{"id":"8e0dd724-0d00-442a-90df-2aaab54ab27d","arxiv_id":"2507.00179","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A mixed average of a quadratically twisted L-value and a quadratically twisted L-derivative is shown to equal a constant times X log X up to a small error.","lead":"This paper proves a new asymptotic formula for averages of products of a twisted central L-value of one modular form with a twisted central L-derivative of another. The result extends recent moment computations and implies that for two elliptic curves there are infinitely many simultaneous quadratic twists with ranks one and zero.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 depends on unproved imports: Li's bilinear form lemma generalized to arbitrary levels and Zhou's shifted T-sum bounds; without proofs, Propositions 2-4 lack support.","rationale":"The reader's weakest assumption—the unproved generalization of Li's bilinear form estimate and reliance on Zhou's preprint estimates—is the same broad soft spot I identify, so I agree with the CONDITIONAL verdict. I add two internal consistency flags that the reader noted in passing but did not make central: the Proposition 1 proof replaces X log Y by X log X with only an O(X) error, and the decomposition of S_J has a suspicious factor 1/4. Neither of these by itself destroys Theorem 1 as an asymptotic statement, because the log-log-X discrepancy is absorbed by the theorem's weaker error term and the factor 1/4 likely reflects a missing factor in the definition of I1, but both need clean correction. The decisive issue remains whether Li's lemma genuinely extends to arbitrary levels with q-independent implied constants and whether Zhou's shifted T-estimates are valid as used; the manuscript provides no proof, so a reader cannot certify Propositions 2-4. Thus the correct verdict is CONDITIONAL: the theorem is plausible and likely correct, but full acceptance requires either proofs of these imported estimates or a rigorous reduction to published results.","tokens_in":27187,"tokens_out":8937,"duration_ms":98083,"concrete_test":"Independently re-derive Lemma 3 for arbitrary level q by tracking the q-dependence through Li's proof of [7, Lemma 6.3]; verify the stated bound d(q)^5 X (1+|t|)^3 log(2+|t|) holds with implied constant independent of q and with only the stated polynomial dependence on t. For the shifted T-sums, reproduce the proof of (5.9) of [15] for T(k1, 1/2+it1, 1/2+it2, Q', c+it4) with c = 1/log X and c = 6; if either fails, the I4 bound in §8.2 is unsupported. Additionally, recompute the constant term in I1 directly from the residue at u=0 to settle whether the main term is X log Y or X log X, and check whether the factor 1/4 in the decomposition S = (1/4)(I1+I2+I3+I4) is a typo by comparing with the definitions of Af and A'_g.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1) is assembled from Propositions 2-4, and each of those proofs leans on estimates that the manuscript states but does not prove. Lemma 3 is labelled as Li [7, Lemma 6.3] 'with minor technical modification to generalize to arbitrary levels', and it is used in Lemma 5 and in §8.2-8.3 with levels q1, q2, q1q2 (through Q). The needed bound must hold with implied constant independent of q and polynomial in |t|; if the level dependence is worse (e.g., q^ε or q^c), the dyadic sums over N1,N2 acquire factors that destroy the X(log log X)^5 bound. The paper does not supply the modification, so this is an unverified load-bearing input. Similarly, the bounds for the T-sums and their shifted variants T(k1, 1/2+it1, 1/2+it2, Q', c+it4) are imported from (5.9) of Zhou's preprint [15] and used for c = 1/log X and c = 6 in Sections 6-8; these shifted cases are not identical to Zhou's setup, and no reproduction is given. If either import fails, the estimates for I2, I3, and I4 collapse, leaving Theorem 1 unproved. There is also an internal gap in Proposition 1: the residue calculation leading to (5.5) gives X log Y, while the proof concludes with X log X inside O_{f,g,ε}(X); the difference X log log X is not O(X). Although the final theorem's error X(log log X)^5 would absorb this discrepancy, Proposition 1's stated error is overstrong and the conclusion as written does not follow from (5.5).