Twisted second moment of primitive cubic L-functions

Zhiyong Zheng; Ziwei Hong

REVIEW 2 major objections 3 minor 1 cited by

Twisted second moment of primitive cubic L-functions

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that, for $q\equiv2\pmod3$ and fixed $h_1,h_2\in\mathbb{F}_q[T]$, the twisted second moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ equals $(1-q^2)\,S\,C(h_1,h_2)\,P(q^{-2})\,q^g +…

desk verdict Solid double Dirichlet series work undone by a sign error in the contour shift; the main theorem as stated is false, but flipping the sign likely repairs it. read the letter →

arxiv 2506.14656 v1 pith:BS4AJU3U submitted 2025-06-17 math.NT

Ziwei Hong , Zhiyong Zheng This is my paper · ORCID

classification math.NT MSC 11M0611M4111N3711L0511L40

keywords centralvaluescubicL-functionstwistedmomentfunctionfielddoubleDirichletseriesPerronformulaprimitivecharactersnon-Kummersetting

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 twisted second moment of primitive cubic $L$-functions over the rational function field $\mathbb{F}_q(T)$ in the non-Kummer setting $q\equiv2\pmod3$. For fixed twisting polynomials $h_1,h_2$, the average of $\chi(h_1)\bar\chi(h_2)|L_q(1/2,\chi)|^2$ over primitive cubic characters of genus $g$ is shown to be an explicit constant times $q^g$, with an error of size $q^{(1/2+\varepsilon)g}$. The constant is built from a twist-independent Euler product $S$, a polynomial-dependent factor $C(h_1,h_2)$, and an Euler factor $P(q^{-2})$, all arising from the residue of a double Dirichlet series at a single pole. This matters because twisted moments are the basic input for non-vanishing statements and zero statistics of this family, and no asymptotic for the twisted cubic second moment had been established before.

What carries the argument

The engine is the double Dirichlet series $A_2(u,v) = \sum_F \chi_F(h_1)\overline{\chi_F}(h_2)|L_q(v,\chi_F)|^2 u^{\deg F}$, where the sum runs over square-free $F$ over $\mathbb{F}_{q^2}[T]$ with no prime divisor in $\mathbb{F}_q[T]$. Perron's formula converts the genus-$g$ character sum into a contour integral of $A_2(u,q^{-1/2})$. The convergence regions $S_1,S_{2,1},S_{2,2},S_3$ are obtained from the Euler product, the bound from Lemma 2.3, and the functional equation; the tube-domain continuation theorem then extends $(u-q^{-2})A_2(u,v)$ to the convex hull of these regions. The residue at $u=q^{-2}$, in the form $(q^{-4}-q^{-2}) S C(h_1,h_2) P(q^{-2})$, becomes the main term after shifting contours.

What would settle it

Set $h_1=h_2=1$ and compute the sign of the claimed main-term constant $(1-q^2)\,S\,C(1,1)\,P(q^{-2})$ for an admissible prime power $q\equiv2\pmod3$. Since the left side of (1.3) is a sum of nonnegative terms, a negative constant would refute the asymptotic as written; the magnitude can be checked against the known untwisted second moment, which fixes both the sign and the size of the main term.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1: for $q\equiv2\pmod3$, fixed $h_1,h_2\in\mathbb{F}_q[T]$, and any $\varepsilon>0$, the sum over primitive cubic characters of genus $g$ of $\chi(h_1)\bar\chi(h_2)|L_q(1/2,\chi)|^2$ equals $(1-q^2)\,S\,C(h_1,h_2)\,P(q^{-2})\,q^g$ plus $O(q^{(1/2+\varepsilon)g})$. The pole of the double Dirichlet series $A_2(u,q^{-1/2})$ at $u=q^{-2}$, with residue $(q^{-4}-q^{-2})S C(h_1,h_2)P(q^{-2})$, produces the main term; the factor $C(h_1,h_2)$ depends only on primes dividing $h_1h_2$ for which the relevant cube-congruence condition fails, while $S$ is independent of the twist.

