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The first moment of central value of primitive quartic $L$-functions with fixed genus

T0 review · 5 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For q≡3 mod 4, the first moment of primitive quartic L-functions over F_q(T) equals an explicit main term plus an error of size q^{(3/5+ε)g}, and many characters in the family are nonzero at the central point.

desk verdict A plausible first-moment asymptotic for quartic L-functions over function fields, but the written proof has a false lemma and a sketched continuation argument; worth serious refereeing, not acceptance as is. read the letter →

arxiv 2504.14291 v5 pith:6AXDKK3C submitted 2025-04-19 math.NT

Ziwei Hong This is my paper · ORCID
classification math.NT MSC 11M0611M4111N3711L0511L40
keywords centralvaluesquarticL-functionsGausssumsfunctionfielddoubleDirichletseriesfirstmomentnonvanishingfixedgenus
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

This paper claims to pin down the average value of primitive quartic Dirichlet L-functions over the rational function field $\mathbb{F}_q(T)$, taken over characters of a fixed genus $g$ in the case $q\equiv 3\pmod 4$. The claimed formula gives a main term $q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2})$ plus an error of size $q^{(3/5+\varepsilon)g}$, which is genuinely smaller than the main term. Because the main term is nonzero, the formula forces many of these L-functions to be nonzero at the central point $s=\tfrac12$, a nonvanishing statement for quartic L-functions in a function-field family where none was known. The proof routes the whole computation through a double Dirichlet series $A_4(u,v)$, using an upper bound for the generating function of quartic Gauss sums to continue the series to a convex hull and read the answer off a residue.

What carries the argument

The load-bearing object is the double Dirichlet series $$A_4(u,v)=\sum_{\substack{F\in H_{$q^{2}$}\\ P\mid F\Rightarrow P\notin\mathbb{F}_q[T]}} L_q(w,\chi_F)\,$u^{{\deg F}}$,\qquad v=$q^{{-w}}$,$$ where $H_{q^2}$ is the set of monic square-free polynomials in $\mathbb{F}_{q^2}[T]$ whose prime divisors do not come from $\mathbb{F}_q[T]$, together with the quartic Gauss-sum generating function $\Psi_q(f,u)=\sum_{F\in\mathbb{F}_q[T]}G_q(f,F)u^{\deg F}$. Perron's formula rewrites the desired family sum as a contour integral of $A_4(u,\tfrac12)\,u^{-g/3-2}$, and the main term is the residue at the pole $u=q^{-2}$. The estimate that makes the whole argument possible is the bound (3.14), $\widetilde\Psi_q(f,u)\ll |f|^{\frac12(\frac32-\sigma)+\varepsilon}$ in the strip $\tfrac12\le\sigma\le\tfrac32$; this comes from a functional equation for $\Psi_q$ and a Phragmén–Lindelöf argument. With that bound in hand, the paper continues $(u-q^{-2})A_4(u,v)$ holomorphically to the convex hull $$S_4=\{(u,v): |u|<$q^{{-1}}$,\ |$u^{{1/2}}$v|<$q^{{-5/4}}$,\ |$u^{5}$$v^{8}$|<$q^{{-13}}$\},$$ and shifting the contour to $|u|=q^{-9/5+\varepsilon}$ produces the error term $q^{(3/5+\varepsilon)g}$.

What would settle it

A direct check would be to compute, for a small prime power $q\equiv3\pmod4$ such as $q=3$ or $q=7$, the left-hand side of Theorem 1.1 for a few small genera $g$ by enumerating the primitive quartic characters and evaluating $L_q(\tfrac12,\chi)$ via the functional equation; any difference from the claimed main term that grows faster than $q^{(3/5+\varepsilon)g}$ would disprove the theorem. A cheaper test targets Lemma 3.1 directly: fix a square-free $f\in\mathbb{F}_q[T]$, evaluate $\Psi_q(f,u)$ at a point with $|u^4-q^{-5}|>\delta$ and $\sigma\in[\tfrac12,\tfrac32]$, and see whether the asserted bound $|f|^{\frac12(\frac32-\sigma)+\varepsilon}$ holds.

