First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

Liangyi Zhao; Peng Gao

REVIEW 2 major objections 5 minor 19 references

First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that, under the Riemann and Lindelöf hypotheses, the smoothed first moment of primitive quadratic Hecke L-functions in the Gaussian field equals X Q1(log X) + X^{1/3} Q2(log X) + O_ε(X^{1/4+ε}) at the central point s=1/2,

desk verdict Solid new result for the Gaussian field; the s→1/2 step is under-explained, but the pole cancellation is real, so the main theorem survives. read the letter →

arxiv 2512.22509 v2 pith:7TZHKKXU submitted 2025-12-27 math.NT

Peng Gao , Liangyi Zhao This is my paper · ORCID

classification math.NT MSC 11M0611M41

keywords quadraticHeckeL-functionsfirstmomentdoubleDirichletseriesGaussianfieldsecondarymaintermLindelöfhypothesisRiemannprimitivecharacters

topics The Riemann Hypothesis

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 evaluates the first moment of the family of primitive quadratic Hecke L-functions over the Gaussian integers, focusing on central values at s=1/2. Under the Riemann hypothesis (which implies the Lindelöf hypothesis for these L-functions), the smoothed sum is shown to have two main terms: a leading term of order X multiplied by a linear polynomial in log X, and a secondary term of order X^{1/3} multiplied by another linear polynomial, with an error of size X^{1/4+ε}. This shape—main term, X^{1/3} secondary term, X^{1/4} error—matches the corresponding families over the rational numbers and over function fields. The proof works by treating the generating series as a double Dirichlet series, establishing its meromorphic continuation, and computing its residues explicitly. If correct, the result gives the sharpest known asymptotic for this family and provides a template for studying higher-order terms in related moments.

What carries the argument

The load-bearing object is the double Dirichlet series A(s,w) = Σ_{d square-free primary} L(s, χ_{(1+i)^5 d}) / N(d)^w. The paper establishes its meromorphic continuation to a tube domain, locates its possible poles, and computes the residues at w=1, w=3/2−s, w=1/2−s/3, and w=2/3−2s/3. The proof chains together the functional equation for quadratic Hecke L-functions, quadratic reciprocity in the Gaussian field, and Bochner's tube theorem to glue together local holomorphic continuations; the residues then give the main terms in the moment after Mellin inversion and contour shift.

What would settle it

For a fixed smooth weight Φ, compute the smoothed first moment at increasing X, subtract X Q1(log X) + X^{1/3} Q2(log X) using the explicit polynomials obtained from the proof, and check that the remainder stays bounded by a constant times X^{1/4+ε}; a single X where the remainder clearly exceeds, say, X^{1/4} log X would disprove the asymptotic. Alternatively, finding a counterexample to the Lindelöf bound |L(1/2+it, χ)| ≪ N(q)^ε for some primitive quadratic Hecke character would invalidate the error term.

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Extended reading notes

Core claim

The central claim is that the first moment of quadratic Hecke L-functions in the Gaussian field admits an asymptotic expansion with a secondary main term whose size is the cube root of the leading term. Specifically, for a smooth compactly supported weight Φ, the sum over square-free primary d of L(1/2, χ_{(1+i)^5 d}) Φ(N(d)/X) equals X Q1(log X) + X^{1/3} Q2(log X) + O_ε(X^{1/4+ε}) under GRH, where Q1 and Q2 are linear polynomials whose coefficients are absolute constants. For general s with 1/3 < Re(s) < 1, the analogous formula contains four main terms arising from four poles of the double Dirichlet series A(s,w), with the error expressed in terms of β, the supremum of the real parts of z

Load-bearing premise

The argument rests on the Lindelöf hypothesis for quadratic Hecke L-functions of trivial infinite type, which is used to show that the double Dirichlet series is polynomially bounded in vertical strips; if it fails, the stated error term and the treatment of the secondary terms as main terms are not justified.

