The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields

J.C. Andrade; J. MacMillan

REVIEW 2 major objections 4 minor 24 references

The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The average of quadratic Dirichlet L-functions over even-degree function fields now has a three-term lower-order expansion with error O(q^{g/2(1+ε)}).

desk verdict A credible Florea-style extension to the even-degree real quadratic family that delivers new secondary main terms, but the error term relies on an imported bound and a missing lemma that need referee attention. read the letter →

arxiv 1908.04078 v1 pith:R5LO5PTC submitted 2019-08-12 math.NT

J.C. Andrade , J. MacMillan This is my paper · ORCID

classification math.NT MSC 11M3811M0611G20

keywords firstmomentquadraticDirichletL-functionsfunctionfieldsrealhyperellipticensembletrivialzeroPoissonsummationasymptoticexpansion

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 derives a sharper asymptotic for the first moment of quadratic Dirichlet L-functions at the central point, averaged over all monic square-free polynomials of even degree, the real quadratic function fields. The main theorem gives the leading term together with three explicitly computable lower-order main terms, of sizes (2g+2)$q^{{(2g+2)/3}}$, $q^{{g/6+⌊g/2⌋}}$, and $q^{{g/6+⌊(g-1)/2⌋}}$, and bounds the error by $q^{{g/2(1+ε)}}$. Such averages govern the distribution of class numbers and zero statistics in hyperelliptic families, so a precise formula is a stepping stone toward moment conjectures over function fields. The result extends a method previously applied to odd-degree (imaginary quadratic) fields to the even-degree case, where a trivial zero forces extra terms.

What carries the argument

The central mechanism is a sequence of changes of variables that converts the averaged approximate functional equation into double contour integrals of the Dirichlet series B(z,w) = Z(z)Z(w)Z($qw^{2}$ z) ∏_P B_P(z,w), where B_P(z,w) is a local Euler factor. The analysis separates the contribution from square V (which produces the main terms) from non-square V (which is shown to be small), and then extracts the main terms by evaluating residues at the poles w = $q^{{-1}}$ and w = qz, with a second contour in z picking up a double pole at z = $q^{{-4/3}}$. The Euler product C(u) = ∏_P (1 - $u^{{d(P)}}$/(|P|+1)) and its analytic continuation supply the explicit constants.

What would settle it

Compute the left-hand side of Theorem 1.4 for a fixed prime q ≡ 1 mod 4 and a range of g by enumerating all monic square-free polynomials of degree 2g+2, evaluating L(1/2, χ_D) for each, and subtracting the three main terms; if the remainder grows faster than $q^{{g/2(1+ε)}}$ (for example, like $q^{{2g/3}}$), the error claim in Theorem 1.4 is false. Alternatively, test the borrowed estimate directly by computing δ_{V;n}(u) for non-square V in the even-degree ranges.

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Extended reading notes

Core claim

For q a prime with q ≡ 1 mod 4, the sum over D ∈ H_{2g+2} of L(1/2, χ_D) is shown to equal P(1)/(2 ζ_A(2)) $q^{{2g+2}}$ [(2g+2) + (4/log q)(P'/P)(1) + 2 ζ_A(1/2)] plus $q^{{(2g+2)/3}}$ R(2g+2) plus C_1 $q^{{g/6+⌊g/2⌋}}$ plus C_2 $q^{{g/6+⌊(g-1)/2⌋}}$ plus O($q^{{g/2(1+ε)}}$). Here P(s) = ∏_P (1 - 1/(|P|^s(|P|+1))), ζ_A is the zeta function of F_q[x], R is an explicit linear polynomial, and C_1, C_2 are explicit constants built from Euler products. The three added terms are the paper's central discovery: they are larger than the error term, hence genuine parts of the asymptotics, and they capture the parity-dependent structure of the even-degree family.

Load-bearing premise

The small error term relies on an estimate for a certain average of Gauss sums that was proved in the odd-degree setting and is assumed, without proof, to hold in all the ranges needed here; if that estimate fails, the error term could be larger than O($q^{{g/2(1+ε)}}$).