Load-bearing premise

The load-bearing premise is that, after the integration contour is moved inward, the simple pole at $u=q^{-2}$ contributes to the main term with the same sign as its residue; standard residue calculus would subtract the residue when reading the integral, and the sign determines whether the main term in (1.3) is positive or negative.

Editorial extensions

If this is right

For each fixed choice of $h_1,h_2$, the twisted second moment grows like a constant multiple of $q^g$, so the family-level central values are not uniformly suppressed by the twist.
The twist dependence is fully factorized: $C(h_1,h_2)$ factors over primes, so the ratio of two twisted moments depends only on the primes dividing $h_1h_2$ and their valuations modulo 3.
The error term $O(q^{(1/2+\varepsilon)g})$ is a power saving relative to the main term, making the asymptotic usable in mollified-moment arguments of the kind used to prove non-vanishing.
Specializing to $h_1=h_2=1$ must reproduce the untwisted second moment, providing a built-in consistency check against the earlier computation [10].
The double-Dirichlet-series route avoids a detailed distribution analysis of cubic Gauss sums, so the same residue computation should transfer to other ratios of twisted characters.

Reading between the lines

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

A direct check not performed in the paper: putting $h_1=h_2=1$ in (1.3) gives a signed main term that must be compared with the nonnegative left-hand side and with the known untwisted second moment; this would determine whether the sign of the residue contribution is correct as stated.
If the sign is adjusted to the standard residue-theoretic value, the asymptotic shape and the factorized coefficient are likely preserved, so the paper's structural conclusions would survive a sign correction.
The same contour-geometry argument could be run in the Kummer setting $q\equiv1\pmod3$, where the cube-residue symbol behaves differently and the analogue of $C(h_1,h_2)$ would encode the extra characters.
One could test the claimed error term by computing the finite character sum for small $q$ and $g$ and comparing deviations from the main term with $q^{(1/2+\varepsilon)g}$.

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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

I agree with the reader's take: the paper has a load-bearing sign error. Theorem 1.1 as stated is contradicted by the nonnegativity of the left-hand side for h1=h2=1, since S, C(1,1), and P(q^{-2}) are all positive while (1-q^2) is negative. The error sits in Section 3.5: when the Perron contour is shifted past u=q^{-2}, the residue must be subtracted, not added. The residue computation itself is correct—(q^{-4}-q^{-2}) S C P is negative—so the main term should be (q^2-1) S C P q^g. This is a fixable mistake, but the theorem as written is false.

What is actually new: the twisted second moment for primitive cubic L-functions over function fields has not been treated before. Reference [10] handles only the untwisted case, and [6] gives a mollified bound. The double Dirichlet series machinery from Gao–Zhao and Hong–Zheng is adapted cleanly, and the convergence regions, functional equation, and residue extraction are largely standard and correct. The explicit Euler product computation of C(h1,h2) and S is a useful and nontrivial piece of work. The derivation is self-contained, with no fitted parameters or post hoc exclusions.

Soft spots beyond the sign: there are typos ("folllowing," "associeted," "exadtly"), and the definition of S4 in (3.29) is missing an inequality sign. Also, the claim that the result is "consistent with the bound in (1.1)" is vague, since (1.1) concerns a mollified second moment. None of these are major.

If the sign is flipped, this becomes a solid contribution to the function-field moments subfield. I would send it to a serious referee rather than desk reject: the error is localized and easily verified, and the core method is competently executed. The authors should be asked to fix the sign and run the sanity check h1=h2=1 before acceptance.

Referee Report

2 major / 3 minor

Summary. The paper studies the twisted second moment of primitive cubic L-functions over F_q(T) in the non-Kummer setting, for fixed polynomials h1,h2 and characters of fixed genus g. The authors use Perron's formula to express the moment as a contour integral of a double Dirichlet series A2(u,v), compute the residue at u=q^{-2}, and use analytic continuation through three convergence regions to obtain the claimed asymptotic in Theorem 1.1: the main term (1-q^2) S C(h1,h2) P(q^{-2}) q^g plus an error O(q^{(1/2+eps)g}).

