN1(2025), Paper No. rnae277, 28

2025

[2] [2]

Gao and L

P. Gao and L. Zhao,One level density of low-lying zeros of quadratic and quartic HeckeL-functions, Canad. J. Math.72(2020), no. 2, 427–454

2020

[3] [14]

Zubrilina,Murmurations, Invent

N. Zubrilina,Murmurations, Invent. Math.241(2025), no. 3, 627–680. School of Mathematical Sciences, Beihang University, Beijing 100191, China Email address:penggao@buaa.edu.cn School of Mathematics and Statistics, University of New South W ales, Sydney NSW 2052, Australia Email address:l.zhao@unsw.edu.au

2025

[4] [1]

Davenport,Multiplicative Number Theory, Third edition, Graduate Texts in Mathematics, vol

H. Davenport,Multiplicative Number Theory, Third edition, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, Berlin, etc., 2000

2000

[5] [3]

I. S. Gradshteyn and I. M Ryzhik,Table of Integrals, Series and Products, Sixth Edition, Academic Press, San Diego, 2000

2000

[6] [4]

He, K.-H

Y.-H. He, K.-H. Lee, T. Oliver, and A. Pozdnyakov,Murmurations of elliptic curves, Exp. Math.34(2025), no. 3, 528–540

2025

[7] [5]

Ireland and M

K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, Second edition, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990

1990

[8] [6]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, 2004

2004