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an unconditional asymptotic formula for the smoothed mixed moment SJ(f,g';X) = sum over sign-selected quadratic twists of L(1/2, f⊗χ8d) L'(1/2, g⊗χ8d) J(8d/X), with main term Cf,g Jhat(0) X log X and error O_{f,g,J}(X(log log X)^5), where Cf,g is an explicit parameter-free product of L-values and an absolutely convergent Euler product. The proof decomposes the moment into four pieces I1,...,I4 (Propositions 1-4), treats I1 by Poisson summation and residue calculus, treats I2 and I3 by the same machinery with truncation error terms, and treats I4 by a combination of dyadic decompositions, Cauchy-Schwarz, a bilinear form estimate (Lemma 3, imported from Li [7]), and bounds for T-sums imported from Zhou [15]. A corollary asserts the existence of infinitely many quadratic twists of two elliptic curves (with nonsquare conductors) for which one curve has rank 1 and the other rank 0.","tokens_in":27576,"tokens_out":3804,"duration_ms":43186,"significance":"If the main theorem is correct, it gives the first mixed moment asymptotic of a twisted central value and a twisted central derivative over quadratic twists, with an explicit constant that is not fitted and with a relatively small error term of size X(log log X)^5. This is a meaningful extension of the works of Li [7], Kumar-Mallesham-Sharma-Singh [6], and Zhou [15], and the elliptic-curve corollary is an attractive arithmetic consequence. The proof is structured in a standard way (approximate functional equations, Poisson summation, residue shifts, dyadic partitions), and the main constant is parameter-free, with no invented auxiliary entities. However, the paper's central claims rest on two imported estimates that are not proved in the manuscript, and Proposition 1 is internally overstrong by an X log log X amount; these issues must be resolved before the theorem can be regarded as fully established.","major_comments":[{"comment":"The residue computation displayed in (5.5) gives S_{1,d}^{Q'}(a≤Z) = (Jhat(0) X log Y / 2π^2) L(1,f⊗g)L(1,Sym^2 f)L(1,Sym^2 g) Z(0,0;Q') + O(X + X(log X)^63/Z). Since Y = X/(log X)^200, this main term is cX(log X - 200 log log X), not cX log X. Proposition 1 states I1 = Cf,g Jhat(0) X log X + O_{f,g,ε}(X). The discrepancy c·X log log X is not O(X), so the stated error in Proposition 1 does not follow from the displayed residue computation. The final theorem's error O(X(log log X)^5) would absorb a weaker bound of O(X log log X), so the argument is likely repairable, but the proposition as stated is wrong and the proof must be corrected or the proposition weakened.","section":"§5, Proposition 1 and equation (5.5)"},{"comment":"Lemma 3 is quoted as Li [7, Lemma 6.3] 'with minor technical modification to generalize to arbitrary levels', and it is used with levels q1, q2, and q1q2 through the parameter Q in Lemma 5 and in the Cauchy-Schwarz estimates of §8.2-8.3. The needed bound must hold with the stated factor d(q)^5 and an implied constant independent of q (or at least with the q-dependence entirely captured by d(q)^5 and fixed form-dependent constants). The manuscript does not state or prove the modification, and it is not enough to cite a parenthetical generalization without giving the details of the level-dependence. If the generalization only holds with an extra factor q^ε or q^c, the dyadic sums over N1 and N2 in Lemma 5 and in §8.2-8.3 would acquire extra powers of q, and the final bound O(X(log log X)^5) could fail. This is a load-bearing input and must be supplied.","section":"§3, Lemma 3, and §7, §8.2-8.3"},{"comment":"Several key estimates in Sections 5, 6, and 8 invoke the bound labeled '(5.9) of [15]' for the sums T(k1, 1/2+it1, 1/2+it2, Q') and for their shifted variants T(k1, 1/2+it1, 1/2+it2, Q', c+it4) with c = 1/log X, c = 6, c = -6, c = 1/2, and combinations such as 1/2 + i(t3+t4). The paper does not state Zhou's (5.9), nor does it reproduce the argument showing that the shifted sums satisfy the same uniform bounds with the same quality. Since these bounds control the off-diagonal contributions to I2, I3, and I4, the absence of a precise statement or proof is a gap that cannot be waved away by citation to an unpublished preprint. The author should either quote the exact form of (5.9) with all hypotheses and verify that each shifted variant falls under it, or provide a self-contained proof (or a detailed appendix) of the shifted estimates.","section":"§5-§8, bounds 'by (5.9) of [15]' for shifted T-sums"}],"minor_comments":[{"comment":"The title as it appears in the PDF contains broken spacing ('A VERAGING QUADRA TICALL Y TWISTED MODULARL-V ALUES AND THEIR DER V ATIVES') and the abstract repeats reference labels [7], [6], [15] instead of formatted citations; these artifacts should be fixed in the final version.","section":"Title and abstract"},{"comment":"The statement 'the Mordell-Weil rank of E(8d)1 is equal to 1, and that of E(8d)2 is equal to 0' has