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Extended reading notes

Core claim

For $q\equiv 3\pmod 4$ and every $\varepsilon>0$, the paper claims that $$\sum_{\substack{\chi\text{ primitive quartic}\\ \$chi^{2}$\text{ primitive}\\ \mathrm{genus}(\chi)=g}} L_q(\tfrac12,\chi) = $q^{{2g/3}}$(1-$q^{2}$)P($q^{{-2}}$)Z($q^{{-2}}$,$q^{{-1/2}}$) + O($q^{{(3/5+\varepsilon)g}}$),$$ where $P(u)$ and $Z(u,v)$ are explicit Euler products introduced in Section 4.3. It then derives the lower bound $\#\{\chi:L_q(\tfrac12,\chi)\neq 0\}\gg q^{2g/3-\varepsilon(g/3+1)}$. The author presents this as the quartic analogue of the cubic first moment over function fields, obtained without approximating the functional equation or computing individual Gauss-sum residues; all of the arithmetic input is concentrated in an estimate for the quartic Gauss-sum generating function.

Load-bearing premise

Everything rests on one estimate: that the generating function of quartic Gauss sums, averaged over polynomials coprime to a fixed index, grows at most like $|f|^{(\frac32-\sigma)/2+\varepsilon}$ in the strip $\tfrac12\le\sigma\le\tfrac32$; if this bound fails, the meromorphic continuation to the convex hull fails and the power-saving error term collapses.

Editorial extensions

If this is right

  • The first moment of the quartic family is asymptotic to $q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2})$, with an error $q^{(3/5+\varepsilon)g}$ that is smaller than the main term.
  • At least $q^{2g/3-\varepsilon(g/3+1)}$ primitive quartic characters of genus $g$ have $L_q(\tfrac12,\chi)\neq0$.
  • The average of $L_q(\tfrac12,\chi)$ over the family is bounded, so the main term has the same order as the family size $q^{2g/3}$; a typical character contributes a bounded amount at the central point.
  • The method shows that explicit residue computations for Gauss sums can be replaced by convex-hull continuation plus a single bound on the Gauss-sum generating function, reducing the computational burden of such moment problems.
  • If the bound (3.14) were improved, the error exponent $3/5$ in Theorem 1.1 would improve accordingly.

Reading between the lines

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

  • The same convex-hull scheme should carry over to the case $q\equiv1\pmod4$ with minor modifications, which the paper notes but does not carry out; spelling out the Gauss-sum bound there would complete the quartic picture.
  • A second moment of the same family would convert the nonvanishing lower bound into an asymptotic count of nonvanishing characters; the first-moment main term here is the natural starting point for such a calculation.
  • Since the error exponent is controlled by the Phragmén–Lindelöf width in Lemma 3.1, a sharper analytic treatment of $\Psi_q$ would directly shrink the error in Theorem 1.1.
  • The same strategy may apply to sextic or higher-order characters over function fields, with the convex hull growing more complicated as the order increases; the bottleneck would be the analogous Gauss-sum generating function bound.
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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

5 major / 4 minor

Summary. The paper studies the first moment of central values of primitive quartic L-functions over F_q(T) in the non-Kummer setting q≡3 mod 4, with characters ordered by genus. The main result (Theorem 1.1) states that the sum over primitive quartic characters with χ^2 primitive and genus g equals an explicit main term of size q^{2g/3} times an Euler product, with an error O(q^{(3/5+ε)g}). The method uses a double Dirichlet series A_4(u,v), a Perron-type contour integral, and a meromorphic continuation to a convex hull S4 obtained by combining three convergence regions. The proof follows the strategy of Gao–Zhao and the cubic work of David–Florea–Lalin and Hong et al., and it centers on bounds for the generating function of quartic Gauss sums (Theorem 3.1 and Lemma 3.1). A nonvanishing corollary for quartic characters of genus g is also claimed.

