First Moment of derivatives of $L$-functions in a nonlinear family

Tinghao Huang; Zhining Wei

REVIEW 2 major objections 4 minor 1 cited by

First Moment of derivatives of $L$-functions in a nonlinear family

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.09056 v2 pith:NMMDSM46 submitted 2026-07-10 math.NT

Tinghao Huang , Zhining Wei This is my paper · ORCID

classification math.NT MSC 11F6711M06

keywords firstmomentL-functionsquadratictwistscentralderivativenonlinearfamilynonvanishingmodularformsEulerproducts

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 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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

Secondary 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[§2.2, §3.2, §4] 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.
[§3.1, §4.1, §4.2] 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.

minor comments (4)

[Remark 2.4] 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.
[§3.2] Near the end of the q′<0 case, the text says 'apply Lemma 2.2' for M2(q′); this should be Lemma 2.3.
[§4.3] 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.
[§3] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main term is obtained by explicit contour residues from Euler products, not by fitting; the only self-citation is background, and the proof gaps flagged by the reader are correctness risks, not circularity.

full rationale

The central derivation is not circular. Theorem 1.1's main term c_{q,D}G(q,D)J̃(1)L(1,χ_D)X log X is computed from the residues at s=0 of the Dirichlet series A1 and A2 after shifting contours; the constants H_i(0,q',D), G(q,D), and L(1,χ_D) are explicit Euler products and L-values, and no parameter is fitted to the moment being predicted. The identities A1(s;q',D) and A2(s;q',D) in §3.2 coincide with the series in Lemma 2.2/2.3 by definition, but those lemmas are then used to obtain Euler product factorizations; the main term is evaluated by contour residue, not imposed. The cited bounds from [Mun11a], [Li24], and [Zho25] are external to this paper and are not self-citations; whether they transfer to the present weight functions is a legitimate correctness/rigor concern, but it is not circular. The single self-citation [Hua25] appears only in the introduction as related work and is not load-bearing. The n-power inconsistency noted in the manuscript's core calculation (A(d;Y) uses √n while the approximate functional equation and Mellin opening use n^{-1/2}) is a serious mathematical gap if the literal definitions are used, since the displayed reduction to A1(w) is then algebraically false; however, an algebraic inconsistency is the opposite of a circular reduction, because the claimed equations do not reproduce their inputs by construction but rather fail to cohere. That issue should be weighed as a correctness risk, not as circularity. Overall, there is no load-bearing self-citation, no fitted input renamed as a prediction, and no imported uniqueness/ansatz that forces the conclusion; the low score reflects only the minor background self-citation and the external, unverified estimates whose transfer is asserted rather than proved.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters: the asymptotic constant is an explicit product of Euler factors and L-values; nothing is fitted. No invented entities: no new arithmetic objects, conductors, or spaces are postulated. The inputs are standard background (approximate functional equation, Rankin–Selberg, Deligne, Landau–Selberg–Delange) plus three deep estimates imported from prior work ([Mun11a] U(N,t) bound; [Li24, Prop 3.2] bilinear forms; [Zho25] dyadic second-moment bounds). The last import — that the Zho25 bounds carry over to the present weight difference and ψ(d,D)²-weighted d-sum — is the ledger's only genuinely ad hoc entry and is asserted, not proven, in the manuscript.

assumptions (8)