Significance. If correct, the asymptotic would be a new result and a clean illustration of the double Dirichlet series method for higher-order characters over function fields. The paper has genuine strengths: the residue computation is explicit and has no fitted parameters, the use of multivariable tube-domain continuation is appropriate in principle, and the claimed error term is sharply falsifiable. However, the central theorem as stated is internally contradicted by the positivity of the left-hand side when h1=h2=1, and the root cause is a sign error in the contour shift in Section 3.5. A second, closely related correctness issue is the false identity 'chi^2=chi' invoked in Section 3.1. Since the main claim is false as written, the current significance of the paper is not realized.

major comments (2)

[Section 3.5 / Theorem 1.1, Eq. (1.3)] The sign of the main term is wrong. The residue computed in (3.22) is (q^{-4}-q^{-2}) S C(h1,h2) P(q^{-2}), so the residue of the integrand in (3.2) at u=q^{-2} is (1-q^2) S C(h1,h2) P(q^{-2}) q^g. When the contour is shifted from |u|=r outward past this pole, the residue theorem gives I_inner = I_outer - 2*pi*i*Res, so the pole contributes with the opposite sign to the one the paper uses. The paper instead takes the pole contribution to be exactly the residue, yielding the negative main term in (1.3). For h1=h2=1, C(1,1)=1 and S and P(q^{-2}) are positive Euler products, while (1-q^2)<0, so the stated right-hand side is negative and of size q^g. The left-hand side is sum_chi |L(1/2,chi)|^2 >= 0, and the error O(q^{(1/2+eps)g}) cannot dominate a negative q^g term for any fixed eps<1/2. This contradicts Theorem 1.1 as stated.
[Section 3.1, between Eqs. (3.3) and (3.7)] The text states 'Since cubic characters satisfy chi^2=chi' and uses this to replace chi_F(h2) and chi_F(N2) by their squares. For a cubic character the correct identity is chi^2 = bar(chi). The algebra leading to the argument h1 h2^2 N1 N2^2 is only valid if (3.3) is read with the conjugate bar(chi_F)(h2) and the expansion of |L|^2 is written as sum chi(N1) bar(chi)(N2) = sum chi(N1) chi(N2)^2. As written, the displayed identity is false and the derivation is invalid; this must be corrected and the notation in (3.3) fixed throughout.

minor comments (3)

[Eq. (3.29)] The displayed definition of S4 is malformed: it reads '|uv^4|^{-3}', which is not a region. It should presumably be |uv^4| < q^{-3}, matching the regions S2,2 and S3.
[Throughout] There are several typographical errors: 'associeted' in Theorem 1.1, 'folllowing' in Section 2.1, 'Eular' in Section 3.2, and 'exadtly' in Section 3.4.
[References] Several references are incomplete: [3] and [4] lack a year, [8] lists no volume or pages, and the title of [12] contains the typo 'L-funcitons'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main term is an explicit residue computation from an independent double Dirichlet series, with no fitted inputs or load-bearing self-citation chain.

full rationale

The derivation is self-contained in the sense required by the circularity pass. The claimed main term in Theorem 1.1 is produced by a concrete residue computation: the sum over primitive cubic characters is rewritten via Perron's formula as a contour integral of the double Dirichlet series A2(u,v), and the residue at u = q^{-2} is evaluated in Section 3.2 using the explicit local factors and the pole of ζ_{q2}(u)/ζ_{q2}(u^2). The quantities P(u), S, and C(h1,h2) are defined from Euler products in equations (3.11), (3.12), (3.20), and (3.21); they are not fitted to the moment being studied. The convergence regions and meromorphic continuation rely on Bochner's tube theorem and on external lemmas from David–Florea–Lalín [8] (character parametrization, functional equation, Lindelöf-type bounds), which are stated as prior results and are not equivalent to Theorem 1.1. The self-citations [12,13] are cited only as origin of the double-Dirichlet-series method, not as the justification of the main term, so they are not load-bearing. A reviewer concern about the sign of the residue contribution in Section 3.5 is a mathematical correctness issue about whether the residue should be added or subtracted after the contour shift, not a circularity of the kind defined by the analysis; per the hard rules, correctness objections are not scored as circularity. No step reduces to its inputs by construction, no fitted parameter is renamed a prediction, and no uniqueness claim is imported from the authors' prior work. Therefore the appropriate circularity score is 0.