Significance. If Theorem 1.1 is correct, it would be the first power-saving first moment result for primitive quartic L-functions over function fields with fixed genus, with an explicit main term and without the heavy residue computations of earlier approaches. The paper also advertises a clean framework (bounds on Gauss-sum generating functions plus convex hulls) that could apply to higher-order characters. The claimed nonvanishing corollary is a natural consequence. However, the significance is conditional: several load-bearing steps in the proof are either invalid as written (Lemma 4.1) or only sketched (Theorem 3.1), and the derivation of the third convergence region in §5.1 contains apparent degree-exponent and region-logic errors. The central idea is promising, but the manuscript needs substantial technical revision before the result is established.

major comments (5)
  1. [§4.2, Lemma 4.1 and equations (4.5)–(4.9)] The proof of Lemma 4.1 is invalid. From σ(Ω(α)) = Ω(α) one may only conclude that Ω(α) lies in F_q, i.e., Ω(α) ∈ {±1}; the nontrivial fourth roots of unity are interchanged by the Frobenius σ. Thus the lemma's conclusion χ_D(F)=1 is not established and is in fact false in general for inert or split primes. This invalidates the simplification used to pass from (4.5) to (4.9): the Euler product should contain characters χ_{P_1}(N) that are quadratic signs, and the factor Y_{P_1|N}(1-|P_1|^{-s}∏(...))^{-1} is not the correct local factor. Since (4.9) underlies the first convergence region and the residue computation at u=q^{-2}, this is a load-bearing error. The argument must either prove that the signs are trivial (which appears not to hold) or track this quadratic twist throughout the subsequent analysis.
  2. [§3.2, Theorem 3.1 and equations (3.8)–(3.12)] The proof of Theorem 3.1 is only a sketch, and the absent details are load-bearing because the bound (3.14) is the key input for the third convergence region in §5.1. Equations (3.10)–(3.12) are asserted without derivation; the dichotomy between the linearly dependent and linearly independent cases of the coefficient pairs is not justified; and the statement that a_1(s) and a_2(s) are absolutely bounded independently of f is not demonstrated, even though W_{f,i} appears in a_2(u) and depends on the Gauss sum of χ_4^{2i-1}χ_f. A complete proof is needed, or the paper should explicitly invoke Corollary 5.4 of [7] and show precisely how the required range 1/2≤σ≤3/2 is obtained.
  3. [§5.1, equations (5.4)–(5.6) and the definition of S3] When Lemma 3.1 is applied over F_q2 to Q=ND^2, the degree of Q over F_q2 is deg N + 2 deg D, not deg ND = deg N + deg D. The bound should read H(ND^2,uv^2) ≪ |u^{1/2}v q^{3/2+ε}|^{deg N + 2 deg D}. With the doubled D-exponent, the D-summation condition changes from |q^{5/2+ε}u^{3/2}v^3|<1 to a different condition, and the claimed region S3 and the convex hull S4 in (5.7) are not established. This is a central technical step: the contour shift in §5.2 and the final error term O(q^{(3/5+ε)g}) both depend on S4.
  4. [§5.1, definition of S3 and its proof] The region S3 is defined by removing exactly the annulus q^{-3/2} ≤ |u^{1/2}v| ≤ q^{-1/2} (equivalently q^{-3} ≤ |uv^2| ≤ q^{-1}), yet the only upper bound derived in the text, from Lemma 3.1, is valid precisely in that annulus. Hence the proof does not establish holomorphy of A_4(u,v) on the stated S3; the logic appears to be backwards. The relation between the region where the estimate applies and the region where holomorphy is claimed must be corrected.
  5. [§5.2, contour shift to |u|=q^{-9/5+ε}] For v=q^{-1/2}, a point with |u|=q^{-9/5+ε} does not satisfy the third defining inequality of S4, |u^5v^8|<q^{-13}, because |u^5v^8|=q^{-13+5ε}. The contour should be at |u|=q^{-9/5-ε} in order to lie inside S4. As written, the integration contour is outside the region of meromorphic continuation, so the proof of Theorem 1.1 is incomplete at this step; if this is a sign typo, it must be corrected and the effect on the ε in the error term checked.
minor comments (4)
  1. [§3.2, Theorem 3.1 statement] In the theorem statement the condition is |u^4-q^{-5}|>δ, but equation (3.9) and Lemma 3.1 also require |u^4-q^{-3}|>δ; the relationship between these exclusions and the poles at u^4=q^{-5}, q^{-3} should be clarified.
  2. [§4.1, equation (4.2)] The Perron formula is written with a denominator (1-u)u^{N+1}; the text should make explicit the contour orientation and the required radius r for the integral to be valid.
  3. [§5.3, proof of Corollary 1.1] There are typographical errors: 'gievs' should be 'gives' and 'Menawhile' should be 'Meanwhile'. More importantly, the final lower bound ≫ q^{2g/3}q^{-ε(g/3+1)} follows only if the implicit constants in the preceding inequalities are independent of g, which should be stated.
  4. [§2.1, Lemma 2.1] The set H_{q^2,g/3+1} is described as monic square-free polynomials of degree g/3+1, which implicitly assumes 3 divides g; the paper should state what happens for other g, even if the moment is then trivially zero or the degree parameter is adjusted.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main term is an explicit residue/Euler product with no fitted parameters, and the only self-citation is motivational rather than load-bearing.