Editorial extensions

If this is right

For prime q ≡ 1 mod 4, the first moment has a complete asymptotic expansion whose three lower-order terms are all larger than the error term, so the formula genuinely distinguishes their sizes.
The explicit constants C_1, C_2, and R allow the expansion to be evaluated numerically for any fixed q, making it testable by computer for small genera.
The error term O(q^{g/2(1+ε)}) matches the quality of the odd-degree result, so the even-degree family is now known with the same precision.
The appearance of the trivial-zero factor ζ_A(1/2) in the leading constant is isolated as part of the main term, clarifying the structural difference from the odd-degree case.

Reading between the lines

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

The same double-residue calculation may generalize to higher moments over real quadratic fields, producing additional secondary main terms whose exponents depend on the parity of g.
The parity-dependent exponents suggest a general pattern: families with a trivial zero at the central point may acquire lower-order terms of size roughly q^{(2g)/3} with coefficients that alternate according to g mod 2, a prediction that could be tested in other even-degree families.
Because the extra main terms come from a double pole at z = q^{-4/3}, analogous double poles in other averages may correspond to new arithmetic features, such as contributions from polynomials with a fixed factorization shape.

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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

Dear [Colleague],

Short version: this is a credible extension of Florea's method to the even-degree real quadratic function field family, and it delivers what it claims: the first secondary main terms for that family, with explicit constants. It deserves a serious referee, not a desk reject. But the error-term control rests on an imported Gauss-sum bound that needs careful checking, and there is a missing lemma reference that has to be fixed.

The new content is real. Jung's theorem had only the leading term and a q^{3/4} error; this paper gets terms of size q^{(2g+2)/3}, q^{g/6+floor(g/2)}, and q^{g/6+floor((g-1)/2)} with error O(q^{g/2(1+epsilon)}). The main terms come from residue computations, with R and C1, C2 explicitly defined as Euler products and rational functions of q. No parameters are fitted; the machinery is Florea's, but the even-degree case has genuinely extra complications (the V-square contributions from odd-degree f, the interplay of the two floors, and a long induction in the appendix). The exposition is structured and mostly careful.

The soft spots are real but localized. The most serious is Proposition 6.1: the bound on delta_{V;n}(u) is simply imported from Florea's paper as (6.6), with a citation. The manuscript does not verify that the ranges of n and non-square V appearing here satisfy the hypotheses of that result. If that bound fails in any of the ranges used, the O(q^{g/2(1+epsilon)}) error term in Theorem 1.4 is unsupported, and the new terms lose their separation. This is not a fitting issue; it's an assumption that needs a proof or at least a careful statement. Second, the text references 'Lemma 6.3' twice in Section 5 for bounding the tail integrals C_{k,ell}, but no such lemma appears. That's a gap in the written proof, though likely fixable. Finally, there are numerous typos and garbled renderings in the abstract and body; the authors should clean those up.

The long residue algebra in Section 5 and the appendix induction I did not fully check. The structure is believable, and the appendix gives a detailed induction argument for the cancellation. This is the kind of paper where the hard part is mechanical verification, not conceptual surprise.

Bottom line: send it to peer review. The referee should be asked to verify (6.6) in the needed ranges and to supply the missing Lemma 6.3. If those checks come out, the paper is a solid contribution to the function-field moment literature. It's not a new framework, but it's a new result in a family that matters for random-matrix comparisons.

Referee Report

2 major / 4 minor

Summary. The paper studies the first moment of quadratic Dirichlet L-functions at the central point, averaged over monic square-free polynomials D of degree 2g+2 over F_q[x] (the real quadratic function field case), with q an odd prime congruent to 1 mod 4. Theorem 1.4 claims an asymptotic expansion whose leading term matches Jung's formula, followed by secondary main terms of sizes q^{(2g+2)/3}R(2g+2), C_1 q^{g/6+floor(g/2)}, and C_2 q^{g/6+floor((g-1)/2)}, with error O(q^{g/2(1+epsilon)}). The proof follows Florea's method: an approximate functional equation reduces the moment to character sums; Poisson summation separates the contributions of square and non-square V; Perron's formula and contour residues evaluate the main and secondary terms; the non-square contribution is bounded using an imported Gauss-sum estimate. The constants R, C_1, C_2 are given by explicit Euler products and residue computations, and no parameter is fitted to data.