the subscripts 1 and 2 set in an ambiguous way; it should be written as E_1^{(8d)} (or a clearly defined twist notation) to avoid confusion.","section":"§1, Corollary 1"},{"comment":"The sentence 'we may directly check as in [10] that Z(0,0;1)+ i^{κ1}η_f Z(0,0;q1) - ... = 0, hence the main term in the moment (1.1) vanishes, if and only if ...' is logically compressed: the displayed combination is the constant Cf,g up to a positive factor, so its vanishing is by definition equivalent to Cf,g = 0. The 'if and only if' clause about i^{κ1}η_f = -1 and q1 a square should be justified explicitly, not attached to the word 'hence'.","section":"§5, around equation (5.5)"},{"comment":"The three objects displayed at the beginning of §8.1 (S4(a≤Z; N1≤Y, N2), S4(a≤Z; N2≤Y, N1), S4(a≤Z; N1≤Y, N2≤Y)) are not mutually exclusive and the second one does not specify the range of N1; the text would be clearer if each dyadic region were defined disjointly and labelled consistently with the subsequent estimates.","section":"§8.1"},{"comment":"The manuscript cites 'Lemma 5.3 of [15]', 'Lemma 7.1 of [15]', and 'section 7.4 of [15]' without reproducing the statements. Since [15] is a preprint, the relevant estimates should be quoted in full or the dependence on them made transparent. Also, (6.1) and (7.1) use exponent 3/2+ε while (8.1) uses 1+ε; the author should confirm that all three bounds follow from the cited propositions with the stated exponents.","section":"References and labelled lemmas"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically conventional and the claimed result is plausible, but the two unproved imports (the level-generalization of Li's bilinear form estimate and the shifted versions of Zhou's T-sum bound) are load-bearing, and Proposition 1's stated error term is internally inconsistent with equation (5.5). These are fixable in principle, but the manuscript as it stands does not completely prove its main theorem. The editor may also wish to verify the availability and correctness of Zhou's preprint [15] before accepting any revision that continues to rely on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this is a serious paper with a genuinely new result—a full asymptotic for the mixed moment L(1/2,f⊗χ8d)·L'(1/2,g⊗χ8d) under sign conditions—but it is currently not certified because two load-bearing estimates are imported from other papers without proof, and Proposition 1 states an error term stronger than the proof yields.\n\nWhat is new: neither the second moment of values (Li) nor of derivatives (Kumar et al.), nor the product of two derivatives (Zhou), covers this mixed value-times-derivative family. The main constant is explicit and parameter-free, with clean vanishing conditions, and the elliptic curve corollary (infinitely many simultaneous rank (1,0) twists) is a nice payoff. The paper is honestly written: the author tells you exactly where the ingredients come from, and the overall architecture—Möbius to drop squarefree, Poisson summation, dyadic decomposition, contour shifts—is the right modern toolkit. Most of the hard bookkeeping looks careful.\n\nThe soft spots are real but not necessarily fatal. First, in Proposition 1, the residue computation leading to (5.5) gives X log Y, not X log X; with Y = X/(log X)^200, the discrepancy is X log log X, which is not O_{f,g,ε}(X). The final theorem's error X(log log X)^5 absorbs this, so the main theorem probably survives, but Proposition 1 as written does not follow from the proof. Second, Lemma 3 is stated as Li's Lemma 6.3 'with minor technical modification to generalize to arbitrary levels,' and no proof is supplied. That lemma is load-bearing for Lemma 5 and the dyadic bounds in Section 8; the needed q-independence of the implied constant is exactly what could undermine the estimates if the modification fails. Similarly, the bounds for the shifted T-sums T(k1, ..., c+it4) are imported from (5.9) of Zhou's preprint without reproduction. These may be routine extensions, but they need to be written out or cited precisely.\n\nMy read is that the central claim is likely correct, though I cannot certify the intricate estimates. The paper should be sent to a referee who can check Lemma 3 and the shifted T-sums carefully; the author should also restate Proposition 1 with an error of size X log log X (or absorb it into the theorem's error).\n\nFor a reading group, it is a useful case study in how modern moment papers assemble ingredients. I would engage with it.