Editorial extensions

If this is right

Under GRH, the smoothed first moment of this family has a non-trivial secondary main term of size X^{1/3}, so the family follows the same asymptotic pattern as the rational and function-field families.
The error term O_ε(X^{1/4+ε}) is the square root of the leading main term up to the power of X; this is currently the smallest error proved for this first moment.
The general- s formula (Theorem 1.1) gives a uniform asymptotic in a vertical strip, which can be differentiated or integrated to yield information on other statistics such as shifted moments.
The explicit residue computations in Theorem 1.4 supply a complete analytic description of the double Dirichlet series, which can serve as a template for computing higher moments in this family.
If the leading main term is positive for sufficiently large X, the first moment being positive implies infinitely many non-vanishing central values among the primitive quadratic Hecke L-functions in the Gaussian field.

Reading between the lines

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

The X^{1/3} secondary term likely originates from the square-conductor contribution inside the double Dirichlet series; isolating that contribution, as the proof does through A_1(s,w), would allow one to predict the next-order term and test it numerically.
The same double Dirichlet series machinery may yield the second moment, or a shifted first moment, potentially leading to a non-vanishing proportion result for central values in this family, given the positivity of the leading main term.
By analogy with the rational case, the X^{1/4} error term may be improvable only under a stronger hypothesis; a numerical check at modest X could suggest the true size of the error and whether the secondary term is indeed X^{1/3}.
The method appears adaptable to other imaginary quadratic fields with class number one, or to twists of higher order, provided a functional equation for non-primitive Hecke characters analogous to the one used here exists.

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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

The paper evaluates the smoothed first moment of primitive quadratic Hecke L-functions over Q(i) and gets the expected asymptotic: a main term X Q1(log X), a secondary term X^{1/3} Q2(log X), and an O(X^{1/4+ε}) error under GRH. That is a genuine extension of Čech's rational-field result and Florea's function-field result to a number field where the double Dirichlet series machinery has to handle primary elements, the unit group, and the group of quadratic characters. The novelty is real, and the main technical work — the meromorphic continuation in Theorem 1.4 and especially the residue computation at w = 1/2 − s/3 — is presented in enough detail to be checked. The result is not circular and contains no fitted parameters.

The soft spots are concentrated in the transition from Theorem 1.1 to Corollary 1.2. The stress-test note is right that each displayed main term in (1.4) has a simple pole at s = 1/2 and that the paper simply says "taking the limit" without comment. But the worry that the limit may not exist is unfounded: the poles cancel pairwise. For the R_{K,1} terms, at s = 1/2 the coefficients are both X Φ̂(1) (since X_K(1/2)=1), and R_{K,1}(1−s) has the opposite residue to R_{K,1}(s); similarly the R_{K,2} terms both carry coefficient X^{1/3} Φ̂(1/3). So the RHS of (1.4) is analytic at s=1/2. The paper should have said this explicitly; as written, a referee has to do the computation. The uniformity of the error term as s→1/2 is also asserted rather than demonstrated, though it looks fixable with a fixed vertical strip.

This is a minor presentational gap, not a load-bearing flaw. The main theorem holds up, the citation pattern is appropriate, and the analytic computations are substantial. A serious referee could verify the residue checks without too much trouble, and the authors should be asked to expand the s→1/2 passage and address the error-term uniformity. I would accept this for peer review and would cite it in work on moments of L-functions over number fields.

Referee Report

2 major / 5 minor

Summary. This paper studies the smoothed first moment of the family of primitive quadratic Hecke L-functions attached to the Gaussian field K=Q(i), with characters χ_{(1+i)^5 d} for odd square-free primary d. Assuming the Lindelöf hypothesis for Hecke L-functions of trivial infinite type, the authors prove an asymptotic formula (Theorem 1.1) for the sum at a general point s in 1/3<Re(s)<1, s≠1/2, with two main terms and an error term. Taking s→1/2 gives Corollary 1.2, and under GRH Corollary 1.3 gives a central-value formula with main terms X Q_1(log X)+X^{1/3}Q_2(log X) and error O(X^{1/4+ε}). The proof is based on the double Dirichlet series A(s,w) defined in (1.7); its meromorphic continuation and residues (Theorem 1.4) are computed using the functional equation of [11, Prop. 2.5] and a large sieve estimate.