Significance. If correct, the result is a genuine improvement over Jung's asymptotic formula and provides an even-degree analogue of Florea's secondary main terms, including additional terms that have no counterpart in the imaginary quadratic case. A notable strength is that the whole computation is parameter-free: the constants are explicit Euler products and logarithmic derivatives of Euler products, and the main terms come from residue calculations rather than from assumed answers. The paper is, however, not fully self-contained at two load-bearing points: the control of the non-square contribution depends on an imported Gauss-sum bound whose hypotheses are not verified for the present ranges, and the proof of the V-square contribution invokes a 'Lemma 6.3' that does not appear in the manuscript. These gaps are local in nature, but they must be repaired before the claimed error term and secondary terms are established.

major comments (2)

[Section 6, Proposition 6.1 and Eq. (6.6)] Proposition 6.1, the entire bound for S(V != square), rests on the estimate |delta_{V;n}(u)| << q^{n/2(1+epsilon)} stated in (6.6) with a citation to [9, Section 7]. The manuscript then applies this estimate in Sections 6.1 and 6.2 to even n up to floor(g/2) and to inner sums over non-square V of degree depending on d(C), with d(C) as large as g; for n close to g/2 and d(C)=g the V-sums have degree about 2n-4, so the range is nontrivial. No argument is supplied that the hypotheses under which the bound is proved in [9] are satisfied for these parities and ranges, nor is it stated whether the quoted bound is uniform in d(V) in the ranges that occur. Since (6.6) is the only estimate feeding into the O(q^{g/2(1+epsilon)}) error term, this is a load-bearing gap. Please provide a proof or a precise transfer argument covering the ranges used here.
[Sections 5.1 and 5.2, after Eqs. (5.12) and (5.18)] In the proofs of Lemmas 5.4 and 5.5, the manuscript writes 'whilst using Lemma 6.3' to conclude that the C-terms are bounded by O(q^{g/2(1+epsilon)}). No Lemma 6.3 is stated anywhere in the manuscript: Section 6 contains Proposition 6.1 and Eqs. (6.5)-(6.6) only. These C-term bounds are needed to reduce S^e(V=square) and S^o(V=square) to the A- and B-integrals, so the omission is not merely a numbering issue. Please add the missing lemma, with a proof or with an explicit statement of the relevant result in [9], and verify that its hypotheses hold for each k and ell to which it is applied.

minor comments (4)

[Section 3, p. 7] The truncation of the C-sums at d(C) <= g is justified by an appeal to Section 4 of [9]; since this truncation is used in all four sums S_{g,1}, S_{g,2}, S_{g-1,1}, S_{g-1,2}, a short derivation of the claimed O(q^{g/2(1+epsilon)}) contribution would make the paper more self-contained.
[Section 6.1, displayed formulas for tilde{S}^e and hat{S}^e] The formulas contain an undefined factor '1/q^{2k}' in the integrands, while the summation variable appearing in the range is m; the notation should be clarified so that the reader can verify the displayed ranges and the subsequent bound.
[Abstract and Introduction] There are several grammatical slips (e.g., 'extra technical difficulties that doesn't appear'); these should be corrected in a final revision.
[Appendix A] The induction proof of the algebraic identity (A.1) is long and hard to check; a short indication of how (5.27) and (5.45) cancel the residue contributions at u=q^{-1} and u=q^{-2} would greatly help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed main and secondary terms are derived from residue computations over an explicit Poisson-summation decomposition, not from fitted inputs or self-citations.

full rationale

The derivation is self-contained in the relevant sense. Theorem 1.4's leading term, the q^{(2g+2)/3} R(2g+2) term, and the C1, C2 terms are obtained by evaluating residues of explicit integrals (Propositions 4.1 and 5.1, formulas (5.28)-(5.30), (5.46)-(5.48)) built from the algebraic factorization B(z,w)=Z(z)Z(w)Z(qw^2 z)\prod_P B_P(z,w) (Lemma 5.2) and the Euler-product identity (5.6). No parameter is fitted to the moment being predicted: C1, C2, and R are given as explicit convergent Euler products and residue coefficients. The reliance on Florea for the factorization lemmas and for the Gauss-sum bound (6.6) is external citation of prior independent work, not self-citation, and the target formula is not an input to those citations. The only flagged issues are proof-completeness concerns, not circularity: (i) the proofs of Lemmas 5.4 and 5.5 invoke a 'Lemma 6.3' to control the tail integrals C_{k,\ell}, but no such lemma appears in Section 6; and (ii) the estimate |\delta_{V;n}(u)| \ll q^{n/2(1+\epsilon)} is imported from Florea's odd-degree paper and applied to even-degree ranges without an explicit transfer check for all n and V arising in Sections 6.1-6.2. These are gaps in the verification of the O(q^{g/2(1+\epsilon)}) error term, but they do not make the main-term derivation circular, because the error bound is not used to produce the main terms and the main terms are not fitted to any data.

