{"id":"4ef0c47e-f78f-4f28-8236-1fbe825a7a6f","arxiv_id":"2607.20853","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditional on GRH, the murmuration density for quadratic Hecke L-functions over Q(i) equals an explicit Fourier-kernel sum, with end limits 0 and −Φ̃(0)/(3ζ_K(2)); the paper also derives the Zubrilina-type kernel for the family.","lead":"Under the generalized Riemann hypothesis, the authors compute the murmuration density for the family of quadratic Hecke L-functions over the Gaussian integers, giving an explicit limit formula and the two end limits. It is the first such computation beyond the rationals, and it lands exactly where the Sarnak framework predicts.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing soft spot is the unproven limit interchange after (3.10): the paper delegates to [10, Lemma 2.9] the step converting the ϖ-prime average into the pointwise limit (3.12), without establishing the needed uniform/Stieltjes error for this l-series.","rationale":"The paper extends the authors' earlier one-level density machinery to murmuration densities for quadratic Hecke L-functions over Q(i). The internal checks I carried out—the cancellation in the y→0+ residue computation, the B(s) Euler product giving Res_{s=1}B(s)=π/(6ζ_K(2)), the tail bounds for S′ and the non-square k contribution, and the y→∞ limit—are consistent with the stated (3.12). The GRH assumption is explicit in Theorem 1.1, so I do not count it as a hidden flaw. The decisive soft spot is the limit interchange after (3.10): the paper does not prove the uniform Stieltjes estimate needed to replace the prime average over [yX,yX+X^δ] by the continuous limit, and instead cites [10, Lemma 2.9] by analogy. This is the place where the exact main term (3.12) is produced; if the analogy fails because of the extra l-series or the Gaussian-field normalizations, the central formula could be wrong even under GRH. Since the gap is likely fillable by a direct estimate (the tail is Z^{-3/2} and the total variation of F_X is O(X^{δ-3/4})), the appropriate verdict remains CONDITIONAL rather than ACCEPT. I agree with the reader that GRH is the main external premise, but the most load-bearing internal concern is this delegated convergence step.","tokens_in":10931,"tokens_out":34452,"duration_ms":269208,"concrete_test":"Write out the full proof of the analogue of [10, Lemma 2.9] for F_X(V) = Σ_{l primary} μ[i](l)/N(l²)(−eΦ(0)+T(X,N(l),V)) on V∈[yX,yX+X^δ]. Specifically, use (3.11) and the uniform tail bound Σ_{N(l)>Z} N(l)^{-5/2} ≪ Z^{-3/2} to bound the Riemann-Stieltjes error |(logX/X^δ)Σ_{ϖ}F_X(N(ϖ)) − (1/X^δ)∫_{yX}^{yX+X^δ}F_X(t)dπ(t)| and then show it is o(1) for δ∈(3/4,1). If such an estimate holds, the delegation is benign; if a non-negligible cross term emerges from the k,l sums, (3.12) and hence Theorem 1.1 need correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is GRH-conditional, as stated, so GRH is an explicit premise rather than a hidden flaw. The less visible but equally load-bearing step is the limit passage from the short-interval prime average to the series in (3.12). Immediately after (3.10), the proof asserts that the l-series F_X(V)=Σ_l μ[i](l)/N(l²)(−eΦ(0)+T(X,N(l),V)) converges uniformly, hence is continuous in V, and then 'argue[s] in a way similar to the proof of [10, Lemma 2.9]' to conclude lim (logX/4X^δ)Σ_{N(ϖ)∈[yX,yX+X^δ]} F_X(N(ϖ)) equals the desired sum. That final equality is the point where the explicit formula (3.12) is born; it is not a routine one-line step for this family. It requires a Riemann-Stieltjes estimate in which the l,k sums are interchanged with the prime average, and the error must be controlled uniformly in V. The paper gives no such estimate. A gap here would not weaken a secondary term; it would remove the exact main term. The same delegation appears again in §3.3 when continuity in y is used. This is the single most load-bearing concern: even granting GRH, the central limit formula is asserted rather than derived.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11196,"tokens_out":21461,"duration_ms":187946,"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":[{"comment":"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":"Theorem 1.1, Eq (1.1); Eq (3.12)"},{"comment":"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.","section":"Section 3.2, after Eq (3.10)"}],"minor_comments":[{"comment":"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.","section":"Eq (3.9)"},{"comment":"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.","section":"Eq (3.30)"},{"comment":"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.","section":"Eq (1.2)"},{"comment":"There are typos in the title ('MURMURA TIONS', 'HECKEL-FUNCTIONS') and in the abstract; please proofread.","section":"Title/Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is a reasonable extension of existing techniques, but the two major issues are independent: the theorem statement discrepancy with the proof's formula would mislead readers even if the proof is correct, and the delegated limit interchange leaves the central derivation unsupported as written. Both are fixable within the scope, so major_revision seems appropriate. The GRH conditionality is explicit and not a defect. The manuscript would benefit from stating the analogue of [10, Lemma 2.9] and proving the uniform convergence estimate explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTwo 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.\n\nOverall: 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.","headline":"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.","tokens_in":11823,"tokens_out":1331,"would_cite":true,"duration_ms":15451,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L37","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under GRH, the murmuration density of quadratic Hecke L-functions over the Gaussian field is an explicit arithmetic sum with no undetermined constants.","keywords":["murmuration density","quadratic Hecke L-functions","Gaussian field","generalized Riemann hypothesis","one-level density","Poisson summation","Hecke characters","low-lying zeros"],"falsifier":"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.","tokens_in":10681,"feed_emoji":"🐦","tokens_out":9832,"duration_ms":91703,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact murmuration density found for quadratic Hecke L-functions","feed_subtitle":"Under GRH, the oscillation averages reduce to one explicit sum over Gaussian integers, with known limits at both ends.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Explicit murmuration density for Gaussian quadratic L-functions","Murmuration density reduces to a single Gaussian integer sum","Exact kernel for quadratic Hecke L-function murmurations","Boundary limits pinned for Gaussian murmuration density","Exact murmuration density under GRH for Gaussian L-functions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit murmuration density for Gaussian quadratic L-functions","Murmuration density reduces to a single Gaussian integer sum","Exact kernel for quadratic Hecke L-function murmurations","Boundary limits pinned for Gaussian murmuration density","Exact murmuration density under GRH for Gaussian L-functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000411,"raw_usage":{"total_tokens":1900,"prompt_tokens":615,"completion_tokens":1285,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":359,"completion_tokens_details":{"reasoning_tokens":1218}},"tokens_in":359,"tokens_out":1285,"duration_ms":9110,"temperature":1.0,"reasoning_tokens":1218,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:09:18.237406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}