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

The paper introduces no fitted constants or new entities. Its inputs are standard function-field L-function facts, complex analysis theorems, and a character parametrization taken from the literature. The central derivation is analytic, not empirical.

assumptions (5)

domain assumption Primitive cubic characters of genus g over F_q(T) are parametrized by squarefree F in H_{q2,g/2+1} with no divisor in A_q.
Quoted from reference [8] in Lemma 2.1; this parametrization is the bridge between character sums and the Dirichlet series over F.
domain assumption The Lindelöf bound |L_q(s,χ)| ≪ q^{ε deg h} holds for ℜ(s) ≥ 1/2.
Restated from [8] as Lemma 2.3; it is used in Section 3.1, 3.3 and 3.4 to bound the double Dirichlet series.
standard math Bochner's Tube Theorem and the modulus continuation bound (Theorem 2.1 and Theorem 2.2).
Standard multivariable complex analysis; used in Section 3.5 to justify holomorphic continuation of A2(u,v) to the convex hull S4.
standard math χ_D(F) = 1 for coprime D,F in A_q.
Proved in Lemma 3.1 using the Frobenius automorphism; it simplifies the inner sum over D.
ad hoc to paper The only relevant singularity of (u−q^(−2)) A2(u,v) inside the shifted contour is the simple pole at u=q^(−2).
Assumed implicitly in Section 3.5; other possible poles from the Euler product factors are not discussed.

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Pith. "Pith review of Twisted second moment of primitive cubic L-functions." pith.science (2026). https://pith.science/paper/BS4AJU3U

@misc{pith2026250614656,
  author       = {Pith},
  title        = {Pith review of: Twisted second moment of primitive cubic L-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BS4AJU3U}},
  note         = {Machine review of arXiv:2506.14656}
}

abstract

We investigate the mean value of the twisted second moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ cubic\\ genus(\chi)=g}}\chi(h_1)\bar{\chi}(h_2)|L_q(\frac{1}{2}, \chi)|^2, \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive cubic character $\chi$. Employing a double Dirichlet series approach, we establish an error term of size

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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

On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
math.NT 2026-07 accept novelty 7.0 of 10

Asymptotics for twisted first and second moments of r-th order Hecke L-functions yield a positive proportion of non-vanishing central values for square-free and r-th power-free ideal families when r≥3.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages · cited by 1 Pith paper

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C. David, A. Florea, and M. Lalin. Nonvanishing of L–functions associated to fixed order characters over function fields. arXiv preprint arXiv:2506.07815 , 2025

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [10]

The second moment of cubic Dirichlet L-functions over function fields

S. Goel and A. Ray. The second moment of cubic Dirichlet L-functions over function fields. arXiv preprint arXiv:2505.12015, 2025

work page Pith review arXiv 2025

[2] [6]

David, A

C. David, A. Florea, and M. Lalin. Nonvanishing for cubic L -functions. Forum of Mathematics, Sigma, 9, 2021

work page 2021

[3] [1]

Baier and M

S. Baier and M. P. Young. Mean values with cubic characters. Journal of Number Theory , 130(4):879–903, 2010

work page 2010

[4] [2]

Bary-Soroker and P

L. Bary-Soroker and P. Meisner. On the distribution of the rational points on cyclic covers in the absence of roots of unity. Mathematika, 2017

work page 2017

[5] [3]

S. Bochner. A theorem on analytic continuation of functions in several variables. Ann. of Math. (2) , 39(1):14–19

work page

[6] [4]

M. ˇCech. The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series. Trans. Amer. Math. Soc. , 377:3487–3528

work page

[7] [5]

David, A

C. David, A. Faveri, A. Dunn, and J. Stucky. Non-vanishing for cubic Hecke L-functions. arXiv preprint arXiv:2410.03048, 2024

arXiv 2024

[8] [7]

David, A

C. David, A. Florea, and M. Lalin. Nonvanishing of L–functions associated to fixed order characters over function fields. arXiv preprint arXiv:2506.07815 , 2025

arXiv 2025