{"id":"e047620a-2e18-4ac6-b42f-d8e5dbbc6271","arxiv_id":"2607.09056","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The ψ(d,D)-weighted first moment of L′(1/2, f×χ_d) over d∈D equals c_{q,D}G(q,D)J̃(1)L(1,χ_D)X log X + O(X(log X)^{1/2}(log log X)^3), with explicit positive constants.","lead":"Analytic number theorists prove an asymptotic formula for the average of central derivatives of quadratic-twist L-functions over a sparse, character-weighted family — the first-derivative case, previously open for non-CM forms. The result extends Munshi's work and supplies an explicit, positive main term useful for elliptic-fibration counting and nonvanishing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main-term proof as written has an n-power inconsistency: A(d;Y) uses √n while the Mellin openings in §3.2 require n^{-1/2}.","rationale":"The reader's CONDITIONAL verdict is appropriate. I checked the main-term arithmetic and agree that, modulo the power-of-n typo, the Euler-product computation for c_{q,D}G(q,D) is internally consistent. The most load-bearing issue I see is not exactly the reader's weakest_assumption: the proof's written definitions of A(d;Y) and B(d;Y) use √n, while every Mellin opening that produces A1/A2 requires n^{-1/2}. If taken literally, the central residue calculation collapses, so the paper is not self-contained as written. However, the surrounding text makes the intended normalization fairly clear, and the typo is likely repairable. The secondary concern about transferring the dyadic second-moment estimates from [Zho25] to the present W(n/(d√q)) and ψ(d,D)² setting remains real and also supports the CONDITIONAL verdict. Neither issue makes me think the theorem is false; both require the authors to supply a corrected, fully-derived proof before acceptance.","tokens_in":18636,"tokens_out":40457,"duration_ms":395944,"concrete_test":"Re-derive Eq. (3.11) from the displayed definition of A(d;Y) in §2.2 without correcting the power of n: expand W(n/Y) via (2.1), interchange sums and identify the Euler product. If the inner Dirichlet series is A1(w-1) rather than A1(w), the residue at w=0 is not H1(0)X logX, confirming that a global normalization fix is required; then re-run §3.1 and §4 with n^{-1/2} to verify the error bounds still hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation of the main term is not self-consistent as written. §2.2 defines A(d;Y) = (1+χ_d(q)) Σ λ_f(n)χ_d(n)√n W(n/Y), but the approximate functional equation stated just above it and used in §1.1 has λ_f(n)χ_d(n)n^{-1/2} W(n/(d√q)). In §3.2, M1(q') is reduced to ∫ Γ(1+w)(Y/(2π))^w A1(w;q',D) dw/w², with A1(w)=Σ_{nq'=□} λ_f(n)n^{-1/2-w}. This equality is algebraically false if the n-sum contains √n: opening W gives n^{1/2-w}=n^{-(w-1/2)}, i.e. A1(w-1), not A1(w). The same √n appears in B(d;Y) and in the dyadic estimates of §4. Only after silently replacing √n by n^{-1/2} does the residue at w=0 produce H1(0) (resp. H2(0)) and the stated c_{q,D}G X logX main term. This is not merely a typo in one line: under the literal definitions, B(d;Y) is not L'-A(d;Y) in the claimed sense, and the main-term formula does not follow from the written proof. The reader's concern about the imported [Zho25]/[Mun11a] bounds is also valid, but the n-power inconsistency is more immediate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an asymptotic formula for the first moment of central derivatives L'(1/2, f × χ_d) over a sparse nonlinear family of quadratic twists, weighted by the generalized divisor function ψ(d,D). The claimed main term is c_{q,D}G(q,D)~J(1)L(1,χ_D)X log X plus an error term of size O(X(log X)^{1/2}(log log X)^3), with c_{q,D}G(q,D) explicitly given by Euler products and shown nonvanishing for (q,6)=1. The strategy is a Munshi-type approximate functional equation and Mellin inversion, with the n-summation split at Y=X/log^{100}X; the main term is extracted from the n≪Y range and the error term is treated by Li/Zhou-style second moment bounds.","tokens_in":18837,"tokens_out":9323,"duration_ms":85238,"significance":"If correct, the result is a substantial improvement over Munshi's nonlinear-family first moments, which previously required derivatives of order at least 8, and it would establish nonvanishing on average for first derivatives in a sparse