domain assumption Approximate functional equation: L′(1/2, f×χ_d) = (1+χ_d(q)) Σ_n λ_f(n)χ_d(n) n^{−1/2} W(n/(d√q)) (§2.2)
Standard for this twist family; assumes conductor scaling d²q and root number ω_f = −1.
standard math Rankin–Selberg local factorization ζ_p(2s)Σ_{ℓ≥0} λ_f(p^{2ℓ}) p^{−ℓs} = L_p(s, sym²f) for p∤q, and λ_f(p)^ℓ = λ_f(p^ℓ) for p|q (Atkin–Lehner) (§2.3 proof of Lemma 2.2)
Used to write A1, A2 as L(1+2s, sym²f)·H_i and to check H_i(0,·) ≠ 0.
standard math Deligne's bound |λ_f(p)| ≤ 2 (§2.3, Remark 2.4)
Shows the D0-factor in the c_{q,D} sum is < 1 in absolute value, hence c_{q,D} > 0.
standard math Landau–Selberg–Delange theorem (§3.1, proof of Lemma 3.1)
Controls Σ_d ψ(a²d,D)² J(a²d/X) ≪ X(logX)^5 / a^{2−ε}.
domain assumption Second-moment bound U(N,t) ≪ N(1+|t|)^3 (log N)^{9/2} for quadratic Dirichlet L-functions ([Mun11a, p.32]) (§3.1)
Powers the error term E in Proposition 2.6; imported without proof.
domain assumption Li's optimal bilinear-form estimate for Hecke eigenvalues twisted by quadratic characters ([Li24, Prop 3.2]) (§1.1, §4)
Handles the n ≫ Y range in Proposition 2.5.
ad hoc to paper The two dyadic second-moment bounds of [Zho25, Sec. 7.5 and Lemma 7.1] transfer verbatim to the weight difference (W(n/(d√q)) − W(n/Y)) and the ψ(d,D)²-weighted d-sum (§4.1, §4.2)
Asserted as 'Similar to...' with no derivation; the variable d inside W is a new feature. The weakest premise of the error-term argument.
domain assumption Hypothesis (q,6)=1 so that c_{q,D} G(q,D) ≠ 0 (§2.3, Remark 2.4; Theorem 1.1)
Part of the theorem statement; positivity is checked via local factor signs.

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Pith. "Pith review of First Moment of derivatives of $L$-functions in a nonlinear family." pith.science (2026). https://pith.science/paper/NMMDSM46

@misc{pith2026260709056,
  author       = {Pith},
  title        = {Pith review of: First Moment of derivatives of $L$-functions in a nonlinear family},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMMDSM46}},
  note         = {Machine review of arXiv:2607.09056}
}

abstract

In this paper, we prove an asymptotic formula for the moment of the first order derivative of modular $L$-functions at the center of the critical strip, weighted by generalized divisor functions formed by primitive quadratic characters. Such moments were previously studied by Munshi, which naturally arise in the study of elliptic fibration. To the best of our knowledge, such asymptotic formulae have only been proven in the setting of higher-order derivatives, or under the specialization to dihedral forms.

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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

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.

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Bump, Daniel and Friedberg, Solomon and Hoffstein, Jeffrey , TITLE =. Invent. Math. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01233440 , URL =

work page doi:10.1007/bf01233440 1990

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Conrey, J. B. and Farmer, D. W. and Keating, J. P. and Rubinstein, M. O. and Snaith, N. C. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 2005 , NUMBER =. doi:10.1112/S0024611504015175 , URL =

doi:10.1112/s0024611504015175 2005

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Goldfeld, D. and Hoffstein, J. and Patterson, S. J. , TITLE =. Number theory related to. 1982 , ISBN =

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Li, Xiannan , TITLE =. Invent. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00222-024-01265-1 , URL =

work page doi:10.1007/s00222-024-01265-1 2024

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Luo, Wenzhi and Ramakrishnan, Dinakar , TITLE =. Invent. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/s002220050189 , URL =

doi:10.1007/s002220050189 1997

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Number theory, analysis and geometry , PAGES =

Michel, Philippe and Ramakrishnan, Dinakar , TITLE =. Number theory, analysis and geometry , PAGES =. 2012 , ISBN =. doi:10.1007/978-1-4614-1260-1\_20 , URL =

doi:10.1007/978-1-4614-1260-1 2012

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work page doi:10.2307/2944316 1991

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Acta Arith

Shen, Quanli , TITLE =. Acta Arith. , FJOURNAL =. 2022 , NUMBER =. doi:10.4064/aa211207-7-11 , URL =

work page doi:10.4064/aa211207-7-11 2022