\n\nRecommendation: send to peer review.","headline":"Genuinely new mixed moment asymptotics with a likely-correct main theorem, but two unproved imported estimates and an overstrong Proposition 1 error term keep it from being certified as written.","tokens_in":28077,"tokens_out":3606,"would_cite":true,"duration_ms":40151,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F66","11F41"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an asymptotic formula for the averaged product of a quadratically twisted central L-value and a quadratically twisted central L-derivative, with an explicit main term X log X and error O(X(log log X)^5).","keywords":["quadratic twists","central L-values","L-derivatives","moments of L-functions","Poisson summation","approximate functional equations","bilinear forms","elliptic curve ranks"],"falsifier":"Compute the left side of Lemma 3 numerically for a weight-2 newform of prime level q=101 at t=0 with N=$X^{{1/2}}$ and X=$10^{6}$, and compare with the claimed bound d(q)^5 X (1+|t|)^3 log(2+|t|); a value exceeding this by a factor that grows with q would invalidate the off-diagonal estimates of Sections 6–8.","tokens_in":26916,"feed_emoji":"🔢","tokens_out":6583,"duration_ms":69776,"temperature":0.7,"pith_summary":"The paper establishes, unconditionally, an asymptotic formula for the mixed moment SJ(f,g';X) = Σ* L(1/2,f⊗χ8d) L'(1/2,g⊗χ8d) J(8d/X), summed over squarefree d with sign conditions on the two root numbers. The leading term is Cf,g ˇJ(0) X log X with an explicit constant built from Rankin–Selberg and symmetric-square L-functions and a convergent Euler product, and the error term is O(X(log log X)^5). The constant vanishes exactly when a root-number condition forces every term in the sum to vanish, so the formula is consistent with the identically zero case. The result extends recent unconditional second-moment theorems and implies that for two elliptic curves with non-square conductors there are infinitely many twists where the first curve has rank one and the second has rank zero.","feed_headline":"Mixed twisted L-functions average to X log X","feed_subtitle":"Unconditional formula with explicit constant extends recent second-moment theorems.","key_machinery":"The proof rests on the approximate functional equations, which express L(1/2,f⊗χ8d) and L'(1/2,g⊗χ8d) as (1±root number) times weighted sums of Fourier coefficients, truncated at Y=X/(log X)^{200}; Poisson summation for real characters, which turns the d-sum into a diagonal term that yields the main term and an off-diagonal term; and two external estimates: Li's bilinear form bound (Lemma 3) for the quadratic-twist sums of Hecke eigenvalues, and Zhou's T-sum bounds (quoted from [15]) for the off-diagonal expressions with an extra shifted variable. The main term is extracted from the zero frequency after a careful residue computation, and the error terms come from bounding the nonzero frequencies by these bilinear and T-sum estimates.","core_discovery":"Theorem 1: for two distinct holomorphic cusp newforms f and g of even weights and odd levels, the smoothed sum over squarefree d of L(1/2,f⊗χ8d) L'(1/2,g⊗χ8d) J(8d/X), restricted to d with ω(f⊗χ8d)=1 and ω(g⊗χ8d)=-1, equals Cf,g times the integral of J times X log X plus an error of order X(log log X)^5. The constant Cf,g is the product of L(1,f⊗g), L(1,$Sym^{2}$ f), L(1,$Sym^{2}$ g), and a combination of four values of a convergent Euler product Z(0,0;Q') evaluated at Q'=1,q1,q2,q1q2; it vanishes precisely when $i^{{κ1}}$ηf=-1 and q1 is a square or $i^{{κ2}}$ηg=1 and q2 is a square, and in those cases the moment is identically zero.","pith_inferences":["The same framework should apply to other sign pairings (ω(f⊗χ)=ω(g⊗χ)=1, etc.) by changing the four-term combination, and the constant would be the corresponding signed sum of Z(0,0;Q').","The error term O(X(log log X)^5) likely can be improved to O(X(log log X)^A) for some A, but the method does not obviously give a power-saving error; matching the conjectured second-moment error would require new input on the T-sums.","The corollary on ranks could be made effective: the proof gives infinitely many d but does not bound the least such d; quantifying that would require tracking the implied constants in the bilinear bounds.","The product of two central values L(1/2,f⊗χ)L(1/2,g⊗χ), which the paper notes remains out of reach, would require a different treatment of the diagonal term, potentially through a large-sieve inequality for the double Dirichlet series."],"forward_implications":["The mixed moment has the expected size X log X with an explicit, computable leading constant, so the average of L'(1/2,g⊗χ8d) over the family where L(1/2,f⊗χ8d) is forced to be nonzero is