{"id":"606c2bfc-1594-4fd8-92d7-65ba96cfceff","arxiv_id":"2512.22509","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH, the first moment of primitive quadratic Hecke L-functions in the Gaussian field at s=1/2 is X Q1(log X) + X^{1/3} Q2(log X) + O(X^{1/4+ε}).","lead":"This paper derives an asymptotic formula for the average of quadratic Hecke L-functions over the Gaussian field, including lower-order main terms and a small error term. The result extends recent breakthroughs for the rational numbers and function fields, relying on unproved standard hypotheses (GRH/Lindelöf).","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.2 is obtained by 'taking the limit s→1/2' in Theorem 1.1, but the displayed main terms have simple poles at s=1/2; the cancellation is not proved, so the central asymptotic is not yet justified.","rationale":"The reader's weakest assumption was the Lindelöf hypothesis used for polynomial boundedness, which is an explicit hypothesis of the conditional theorem and therefore not an internal flaw. The more load-bearing step is the passage from Theorem 1.1 to Corollary 1.2, since Corollary 1.3 is the central advertised result and depends entirely on that limiting argument. The main terms in (1.4) display simple poles at s = 1/2 because both R_{K,1}(s) and R_{K,2}(s) contain ζ_K(2s). The paper does not show the cancellation, even though a valid limit would require it. This is concrete, checkable, and closer to the heart of the paper than the conditional hypothesis. If the cancellation holds, the result is likely correct and the paper merely omits a standard but nontrivial computation; if it fails, the central asymptotic is false or unproved. Thus the existing CONDITIONAL verdict remains appropriate, but the verification target should be this limit, not the Lindelöf hypothesis.","tokens_in":18560,"tokens_out":10716,"duration_ms":91345,"concrete_test":"Take the explicit expressions for R_{K,1} and R_{K,2} from Theorem 1.4(2),(5), set s = 1/2 + z, and compute the Laurent expansion of the sum of the four main terms in (1.4) at z = 0, treating X as a fixed parameter. Verify that the z^{-1} coefficients cancel pairwise (term 1 with term 2, term 3 with term 4) and identify the resulting finite coefficients; if they do not cancel, Corollary 1.2 is not a consequence of Theorem 1.1. This is a purely analytic computation using the stated residues.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 1, after Theorem 1.1, states: 'Upon taking the limit s→1/2 in (1.4) ... we obtain Corollary 1.2.' However, each of R_{K,1}(s) (Theorem 1.4(2)) and R_{K,2}(s) (Theorem 1.4(5)) contains ζ_K(2s) as a factor. Therefore at s = 1/2 the first term X Φ̂(1) R_{K,1}(s) and the second term X^{3/2-s} Φ̂(3/2-s) X_K(s) R_{K,1}(1-s) each have a simple pole; similarly the X^{1/2-s/3} and X^{(2-2s)/3} terms involving R_{K,2} have simple poles. For the limit to exist these poles must cancel exactly. The paper gives no computation of the Laurent coefficients and merely asserts that the error term is uniformly bounded. If the cancellation is not exact, the right-hand side of (1.4) is not meromorphic in a neighborhood of s=1/2, and Corollary 1.2 (hence Corollary 1.3) does not follow from Theorem 1.1 as stated. This is an internal gap in the deduction of the headline result, independent of the stated conditional hypotheses.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the smoothed first moment of the family of primitive quadratic Hecke L-functions attached to the Gaussian field K=Q(i), with characters χ_{(1+i)^5 d} for odd square-free primary d. Assuming the Lindelöf hypothesis for Hecke L-functions of trivial infinite type, the authors prove an asymptotic formula (Theorem 1.1) for the sum at a general point s in 1/3<Re(s)<1, s≠1/2, with two main terms and an error term. Taking s→1/2 gives Corollary 1.2, and under GRH Corollary 1.3 gives a central-value formula with main terms X Q_1(log X)+X^{1/3}Q_2(log X) and error O(X^{1/4+ε}). The proof is based on the double Dirichlet series A(s,w) defined in (1.7); its meromorphic continuation and residues (Theorem 1.4) are computed using the functional equation of [11, Prop. 2.5] and a large sieve estimate.","tokens_in":18901,"tokens_out":21426,"duration_ms":177854,"significance":"If the proof is completed, this is a solid contribution: it extends to a number field the X^{1/3} secondary main term and