family. The main-term constants are explicit Euler products and L-values, with no fitted parameters, and the claimed error term is sharper than what a direct application of known large-sieve estimates would give. The significance is real, but it is conditional: the main-term computation as written contains a power-of-n inconsistency, and the error-term half relies on imported bounds whose transfer is not demonstrated.","major_comments":[{"comment":"The proof of the main term is not self-consistent. The approximate functional equation displayed in §2.2 has coefficient n^{-1/2}W(n/(d√q)), but A(d;Y) is defined with √n W(n/Y), and §4 expands B with √n(W(n/(d√q))−W(n/Y)). In §3.2, M1(q′) is reduced to XĴ(1)L(1,χ_D)G(q,D)∫(Y/2π)^w Γ(1+w)A1(w;q′,D)dw/w² with A1(w)=Σ_{nq′=□}λ_f(n)n^{-1/2−w}. Under the literal definitions, opening W(n/Y) gives a factor n^{1/2−w}, i.e. the sum is A1(w−1), not A1(w). The residue at w=0 claimed to produce H1(0)X logX is therefore not obtained from the written formulas. The same substitution is used for A2 and in §4.3. Replacing √n by n^{-1/2} throughout would repair the algebra, and is what the approximate functional equation requires, but as written the main term does not follow.","section":"§2.2, §3.2, §4"},{"comment":"Proposition 2.5 is the whole error term, but the bounds that carry it are imported rather than proved or precisely transferred. §3.1 uses U(N,t)≪N(1+|t|)^3(logN)^{9/2}, citing [Mun11a, p.32]; §4.1 and §4.2 respectively assert second moment bounds ≪(X/N)^6 X and ≪(X/N)^{2/logX}X(loglogX)^4, described as 'similar to' [Zho25, Sec. 7.5] and [Zho25, Lemma 7.1]. The present sums have additional features — the ψ(d,D)² weight in the first factor and the d-depending argument W(n/(d√q)) inside the second moment — that are not present in the cited settings. If these bounds fail to transfer, the claimed O(X(logX)^{1/2}(loglogX)^3) is unsupported. The transfer needs to be written out.","section":"§3.1, §4.1, §4.2"}],"minor_comments":[{"comment":"The displayed formula should have D0^{1/2} in the denominator of λ_f(D0)/D0∏(1+p^{-1}); the inequality that follows uses the square-root normalization.","section":"Remark 2.4"},{"comment":"Near the end of the q′<0 case, the text says 'apply Lemma 2.2' for M2(q′); this should be Lemma 2.3.","section":"§3.2"},{"comment":"In the final estimate for S_{N≤Y}, the appearance of the term X^{15/16}Σ N^{-1/16} is unexplained. Please specify how this term arises and how it is bounded by X log log X.","section":"§4.3"},{"comment":"The notation Σ^* is introduced as summation over squarefree integers, but in equation (3.1) it is used for a sum over fundamental discriminants with additional restrictions; define the restricted sum explicitly to avoid ambiguity.","section":"§3"}],"recommendation":"major_revision","confidential_remarks":"I suspect the n-power inconsistency is a typographical slip rather than a fundamental obstruction, since replacing √n by n^{-1/2} makes the main-term computation internally consistent. However, the error-term bounds in §4 need to be proved or precisely transferred, and the manuscript must be revised throughout for consistency. The paper fits the journal's scope if these issues are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real result waiting for a normalization fix. The genuinely new thing is the l=1 first moment in the ψ(d,D) nonlinear family for non-CM f, where prior work covered l≥8 or CM forms. Credit where it is due: the Euler-product computation in §3.2 is mostly right. I checked that the local factors match Lemmas 2.2/2.3, the G(q,D) factors cancel correctly, and the residue computation gives the stated H1/H2 combination; positivity via Remark 2.4 works modulo the D0^{1/2} display typo. No fitted parameters, no invented entities, no circularity. The strategy—split the n-sum at Y=X/log^{100}X, use Munshi on the short range and Li/Zhou on the long range—is credible and worth engaging.\n\nThe load-bearing problem is the n-power. The approximate functional equation gives n^{-1/2}; A(d;Y) is defined with √n; B(d;Y)=L'−A then compares n^{-1/2} W(n/(d√q)) with √n W(n/Y). Later, in §3.2, opening W in M1(q') produces n^{1/2−w}, and the paper calls that A1(w). It is actually A1(w−1). The residue at w=0 therefore gives A1(−1), not H1(0), and the claimed X log X main term does not follow from the written formulas. The same √n appears in §4's expression for B, so the error-term analysis is affected too. If the intended definition was n^{-1/2} throughout, the proof probably goes through, but that is not the text. This is not a one-line typo; it breaks the main theorem as stated.