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

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

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Pith's one-line read This paper proves that the second moment of cubic Dirichlet $L$-functions over $\mathbb{F}_q(T)$ for $q \equiv 2 \pmod{3}$ equals an explicit constant times $g^2 q^{g+2}$, with error $O(q^g)$.

desk verdict The main theorem is probably right and the reader's Lemma 2.5 objection misfires, but the paper needs a notation cleanup before publication. read the letter →

arxiv 2505.12015 v2 pith:VM2AWBUM submitted 2025-05-17 math.NT

Shivani Goel , Anwesh Ray This is my paper · ORCID
classification math.NT MSC 11R5911M3811R58
keywords cubicDirichletL-functionsfunctionfieldssecondmomentgenusGausssumsapproximatefunctionalequationPerronformulanon-Kummercase
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 second moment of cubic Dirichlet $L$-functions at the central point $s=1/2$, summed over primitive cubic characters of genus $g$ over the rational function field $\mathbb{F}_q(T)$, in the non-Kummer case $q \equiv 2 \pmod{3}$. The main term is an explicit constant depending only on $q$, times $g^2 q^{g+2}$, and the error is $O(q^g)$. This is the first exact second-moment result for cubic characters over function fields in the non-Kummer case, extending the earlier first-moment formula and showing that only perfect cubes contribute to the main term.

What carries the argument

The load-bearing object is the character $\chi_F$ attached to a squarefree conductor $F$ in $\mathbb{F}_{q^2}[T]$, together with the identity (Lemma 2.5) that $\chi_F(f)=1$ for every squarefree $f$ coprime to $F$. This identity lets the principal sum be split into a perfect-cube part whose generating function is $B_2(u,z)=Z_q(u)^4 Z_{q^2}(z) Z_{q^2}(z^2)^{-1} A_q(z,u)$, and a non-cube part bounded by $O(q^{(g+t)/2})$. The dual sum is handled by expressing the sign of the functional equation through Gauss sums and applying the cited average estimates for those Gauss sums.

What would settle it

For $q=5$ and a degree-$2$ prime $P$, compute the cubic residue symbol $\chi_P(T)$ directly from the definition $T^{(q^{2\cdot 2}-1)/3} \bmod P$; if the result is not $1$, Lemma 2.5 is false and the bound $O(q^{(g+t)/2})$ in (3.18) is unsupported.

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Extended reading notes

Core claim

The central claim is Theorem A: for $q \equiv 2 \pmod{3}$, the sum over primitive cubic characters of genus $g$ of $L_q(1/2,\chi)^2$ equals $\frac{g(g+2)}{8} A_q(1/q^2, 1/q^{3/2}) \zeta_q(3/2)^2 \zeta_q(3)^{-1} q^{g+2} + O(q^g)$, where $A_q$ is the explicit Euler product defined in (3.9). The decisive step is that the non-cube part of the principal sum and the dual sum are both lower order: after switching to squarefree conductors $F$ over $\mathbb{F}_{q^2}[T]$ and applying Lemma 2.5, every relevant character value $\chi_F(f)$ equals $1$, so the sum over $f$ collapses to the perfect cubes $l^3$, which is then evaluated by a double Perron integral and a residue computation.

Load-bearing premise

The proof requires that every character $\chi_F$ take the value $1$ on every squarefree polynomial $f$ coprime to $F$; if even one such value is not $1$, the non-cube contribution to the main term is not controlled.

Editorial extensions

If this is right

  • With the cutoff $A=g/2$, both the principal and dual sums fall inside the error $O(q^g)$, leaving the single constant $A_q(1/q^2,1/q^{3/2})$ as the entire main term.
  • Because the non-cube contribution is absorbed into the error, the second moment of this family is determined entirely by the diagonal contribution from $f=l^3$.
  • The formula shows the average of $L_q(1/2,\chi)^2$ over the family grows like $g^2$, so the central values are not concentrated on a single scale.
  • The error $O(q^g)$ matches the quality of the known first-moment asymptotic, so the second moment is known up to the same relative precision.