X^{1/4} error term previously known for the rational function field (Florea) and for quadratic Dirichlet L-functions (Čech). The residue computations in Theorem 1.4 are explicit and involve no free parameters; the main terms are given by convergent Euler products and gamma factors. The paper also gives a useful large-sieve estimate for quadratic Hecke L-functions (Lemma 2.7). The central technical issue is the passage from Theorem 1.1 to Corollary 1.2, as detailed below.","major_comments":[{"comment":"The deduction of Corollary 1.2 from Theorem 1.1 is not justified. In (1.4) each of the four main terms has a simple pole at s=1/2: by Theorem 1.4(2),(5), R_{K,1}(s) and R_{K,2}(s) contain ζ_K(2s), while R_{K,1}(1-s) and R_{K,2}(1-s) contain ζ_K(2-2s). The sentence 'Upon taking the limit s→1/2 ...' simply asserts the limit exists and that the error term is uniformly bounded, but no Laurent expansion is given. A residue calculation using G_K(1/2)=1 in (1.5) does show that the singular parts cancel; however, this is not written down. The paper must provide this computation and a uniform bound for the error term in a neighbourhood of s=1/2 for Corollary 1.2 to follow. As stated, Corollary 1.2 and hence Corollary 1.3 are not established by the arguments in the paper.","section":"Section 1, after Theorem 1.1"},{"comment":"The proof of Theorem 1.1 is only a sketch. The contour shift from (1.6) to the line Re(w)=c requires a detailed pole count, including the verification that no poles of the types s+(2j+1)w=3/2 or 2js+(2j+1)w=j+1 for j≥1 fall to the right of the new contour for 1/3<Re(s)<1, and a bound for A(s,w) on the shifted line. The paper refers to [6, Theorem 1.2] for these arguments. Given that Theorem 1.1 is the main analytic engine and the uniformity in s is essential for the limit in Corollary 1.2, the sketch should be expanded to a full proof or to a precise lemma stating the uniformity.","section":"Section 4"}],"minor_comments":[{"comment":"The factor is written as 'π2 25' in the text, which is ambiguous; it should presumably be (π^2/32)^{s-1/2}. Please correct the typesetting.","section":"Equation (1.5)"},{"comment":"The multiplication rule for the group CG is hard to parse; consider presenting it as a small Cayley table.","section":"Section 2.1"},{"comment":"The line 'c = max{...}' uses both β and ε; it would help to state explicitly that the implied constants in the error term of (1.4) may depend on s and ε, and to specify the uniformity in s needed near s=1/2.","section":"Section 4"},{"comment":"The phrase 'under the Riemann hypothesis and the Lindelöf hypothesis' is slightly misleading: Theorem 1.1 uses Lindelöf only; the Riemann hypothesis is invoked only for Corollary 1.3.","section":"Abstract and Introduction"},{"comment":"Typo: 'the an asymptotical formula' should be 'an asymptotic formula'.","section":"After Corollary 1.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically ambitious and appears to be correct in its main line, but the missing verification of the limit s→1/2 is a real gap that must be fixed before publication. If the authors supply the omitted Laurent computation and the uniformity statement, I would support acceptance. The reliance on preprints [6] and [11] for key functional equations is acceptable, but the referee should check those references."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper evaluates the smoothed first moment of primitive quadratic Hecke L-functions over Q(i) and gets the expected asymptotic: a main term X Q1(log X), a secondary term X^{1/3} Q2(log X), and an O(X^{1/4+ε}) error under GRH. That is a genuine extension of Čech's rational-field result and Florea's function-field result to a number field where the double Dirichlet series machinery has to handle primary elements, the unit group, and the group of quadratic characters. The novelty is real, and the main technical work — the meromorphic continuation in Theorem 1.4 and especially the residue computation at w = 1/2 − s/3 — is presented in enough detail to be checked. The result is not circular and contains no fitted parameters.