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

No fitted parameters are introduced. The proof relies on standard function-field background and on lemmas from Florea's paper: the B(z,w) factorization, the Euler product manipulations, and the non-square Gauss-sum bound. These are external working assumptions, not new entities invented by this paper.

assumptions (5)

domain assumption Quadratic reciprocity for F_q[x] with q ≡ 1 mod 4
Used in Section 2.1 to simplify the reciprocity law and define the quadratic characters; this restricts the theorem to primes q ≡ 1 mod 4.
domain assumption Approximate functional equation for L(1/2, chi_D), Lemma 2.4
Taken from Jung [13] and Rubinstein-Wu [20]; it is the starting point for the moment expansion in Section 3.
standard math Poisson summation formula over function fields, Lemma 2.6
Core transformation from sums over h to Gauss sums; proof is cited to Florea [9, Proposition 3.1].
domain assumption Non-square Gauss sum bound, equation (6.6)
Imported from Florea [9, Section 7] and used to bound S(V ≠ square); its transfer to the even-degree setting is the load-bearing step behind the O(q^(g/2(1+eps))) error term.
standard math Analytic continuation of C(u) via Euler product, equation (4.3)
Used to compute residues at u = q^{-1} and to justify C(1) = 0; standard Euler product manipulation.

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Pith. "Pith review of The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields." pith.science (2026). https://pith.science/paper/R5LO5PTC

@misc{pith2026190804078,
  author       = {Pith},
  title        = {Pith review of: The First Moment of $L(\frac12,\chi)$ for Real Quadratic Function Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R5LO5PTC}},
  note         = {Machine review of arXiv:1908.04078}
}

abstract

In this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of even degree at the central point. With some extra technical difficulties that doesn't appear in Florea's paper, we prove that there are extra main terms of size $(2g+2)q^{\frac{2g+2}{3}}, q^{\frac{g}{6}+\left[\frac{g}{2}\right]}$ and $q^{\frac{g}{6}+\left[\frac{g-1}{2}\right]}$, whilst bounding the error term by $q^{\frac{g}{2}(1+\epsilon)}$.

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Works this paper leans on

24 extracted references · 24 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

A. Florea. Improving the Error Term in the Mean Value of L(1/slash.left 2, χ) in the Hyperelliptic Ensemble. Int. Math. Res. Not. IMRN , 20:6119–6148, 2017

work page 2017

[2] [1]

J.C. Andrade. A Note on the Mean Value of L-Functions in Fu nction Fields. Int. J. Number Theory, 08(07):1725–1740, 2012

work page 2012

[3] [2]

Andrade and H

J.C. Andrade and H. Jung. Mean Value of the Class Number Re visited. Monatsh. Math. , 187(4):577–602, 2018

work page 2018

[4] [3]

Andrade and J.P

J.C. Andrade and J.P. Keating. The mean value of L(1/2, χ) in the hyperelliptic ensemble. J. Number Theory, 132:2793–2816, 2012

work page 2012

[5] [4]

Andrade and J.P

J.C. Andrade and J.P. Keating. Conjectures for the integ ral moments and ratios of L-functions over function ﬁelds. J. Number Theory , 142:102–148, 2014

work page 2014

[6] [5]

Rajagopal

J.C Andrade and S. Rajagopal. Mean values of derivatives of L-functions in function ﬁelds: I. J. Math. Anal. Appl. , 443:526–541, 2016

work page 2016

[7] [6]

E. Artin. Quadratische K¨ orper in Geibiet der H¨ oheren Kongruzzen I and II. Math. Z. , 19:153– 296, 1924

work page 1924

[8] [7]

Bae and H

S. Bae and H. Jung. Note on the mean values of derivatives o f quadratic Dirichlet L-functions in function ﬁelds. Finite Fields Appl. , 57:249–267, 5 2019

work page 2019