Reading between the lines

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

  • If Lemma 2.5 failed for even-degree primes, the non-cube sum would not vanish and the main term would likely pick up extra Euler factors; checking $\chi_P(T)$ for $q=5$ and a degree-$2$ prime $P$ is a direct computation that would settle this.
  • The same double-Perron framework, with cubic Gauss sums replaced by higher-order ones, suggests a route to third and fourth moments, provided the non-cube cancellation is re-examined for those orders.
  • The constant $A_q(1/q^2,1/q^{3/2})$ is an Euler product over $\mathbb{F}_q[T]$ and can be evaluated numerically for small $q$, giving a cheap check against brute-force enumeration of $L$-values for small genus.
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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 / 3 minor

Summary. The paper claims an asymptotic formula for the second moment at the central point of cubic Dirichlet L-functions over the rational function field F_q(T) in the non-Kummer case q≡2 mod 3. The main result, Theorem A, states that the sum over primitive cubic characters of genus g of L_q(1/2,χ)^2 equals (g(g+2)/8) A_q(1/q^2,1/q^{3/2}) ζ_q(3/2)^2 ζ_q(3)^{-1} q^{g+2} + O(q^g). The method follows David–Florea–Lalin: primitive cubic characters are parametrized by squarefree polynomials over F_{q^2}[T], an approximate functional equation is applied, the principal term is split into cube and non-cube parts, and the dual term is estimated through averages of Gauss sums. The paper is essentially an adaptation of the first-moment machinery of DFL22 to the second moment.

Significance. If the result is correct, it is a meaningful advance: it appears to give the first asymptotic for the second moment of cubic Dirichlet L-functions in the non-Kummer function-field case, with an explicit main term and no fitted parameters. The paper builds in a transparent way on the established DFL22 framework and identifies a concrete main term. A key point raised in one reading of the paper, the status of Lemma 2.5, is in fact not an error: for F and f in F_q[T] with (f,F)=1, the F_{q^2}[T] cubic character is the product of the two conjugate-prime symbols, and q≡2 mod 3 makes that product equal to 1; the apparent counterexample with q=5 and f=T evaluates the different F_q[T] symbol, not the F_{q^2}[T] character used in the application. The proof does, however, contain a serious indexing inconsistency in the dual term and an unjustified aggregation of error terms, so the stated theorem is not proved as written.

major comments (3)
  1. [§3.1, Eqs. (1.3), (3.1)–(3.2), Prop. 2.3] The dual sum in the approximate functional equation is indexed incorrectly. Proposition 2.3 with k=2 gives, for the second term, a sum over i of (i+1)ω(χ)^2 over f∈M_{≤2g-A-1-i}. With S_{t,dual} defined in (3.2) by f∈M_{≤2g-t-1}, the matching choice is t=A+i. Equation (1.3) instead uses S_{2g-A-1-i,dual}, which gives the f-range A+i, and equation (3.1) uses S_{2g-A-1+i,dual}, which gives the f-range A-i; neither equals the required 2g-A-1-i. Consequently the estimate (3.29) is applied in §3.5 to a sum that is not the dual part of the second moment, so the proof of Theorem A does not currently establish the claimed identity for the moment.
  2. [§3.5, proof of Theorem A] The summation over i aggregates the error terms incorrectly. For t=A-i with A=g/2, the lower-order terms q^{g-t_0/2-2}(P_2(q)+Q_3(q^{3/2})) in (3.12) contribute after summation a quantity of size g q^g: writing j=A-i, one has ∑_{j=0}^A (j+1)q^{-⌊j/3⌋/2}=O(g). Similarly, the first error term in (3.19) after multiplication by q^{g-A/2+εA}∑(i+1)q^{i/2-εi} is O(g q^g) rather than O(q^g). The displayed simplification to O(q^g+...) is therefore not justified. The final asymptotic may still hold with the weaker error O(g q^g), but Theorem A as stated with O(q^g) is not supported by the given estimates.
  3. [§3.4, Eq. (3.26) to Mt] The transition to the main dual term changes the range of f from M_{≤2g-t-1} in (3.22) and (3.26) to M_{q,≤g-t-1} in the definition of Mt. No justification is given for this reduction, and it affects the size of the dual contribution by a factor of roughly q^{(g-t)/6} in the generating-function estimate. Please either derive this restriction from the ρ-function or the δ_{f_2=1} condition, or correct the range and recompute Mt.
minor comments (3)
  1. [Lemma 2.5] The statement should include the hypothesis (f,F)=1. As written, χ_F(f)=0 when a prime divides both f and F, so the assertion fails for such f. The application at (3.13) satisfies (D,f)=1, so the intended statement suffices.
  2. [Eq. (3.3)] The inner sum over F in the definition of S_{t,prin,=} should have the condition (F,l)=1; otherwise χ_F(l^3)=0 for primes common to F and l. The later generating function in (3.5)–(3.6) correctly imposes this condition.
  3. [Notation for χ_F] Please unify the notation: in Lemma 2.5, F is a squarefree polynomial in F_q[T], while in Lemma 2.4 and in Section 3, F ranges over F_{q^2}[T]. This ambiguity contributed to the confusion about which cubic residue symbol is being evaluated and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a standard moment computation with external cited inputs, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim (Theorem A) is derived by expressing the second moment as an exact sum over primitive cubic characters (Lemma 2.4), applying an approximate functional equation (Proposition 2.3), and then estimating the resulting principal and dual sums. The main term is obtained by a residue computation of an explicit generating function (equations (3.5)-(3.12)); the error terms are bounded using Perron's formula and the cited Gauss-sum averages of DFL22. No parameter is fitted to a subset of the data and then renamed as a prediction. The constant A_q(1/q^2,1/q^{3/2}) is a convergent Euler product defined by equation (3.9), and the leading factors q^{g+2} and g(g+2)/8 come directly from the generating-function calculation, not from an ansatz chosen to match the claimed result. The cited results from DFL22 (Propositions 2.7 and 2.8, Lemmas 2.2 and 4.4, and the L-function bounds) are external published theorems rather than self-citations, and they are invoked as lemmas with stated hypotheses, not as a uniqueness theorem that forces the paper's choice. The only delicate internal step, Lemma 2.5 (used at equation (3.13) to remove the non-cube contribution), is proved from the definition of the F_{q^2}[T] cubic character as a product chi_Q chi_{\bar Q}. Whether that proof is correct is a mathematical correctness question, not a circularity question: if the lemma failed, Theorem A would be unsupported, but the argument still would not be equivalent to its own input. In particular, there is no self-definitional relation, no fitted parameter renamed as a prediction, and no load-bearing self-citation chain. The paper is not circular; its correctness risk lies instead in the validity of the quoted lemmas and the error-term bounds.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles or abstract objects. It relies on standard zeta-function machinery, the DFL22 Gauss-sum average, and most importantly on Lemma 2.5, whose proof appears to misidentify the cubic residue symbol. That lemma is the main unsupported input.