\n\nThe soft spots are concentrated in the transition from Theorem 1.1 to Corollary 1.2. The stress-test note is right that each displayed main term in (1.4) has a simple pole at s = 1/2 and that the paper simply says \"taking the limit\" without comment. But the worry that the limit may not exist is unfounded: the poles cancel pairwise. For the R_{K,1} terms, at s = 1/2 the coefficients are both X Φ̂(1) (since X_K(1/2)=1), and R_{K,1}(1−s) has the opposite residue to R_{K,1}(s); similarly the R_{K,2} terms both carry coefficient X^{1/3} Φ̂(1/3). So the RHS of (1.4) is analytic at s=1/2. The paper should have said this explicitly; as written, a referee has to do the computation. The uniformity of the error term as s→1/2 is also asserted rather than demonstrated, though it looks fixable with a fixed vertical strip.\n\nThis is a minor presentational gap, not a load-bearing flaw. The main theorem holds up, the citation pattern is appropriate, and the analytic computations are substantial. A serious referee could verify the residue checks without too much trouble, and the authors should be asked to expand the s→1/2 passage and address the error-term uniformity. I would accept this for peer review and would cite it in work on moments of L-functions over number fields.","headline":"Solid new result for the Gaussian field; the s→1/2 step is under-explained, but the pole cancellation is real, so the main theorem survives.","tokens_in":19344,"tokens_out":4536,"would_cite":true,"duration_ms":42512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M41"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, under the Riemann and Lindelöf hypotheses, the smoothed first moment of primitive quadratic Hecke L-functions in the Gaussian field equals X Q1(log X) + X^{1/3} Q2(log X) + O_ε(X^{1/4+ε}) at the central point s=1/2,","keywords":["quadratic Hecke L-functions","first moment","double Dirichlet series","Gaussian field","secondary main term","Lindelöf hypothesis","Riemann hypothesis","primitive characters"],"falsifier":"For a fixed smooth weight Φ, compute the smoothed first moment at increasing X, subtract X Q1(log X) + X^{1/3} Q2(log X) using the explicit polynomials obtained from the proof, and check that the remainder stays bounded by a constant times X^{1/4+ε}; a single X where the remainder clearly exceeds, say, X^{1/4} log X would disprove the asymptotic. Alternatively, finding a counterexample to the Lindelöf bound |L(1/2+it, χ)| ≪ N(q)^ε for some primitive quadratic Hecke character would invalidate the error term.","tokens_in":18457,"feed_emoji":"📐","tokens_out":5436,"duration_ms":47588,"temperature":0.7,"pith_summary":"The paper evaluates the first moment of the family of primitive quadratic Hecke L-functions over the Gaussian integers, focusing on central values at s=1/2. Under the Riemann hypothesis (which implies the Lindelöf hypothesis for these L-functions), the smoothed sum is shown to have two main terms: a leading term of order X multiplied by a linear polynomial in log X, and a secondary term of order X^{1/3} multiplied by another linear polynomial, with an error of size X^{1/4+ε}. This shape—main term, X^{1/3} secondary term, X^{1/4} error—matches the corresponding families over the rational numbers and over function fields. The proof works by treating the generating series as a double Dirichlet series, establishing its meromorphic continuation, and computing its residues explicitly. If correct, the result gives the sharpest known asymptotic for this family and provides a template for studying higher-order terms in related moments.","feed_headline":"Averages of Hecke L-functions reduce to X and X^{1/3} main terms","feed_subtitle":"Under GRH, the smoothed first moment in the Gaussian field has a leading term and a cube-root correction, with error X^{1/4+ε}.","key_machinery":"The load-bearing object is the double Dirichlet series A(s,w) = Σ_{d square-free primary} L(s, χ_{(1+i)^5 d}) / N(d)^w. The paper establishes its meromorphic continuation to a tube domain, locates its possible poles, and computes the residues at w=1, w=3/2−s, w=1/2−s/3, and w=2/3−2s/3. The proof chains together the functional equation for quadratic Hecke L-functions, quadratic reciprocity in the Gaussian field, and Bochner's tube theorem to glue together local holomorphic continuations; the residues then give the main terms in the moment after Mellin inversion and contour shift.","core_discovery":"The central claim is that the first moment of quadratic Hecke L-functions in the Gaussian field admits an asymptotic expansion with a secondary main term whose size is the cube root of the leading term. Specifically, for a smooth compactly supported weight Φ, the