\n\nSecondary but real: Proposition 2.5's error term rests on three imported bounds asserted 'similar to' [Zho25] or [Mun11a, p.32] without derivation—the U(N,t) bound, the (X/N)^6 X bound, and the (X/N)^{2/log X} X (log log X)^4 bound. With the new d-dependence inside W(n/(d√q)) and the ψ² weight, transfer is plausible but not automatic. A referee should verify those.\n\nWho this is for: analytic number theorists working on moments of quadratic twists. I would not desk-reject it; the idea is right and the main-term constants are honestly computed. But the paper needs major revision: fix the normalization, make the transferred bounds precise, and clean up the display typos. As written, I would not rely on the theorem.","headline":"A promising proof of the missing l=1 case, but the written main-term computation has a load-bearing √n vs n^{-1/2} normalization error; fix that, fill in the transferred bounds, and the theorem is likely sound.","tokens_in":19555,"tokens_out":7676,"would_cite":false,"duration_ms":72006,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the first moment of central derivatives of quadratic-twist modular L-functions, averaged over a sparse set of fundamental discriminants weighted by a divisor function, has a positive main term of size X log X with an","keywords":["first moment","L-functions","quadratic twists","central derivative","nonlinear family","nonvanishing","modular forms","Euler products"],"falsifier":"Take a fixed f, D, q, J and evaluate numerically the dyadic second-moment sum in §4.1 for N just above X; if it grows faster than X (X/N)^6, the claimed error bound in Proposition 2.5 fails. Alternatively, compute the left-hand side of Theorem 1.1 for moderately large X against the predicted X log X main term and check the difference is compatible with the stated error.","tokens_in":18239,"feed_emoji":"🧮","tokens_out":9850,"duration_ms":79058,"temperature":0.7,"pith_summary":"The paper proves an asymptotic formula for the first moment of derivatives of quadratic-twist modular L-functions, where the averaging is over a sparse set of fundamental discriminants d such that every prime dividing d splits in an imaginary quadratic field, and the sum is weighted by ψ(d,D) = Σ_{r|d} χ_D(r). Previously, such asymptotic formulas were known only for higher-order derivatives, or under a restriction to CM forms; this paper handles the first-derivative case in the full nonlinear family. The main term is c_{q,D} G(q,D) J̃(1) L(1,χ_D) X log X with c_{q,D} G(q,D) ≠ 0, so the average of L′(1/2, f×χ_d) over this thin family does not vanish. The proof combines an approximate functional equation, a Mellin-integral evaluation of the d-sum, a dyadic split at Y = X/(log X)^{100}, short-range estimates, and recent bilinear-form bounds for quadratic characters. The result matters because nonvanishing of the first derivative at the central point corresponds to positive rank under the Birch–Swinnerton-Dyer conjecture, and the formula gives the average order of L′ in a family too sparse for standard equidistribution heuristics.","feed_headline":"X log X main term for first derivative of twisted L-functions","feed_subtitle":"A sparse family of quadratic twists is shown to have positive average first derivative, with an explicit constant.","key_machinery":"The argument runs on two devices. (1) The weight ψ(m,D)=Σ_{r|m} χ_D(r) is nonzero only when every prime factor of m splits in Q(√D), making the family sparse; the identity ψ(m,D)ψ(n,D)=Σ_{ℓ|(m,n)} χ_D(ℓ) ψ(mn/ℓ²,D) and its Möbius-inverted form let the d-sum be uncoupled from the n-sum. (2) The approximate functional equation splits the moment into a short-range term A(d;Y), where the main term is extracted by shifting Mellin contours and evaluating residues at s=1, and a long-range term B(d;Y), bounded by a dyadic decomposition into N>X, Y<N≤X, and N≤Y, combined with Cauchy–Schwarz and estimates for bilinear forms of Hecke eigenvalues twisted by quadratic characters. The constants