free parameters (1)
  • A (approximate functional equation cutoff) = g/2
    Chosen at the end of Section 3.5 to balance the error terms in the principal and dual sums. It is a standard truncation parameter, not fitted to data, but it is selected by hand to make the claimed error O(q^g) come out.
assumptions (4)
  • standard math Weil conjectures and the expression of L-functions as numerator polynomials in the zeta function of a curve
    Used in Section 2.3 to write the L-polynomial, define the genus, and state the functional equation.
  • domain assumption Cubic reciprocity and splitting behavior of primes in F_{q^2}(T)/F_q(T), specifically the identification of cubic characters over F_q[T] with characters over F_{q^2}[T]
    Used in Section 2.5 and Lemma 2.4 to parametrize primitive cubic characters of genus g by squarefree F of degree g/2+1 over F_{q^2}[T]. Lemma 2.5 relies on a further claim about the cubic residue symbol that appears false.
  • domain assumption Proposition 2.8, quoted from DFL22 Proposition 3.1, on averages of Gauss sums
    Imported without proof and used in Section 3.4 to control the dual term. It is an external result, not machine-checked and not proved in this paper.
  • ad hoc to paper Lemma 2.5, asserting χ_F(f) = 1 for f ∈ F_q[T] coprime to F
    This is the load-bearing simplification used to discard non-cube contributions. The proof appears to conflate the Section 2.5 symbol χ_P with the product χ_Q χ_{\bar Q}, which are not equal, so the axiom is not supported.

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Cite this review

Pith. "Pith review of The second moment of cubic Dirichlet L-functions over function fields." pith.science (2026). https://pith.science/paper/VM2AWBUM

@misc{pith2026250512015,
  author       = {Pith},
  title        = {Pith review of: The second moment of cubic Dirichlet L-functions over function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VM2AWBUM}},
  note         = {Machine review of arXiv:2505.12015}
}
abstract

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result extends prior work of David, Florea and Lalin, who obtained an asymptotic formula for the first moment. Our approach relies on analytic techniques (Perron's formula, approximate functional equation, etc), adapted to the function field context. A key step in the construction is to relate second moment to certain averages of Gauss sums, which are estimated in loc. cit. using results of Kubota and Hoffstein.

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Forward citations

Cited by 2 Pith papers

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

  1. Upper bound for the moment of shifted values of cubic $L$-functions over function fields

    math.NT 2026-08 conditional novelty 6.0 of 10

    For non-Kummer cubic L-functions over F_q[t], the mixed shifted moments are at most |C_g| g^{k_1^2+...+k_m^2} times pairwise shift factors, matching the conjectured unitary order.

  2. 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

Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 2 Pith papers

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