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Averaging quadratically twisted $L$-values and their derivatives

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Pith's one-line read 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).

desk verdict 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. read the letter →

arxiv 2507.00179 v1 pith:W7BG3I2A submitted 2025-06-30 math.NT

Tinghao Huang This is my paper · ORCID
classification math.NT MSC 11F6611F41
keywords quadratictwistscentralL-valuesL-derivativesmomentsofL-functionsPoissonsummationapproximatefunctionalequationsbilinearformsellipticcurveranks
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§5, Proposition 1 and equation (5.5)] 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.
  2. [§3, Lemma 3, and §7, §8.2-8.3] 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.
  3. [§5-§8, bounds 'by (5.9) of [15]' for shifted T-sums] 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.
minor comments (5)
  1. [Title and abstract] 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.
  2. [§1, Corollary 1] 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.
  3. [§5, around equation (5.5)] 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'.
  4. [§8.1] 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.
  5. [References and labelled lemmas] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main asymptotic is derived from independent external lemmas and contains no fitted parameter or self-referential definition.

full rationale

The claimed derivation of Theorem 1 from Propositions 1-4 is a standard asymptotic expansion: the main term in Proposition 1 emerges from a residue computation (5.5) applied to a Poisson-summed diagonal term, while the off-diagonal terms are bounded using bilinear form estimates imported from Li [7] and shifted T-sum bounds from Zhou [15]. None of these imports is by the present author, and none is defined in terms of the target moment; the constant C_{f,g} is an explicit, parameter-free product of L-functions and Euler products, not fitted to the moment. The unsupported 'minor technical modification' in Lemma 3 and the use of (5.9) of [15] for shifted variants are genuine correctness risks (an unverified generalization could break Propositions 2-4), and Proposition 1 as written overclaims X log X from the X log Y residue (5.5), but these are gaps or overstrong statements, not circular reductions: the target asymptotic formula does not equal the inputs by construction, and no fitted parameter is renamed as a prediction. Hence no circularity step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new objects: the constant Cf,g is a parameter-free product of known L-functions and Euler products, and no ad hoc parameters are fitted. The load-bearing inputs are external lemmas from Li and Zhou, some of which are extended to arbitrary levels without proof. The central claim is therefore dependent on the validity of those external estimates rather than on circular reasoning.

assumptions (5)
  • standard math Deligne's bound for Hecke eigenvalues λ_f(n), λ_g(n) ≪ n^ε
    Used in (5.4) and the error estimates for the diagonal terms.
  • domain assumption The approximate functional equations (4.1)-(4.2) for L(1/2, f⊗χ8d) and L'(1/2, g⊗χ8d) hold with the stated weight functions W1, W2.
    These are standard consequences of the functional equations and holomorphy of the completed L-functions; the paper cites [4, Theorem 5.3] and [10, Lemma 3.1].
  • ad hoc to paper Li's bilinear form estimate (Lemma 3) remains valid, with the same quality, for arbitrary positive integer level q after 'minor technical modification'
    Stated without proof in Section 3; the paper explicitly says it is a modification and gives no details. This generalization underpins Propositions 2-4.
  • ad hoc to paper The estimates labelled (5.9) of [15] apply to the shifted T(k1, ..., c+it4) sums with c up to 6 and (log X)^{-1} in the dyadic ranges used in Sections 6-8.
    Imported from Zhou's preprint without reproduction; the paper routinely writes 'by (5.9) of [15]' to finish the off-diagonal bounds.
  • domain assumption The Poisson summation formula Lemma 1 and the evaluation of Gauss sums Lemma 2 are correct for the square-free d sums with the (d, 2q1q2) = 1 condition.
    Standard results cited from Soundararajan [12] and Li [7]; the paper applies them after Möbius inversion.

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Pith. "Pith review of Averaging quadratically twisted $L$-values and their derivatives." pith.science (2026). https://pith.science/paper/W7BG3I2A

@misc{pith2026250700179,
  author       = {Pith},
  title        = {Pith review of: Averaging quadratically twisted $L$-values and their derivatives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7BG3I2A}},
  note         = {Machine review of arXiv:2507.00179}
}
abstract

In this paper, we unconditionally establish an asymptotic formula for the product of the quadratically twisted central $L$-value associated to a holomorphic cusp form $f$, and the quadratically twisted central $L$-derivative to a distinct holomorphic cusp form $g$. This result may be viewed as an extension of \cite{Li-MR4768632}, \cite{Kumar.etc-MR4765788} and \cite{zhou2025momentderivativesquadratictwists}.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Binary quadratic forms and elliptic curves with analytic rank one

    math.NT 2026-07 reject novelty 6.0 of 10

    Infinitely many quadratic twists of an arbitrary elliptic curve by values in a genus of binary quadratic forms are claimed to have analytic rank one.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages · cited by 1 Pith paper

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