full rationale

The paper derives Theorem 1.1 by constructing a double Dirichlet series A4(s,w) from the definition of quartic L-functions plus standard Perron, Mobius, and functional-equation manipulations. The main term is computed explicitly as the residue at u=q^{-2}, yielding q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2}); no parameter is fitted to the target moment and no quantity that appears in the conclusion is fed back into the derivation. The convergence regions S1, S2, S3, and their convex hull S4 rest on external results: the function-field Lindelof bound from David-Florea-Lalin, and the quartic Gauss-sum generating-function bounds obtained from Hoffstein's functional equation and Patterson's framework. The proof does not invoke the quartic first moment to prove its own continuation. The only self-citation is [13], used in the introduction to motivate the technique and to confirm a remark in [6] in the cubic case; it plays no role in the proof of the quartic theorem. The skeptical concerns about the proof of Theorem 3.1 or the degree count in the S3 estimate are possible correctness gaps, not circularity: they question whether an asserted bound is valid, not whether the theorem is assumed as an input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters and no invented entities. The central claim rests on external theorems about L-functions, Gauss sums, and metaplectic forms, plus one lemma whose written proof is incorrect but whose statement is true. The heaviest unverified dependency is the bound (3.14) for quartic Gauss sum generating functions.

assumptions (4)
  • standard math Functional equation for even quartic L-functions over F_q(T) (Lemma 2.2) and the standard L-function theory for function fields, including the Lindelöf-type bound (Lemma 2.3) which follows from Weil's Riemann hypothesis for curves.
    Used throughout to compute A_4(u,v) on various regions; proofs cited to [6] and [14].
  • standard math Hoffstein's functional equation for metaplectic theta functions (Proposition 3.1) and Patterson's results on general Gauss sums.
    Foundation for the bound on Ψ_q(f,u) in Theorem 3.1; external results not re-derived in this paper.
  • domain assumption Evaluation of quartic Gauss sums G_q(V,P^i) (Lemma 2.5), stated for q≡1 mod 8.
    Used in Lemma 3.1 and Section 5; proof omitted and cited to [17] and [9]; applied to F_{q^2} where q^2≡1 mod 8.
  • domain assumption Lemma 4.1: χ_D(F)=1 for coprime D,F∈A_q.
    Used to simplify the double Dirichlet series in Section 4.2; the paper's proof is flawed (σ fixes both 1 and -1), but the statement is true via the product over conjugate primes.

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Pith. "Pith review of The first moment of central value of primitive quartic $L$-functions with fixed genus." pith.science (2026). https://pith.science/paper/6AXDKK3C

@misc{pith2026250414291,
  author       = {Pith},
  title        = {Pith review of: The first moment of central value of primitive quartic $L$-functions with fixed genus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AXDKK3C}},
  note         = {Machine review of arXiv:2504.14291}
}
abstract

We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ quartic\\ \chi^2 primitive\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi), \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive quartic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.

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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. Twisted second moment of primitive cubic L-functions

    math.NT 2025-06 reject novelty 6.0 of 10

    A claimed twisted second moment asymptotic for primitive cubic L-functions over F_q(T) is invalidated by a sign error in the residue-theoretic main term.

Reference graph

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