Significance. If the proof is completed, this is a solid contribution: it extends to a number field the X^{1/3} secondary main term and X^{1/4} error term previously known for the rational function field (Florea) and for quadratic Dirichlet L-functions (Čech). The residue computations in Theorem 1.4 are explicit and involve no free parameters; the main terms are given by convergent Euler products and gamma factors. The paper also gives a useful large-sieve estimate for quadratic Hecke L-functions (Lemma 2.7). The central technical issue is the passage from Theorem 1.1 to Corollary 1.2, as detailed below.

major comments (2)

[Section 1, after Theorem 1.1] The deduction of Corollary 1.2 from Theorem 1.1 is not justified. In (1.4) each of the four main terms has a simple pole at s=1/2: by Theorem 1.4(2),(5), R_{K,1}(s) and R_{K,2}(s) contain ζ_K(2s), while R_{K,1}(1-s) and R_{K,2}(1-s) contain ζ_K(2-2s). The sentence 'Upon taking the limit s→1/2 ...' simply asserts the limit exists and that the error term is uniformly bounded, but no Laurent expansion is given. A residue calculation using G_K(1/2)=1 in (1.5) does show that the singular parts cancel; however, this is not written down. The paper must provide this computation and a uniform bound for the error term in a neighbourhood of s=1/2 for Corollary 1.2 to follow. As stated, Corollary 1.2 and hence Corollary 1.3 are not established by the arguments in the paper.
[Section 4] The proof of Theorem 1.1 is only a sketch. The contour shift from (1.6) to the line Re(w)=c requires a detailed pole count, including the verification that no poles of the types s+(2j+1)w=3/2 or 2js+(2j+1)w=j+1 for j≥1 fall to the right of the new contour for 1/3<Re(s)<1, and a bound for A(s,w) on the shifted line. The paper refers to [6, Theorem 1.2] for these arguments. Given that Theorem 1.1 is the main analytic engine and the uniformity in s is essential for the limit in Corollary 1.2, the sketch should be expanded to a full proof or to a precise lemma stating the uniformity.

minor comments (5)

[Equation (1.5)] The factor is written as 'π2 25' in the text, which is ambiguous; it should presumably be (π^2/32)^{s-1/2}. Please correct the typesetting.
[Section 2.1] The multiplication rule for the group CG is hard to parse; consider presenting it as a small Cayley table.
[Section 4] The line 'c = max{...}' uses both β and ε; it would help to state explicitly that the implied constants in the error term of (1.4) may depend on s and ε, and to specify the uniformity in s needed near s=1/2.
[Abstract and Introduction] The phrase 'under the Riemann hypothesis and the Lindelöf hypothesis' is slightly misleading: Theorem 1.1 uses Lindelöf only; the Riemann hypothesis is invoked only for Corollary 1.3.
[After Corollary 1.2] Typo: 'the an asymptotical formula' should be 'an asymptotic formula'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotic is obtained by residue computation, and the self-citations used are prior proved results, not fitted inputs or target-equivalent assumptions.