positive.","When either form satisfies the square-conductor root-number obstruction, the whole moment vanishes identically, confirming the structural vanishing predicted by the approximate functional equation.","For elliptic curves E1,E2 over Q with non-square conductors, infinitely many fundamental discriminants 8d produce a rank-one twist of E1 and a rank-zero twist of E2 (Corollary 1).","The result extends the unconditional second-moment theorems of Li and of Kumar–Mallesham–Sharma–Singh to a mixed moment of a value and a derivative."],"supporting_citations":[{"why":"Supplies Lemma 3, the bilinear form bound for quadratic-twist sums of Hecke eigenvalues, and Proposition 6.2 used to control large a in Sections 5–7.","marker":"[7]"},{"why":"Supplies the T-sum estimates (5.9) and related lemmas (Lemma 3.5, 5.3, 7.1 and Section 7.4) that bound the off-diagonal terms in Sections 5–8.","marker":"[15]"},{"why":"Supplies Propositions 3.1 and 3.2 for moments of twisted central derivatives, used together with Cauchy–Schwarz for the a>Z contributions.","marker":"[6]"},{"why":"Produces the Poisson summation formula for real characters (Lemma 1) and the evaluation of Gauss-type sums (Lemma 2), which separate the diagonal main term from the off-diagonal remainder.","marker":"[12]"},{"why":"Provides the approximate functional equations (Lemma 4) and the divisor-sum bound used to estimate the diagonal term's error.","marker":"[4]"},{"why":"Kolyvagin's theorem, with Gross–Zagier, is used to convert the order of vanishing of a twist's L-function into the Mordell–Weil rank for the elliptic curve corollary.","marker":"[5]"},{"why":"The Gross–Zagier theorem, together with Kolyvagin, is the rank-one converse used in Corollary 1.","marker":"[2]"}],"fun_headline_variants":["Quadratic twists: mixed L-values average to X log X","Mixed L-twist products hit X log X","Twisted L-value times L-derivative: X log X","Unconditional X log X for mixed twisted L-functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that Li's bilinear form estimate, originally stated for level 1, remains valid for arbitrary level q with an implied constant independent of q, and that Zhou's T-sum bounds hold for the shifted variants used here; the paper asserts both without proving them.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic twists: mixed L-values average to X log X","Mixed L-twist products hit X log X","Twisted L-value times L-derivative: X log X","Unconditional X log X for mixed twisted L-functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000813,"raw_usage":{"total_tokens":3506,"prompt_tokens":827,"completion_tokens":2679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":2612}},"tokens_in":443,"tokens_out":2679,"duration_ms":24216,"temperature":1.0,"reasoning_tokens":2612,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:22:27.990529+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the left side of Lemma 3 numerically for a weight-2 newform of prime level q=101 at t=0 with N=$X^{{1/2}}$ and X=$10^{6}$, and compare with the claimed bound d(q)^5 X (1+|t|)^3 log(2+|t|); a value exceeding this by a factor that grows with q would invalidate the off-diagonal estimates of Sections 6–8.","supporting_citations":[{"cited_title":"Moments of quadratic twists of modular L-functions","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3, the bilinear form bound for quadratic-twist sums of Hecke eigenvalues, and Proposition 6.2 used to control large a in Sections 5–7."},{"cited_title":"Moment of derivatives of quadratic twists of modular L-functions, 2025","cited_arxiv_id":null,"evidence_quote":"Supplies the T-sum estimates (5.9) and related lemmas (Lemma 3.5, 5.3, 7.1 and Section 7.4) that bound the off-diagonal terms in Sections 5–8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Propositions 3.1 and 3.2 for moments of twisted central derivatives, used together with Cauchy–Schwarz for the a>Z contributions."},{"cited_title":"Soundararajan","cited_arxiv_id":null,"evidence_quote":"Produces the Poisson summation formula for real characters (Lemma 1) and the evaluation of Gauss-type sums (Lemma 2), which separate the diagonal main term from the off-diagonal remainder."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kolyvagin's theorem, with Gross–Zagier, is used to convert the order of vanishing of a twist's L-function into the Mordell–Weil rank for the elliptic curve corollary."},{"cited_title":"Gross and Don B","cited_arxiv_id":null,"evidence_quote":"The Gross–Zagier theorem, together with Kolyvagin, is the rank-one converse used in Corollary 1."}],"review_version":1}