sum over square-free primary d of L(1/2, χ_{(1+i)^5 d}) Φ(N(d)/X) equals X Q1(log X) + X^{1/3} Q2(log X) + O_ε(X^{1/4+ε}) under GRH, where Q1 and Q2 are linear polynomials whose coefficients are absolute constants. For general s with 1/3 < Re(s) < 1, the analogous formula contains four main terms arising from four poles of the double Dirichlet series A(s,w), with the error expressed in terms of β, the supremum of the real parts of z","pith_inferences":["The X^{1/3} secondary term likely originates from the square-conductor contribution inside the double Dirichlet series; isolating that contribution, as the proof does through A_1(s,w), would allow one to predict the next-order term and test it numerically.","The same double Dirichlet series machinery may yield the second moment, or a shifted first moment, potentially leading to a non-vanishing proportion result for central values in this family, given the positivity of the leading main term.","By analogy with the rational case, the X^{1/4} error term may be improvable only under a stronger hypothesis; a numerical check at modest X could suggest the true size of the error and whether the secondary term is indeed X^{1/3}.","The method appears adaptable to other imaginary quadratic fields with class number one, or to twists of higher order, provided a functional equation for non-primitive Hecke characters analogous to the one used here exists."],"forward_implications":["Under GRH, the smoothed first moment of this family has a non-trivial secondary main term of size X^{1/3}, so the family follows the same asymptotic pattern as the rational and function-field families.","The error term O_ε(X^{1/4+ε}) is the square root of the leading main term up to the power of X; this is currently the smallest error proved for this first moment.","The general- s formula (Theorem 1.1) gives a uniform asymptotic in a vertical strip, which can be differentiated or integrated to yield information on other statistics such as shifted moments.","The explicit residue computations in Theorem 1.4 supply a complete analytic description of the double Dirichlet series, which can serve as a template for computing higher moments in this family.","If the leading main term is positive for sufficiently large X, the first moment being positive implies infinitely many non-vanishing central values among the primitive quadratic Hecke L-functions in the Gaussian field."],"fun_headline_variants":["Hecke moment: X main term plus cube-root correction","Quadratic Hecke L-function average: X and X^{1/3} terms","First Hecke moment in Gaussian field has X^{1/3} secondary term","GRH gives Hecke L-function first moment with X and X^{1/3}","Hecke L-function average: leading X, sub X^{1/3}, error X^{1/4}"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument rests on the Lindelöf hypothesis for quadratic Hecke L-functions of trivial infinite type, which is used to show that the double Dirichlet series is polynomially bounded in vertical strips; if it fails, the stated error term and the treatment of the secondary terms as main terms are not justified.","fun_headline_variants_meta":{"raw":{"variants":["Hecke moment: X main term plus cube-root correction","Quadratic Hecke L-function average: X and X^{1/3} terms","First Hecke moment in Gaussian field has X^{1/3} secondary term","GRH gives Hecke L-function first moment with X and X^{1/3}","Hecke L-function average: leading X, sub X^{1/3}, error X^{1/4}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1335,"prompt_tokens":625,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":369,"completion_tokens_details":{"reasoning_tokens":600}},"tokens_in":369,"tokens_out":710,"duration_ms":6525,"temperature":1.0,"reasoning_tokens":600,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:50:35.748810+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed smooth weight Φ, compute the smoothed first moment at increasing X, subtract X Q1(log X) + X^{1/3} Q2(log X) using the explicit polynomials obtained from the proof, and check that the remainder stays bounded by a constant times X^{1/4+ε}; a single X where the remainder clearly exceeds, say, X^{1/4} log X would disprove the asymptotic. Alternatively, finding a counterexample to the Lindelöf bound |L(1/2+it, χ)| ≪ N(q)^ε for some primitive quadratic Hecke character would invalidate the error term.","supporting_citations":[],"review_version":1}