full rationale

The derivation is self-contained as an analytic computation. The moment is Mellin-inverted to A(s,w); Theorem 1.4 establishes meromorphic continuation, pole locations, and explicit residues; Theorem 1.1 follows by contour shift and residue collection; Corollaries 1.2 and 1.3 are specializations at s=1/2. No constant is fitted to the target family, and no displayed equation is reused as both input and output. The cited results from the authors' own earlier work ([9], [10], [11]) supply Gauss-sum evaluations and a functional equation for nonprimitive Hecke L-functions; these are parameter-free, previously proved statements that do not include the first-moment formula among their assumptions, so they count as independent support rather than circular self-citation. The only concerns are rigor/writing issues, not circularity: the passage s→1/2 in Corollary 1.2 suppresses the Laurent-coefficient cancellation of the ζ_K(2s) factors (the paired R_{K,1}(s)/R_{K,1}(1-s) and R_{K,2}(s)/R_{K,2}(1-s) terms cancel), and (1.6) appears to use the Mellin variable inconsistently. These would be correctness-exposition issues; they do not make the claimed result equivalent to its inputs.

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

The paper's central formula rests on standard conjectures (Lindelöf/GRH) and on cited results from the authors' previous work (functional equations, large sieve). There are no fitted parameters or invented entities; the main terms are computed explicitly as residues.

assumptions (5)

domain assumption Lindelöf hypothesis for Hecke L-functions of trivial infinite type
Assumed in Theorem 1.1 to ensure polynomial boundedness of A(s,w) and to justify the contour shift and error term (Theorem 1.4).
domain assumption Generalized Riemann Hypothesis (for Corollary 1.3)
Used to set β=1/2 and simplify the error term; it is a standard unproved conjecture.
standard math Functional equation for L(s,fχ_n) (Prop. 2.5) from [11]
Proven in previous work by the authors; essential for the double Dirichlet series functional equation.
standard math Large sieve estimate for quadratic Hecke L-functions (Lemma 2.7, from [3, Cor 1.4])
Proven bound used to establish convergence and average estimates.
standard math Bochner tube theorem for meromorphic continuation (Theorem 2.12)
Gives holomorphic continuation to convex hull of tube domains.

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Pith. "Pith review of First Moment of Quadratic Hecke $L$-Functions with Lower Order Term." pith.science (2026). https://pith.science/paper/7TZHKKXU

@misc{pith2026251222509,
  author       = {Pith},
  title        = {Pith review of: First Moment of Quadratic Hecke $L$-Functions with Lower Order Term},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TZHKKXU}},
  note         = {Machine review of arXiv:2512.22509}
}

abstract

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.

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Reference graph

Works this paper leans on

19 extracted references · 2 linked inside Pith

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Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

B. C. Berndt, R. J. Evans, and K. S. Williams,Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1998

1998

[2] [2]

Blomer,Subconvexity for a double Dirichlet series, Compos

V. Blomer,Subconvexity for a double Dirichlet series, Compos. Math.147(2011), no. 2, 355–374

2011

[3] [3]

Blomer, L

V. Blomer, L. Goldmakher, and B. Louvel,L-functions with n-th order twists, Int. Math. Res. Not. IMRN2014(2014), no. 7, 1925– 1955

2014

[4] [4]

Bochner,A theorem on analytic continuation of functions in several variables, Ann

S. Bochner,A theorem on analytic continuation of functions in several variables, Ann. of Math. (2)39(1938), no. 1, 14–19

1938

[5] [5]

ˇCech,The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series, Trans

M. ˇCech,The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series, Trans. Amer. Math. Soc.377(2024), 3487– 3528

2024

[6] [6]

arXiv:2509.03197

,First moment of quadratic DirichletL-functions with secondary terms(Preprint). arXiv:2509.03197

arXiv

[7] [7]

Davenport,Multiplicative Number Theory, Third edition, Graduate Texts in Mathematics, vol

H. Davenport,Multiplicative Number Theory, Third edition, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, Berlin, etc., 2000

2000

[8] [8]

Florea,Improving the error term in the mean value ofL( 1 2 , χ)in the hyperelliptic ensemble, Int

A. Florea,Improving the error term in the mean value ofL( 1 2 , χ)in the hyperelliptic ensemble, Int. Math. Res. Not. IMRN20(2017), 6119–6148

2017