c_{q,D} an","core_discovery":"The central claim is Theorem 1.1: for a weight-2 Hecke eigenform f of squarefree level q with global root number −1, a negative fundamental discriminant D with 4|D, and a smooth compactly supported test function J, the sum over positive fundamental discriminants d ≡ 1 mod 4 with (d, qD)=1 and χ_d(q)=1, of ψ(d,D) L′(1/2, f×χ_d) J(d/X), equals c_{q,D}G(q,D) J̃(1) L(1,χ_D) X log X plus an error of size X (log X)^{1/2}(log log X)^3. The constants c_{q,D} and G(q,D) are explicit Euler products, and c_{q,D}G(q,D) is nonzero. Equivalently, the first moment in this nonlinear family has a genuine main term, so the first derivatives of the twisted L-functions are nonvanishing on average over the spars","pith_inferences":["The averaging set has zero natural density among fundamental discriminants; an X log X main term therefore implies that the average of L′ over the admissible discriminants grows like (log X)² if one normalizes by the number of terms, suggesting vanishing is rare even on very thin families.","The same contour-shift and dyadic-split scheme should yield an asymptotic for the first moment of central values L(1/2, f×χ_d) over the same family, with main term X (no log factor) and the same Euler-product constant; this is a testable prediction.","The bottleneck is the transfer of three analytic bounds to the present context, where the variable d now occurs inside the weight W(n/(d√q)) and the ψ(d,D)² weight appears; verifying those bounds from first principles would put the error term on solid ground.","The method should extend to higher-order derivatives, predicting main terms X (log X)^k, which would unify the earlier higher-derivative asymptotics with the first-derivative case."],"forward_implications":["Averaging with nonnegative weights would rule out a zero sum: a positive X log X main term forces L′(1/2, f×χ_d) to be nonzero for a positive proportion of d in the sparse family.","The error term X (log X)^{1/2} (log log X)^3 is smaller than the main term by about (log X)^{1/2}/(log log X)^3, which diverges as X grows, so the asymptotic genuinely detects the main term.","For D = −4, the theorem improves the earlier high-derivative result of the same family by treating the first derivative directly, with a stronger error term.","The nonzero Euler-product constant makes the leading term fully explicit, so the asymptotic can be compared with arithmetic heuristics such as the Birch–Swinnerton-Dyer conjecture on average."],"fun_headline_variants":["First derivative of L-functions: positive average in sparse twist family","Nonvanishing on average: first derivative of twisted L-functions","Explicit main term for first moment of L′ in quadratic twists","Average first derivative of modular L-functions: nonzero constant","Sparse quadratic twists yield positive average L′-values"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The error term is proved only if three estimates taken from earlier papers remain valid when an extra variable d now appears inside a weight function and a new divisor-function weight is present; the paper asserts these transfer without proving them.","fun_headline_variants_meta":{"raw":{"variants":["First derivative of L-functions: positive average in sparse twist family","Nonvanishing on average: first derivative of twisted L-functions","Explicit main term for first moment of L′ in quadratic twists","Average first derivative of modular L-functions: nonzero constant","Sparse quadratic twists yield positive average L′-values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":2848,"prompt_tokens":664,"completion_tokens":2184,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":2098}},"tokens_in":408,"tokens_out":2184,"duration_ms":13745,"temperature":1.0,"reasoning_tokens":2098,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:43:35.621477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed f, D, q, J and evaluate numerically the dyadic second-moment sum in §4.1 for N just above X; if it grows faster than X (X/N)^6, the claimed error bound in Proposition 2.5 fails. Alternatively, compute the left-hand side of Theorem 1.1 for moderately large X against the predicted X log X main term and check the difference is compatible with the stated error.","supporting_citations":[],"review_version":2}