{"id":"b8112aa3-e7fb-4cd3-bad7-a57537224ae0","arxiv_id":"1908.04078","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For monic square-free polynomials of even degree over F_q[x], the first moment of L(1/2, chi_D) has secondary main terms of sizes q^((2g+2)/3), q^(g/6+floor(g/2)) and q^(g/6+floor((g-1)/2)), with error O(q^(g(1+eps)/2)).","lead":"This paper computes a sharper asymptotic formula for the average of quadratic Dirichlet L-functions at the central point over real quadratic function fields. It finds three extra main terms beyond the known leading term and reduces the error to a smaller power of q.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The final error term depends on the imported non-square Gauss-sum bound (6.6); if it does not transfer to the even-degree n and V ranges, Theorem 1.4's O(q^{g/2(1+epsilon)}) is unsupported.","rationale":"The reader identified the imported estimate (6.6) as the weakest assumption, and my reading agrees. The main term constants and the secondary main terms are derived through explicit Euler products and residue calculations that are coherently structured, and the appendix gives a detailed induction for the delicate cancellation in Lemma 5.6. The genuinely under-supported step is the error term in Proposition 6.1, where the decisive Gauss-sum estimate is quoted rather than proved or checked for transfer. The paper is a credible extension of Florea's method, but the final error term is not independently established within the manuscript. There are also typographical and cross-reference problems, such as the missing 'Lemma 6.3', that make verification harder, but the substantive risk remains the (6.6) transfer. Since the reader's CONDITIONAL verdict already reflects exactly this uncertainty, no change to the verdict is needed; the condition should be that the transfer is confirmed.","tokens_in":41941,"tokens_out":23081,"duration_ms":232433,"concrete_test":"Extract from Florea [9] the exact statement and proof of the bound (6.6): record its hypotheses on d(V), n, uniformity in u, and any parity or degree restrictions. Then substitute the ranges actually used in Section 6.1 and 6.2, e.g. for the even f terms n <= floor(g/2) and d(V) <= 2n - 2g - 4 + 2m with g - n + 2 <= m <= g, and for the odd f terms the analogous ranges with d(V) = 2n - 2g - 3 + 2m or 2n - 2g - 1 + 2m. If every pair (n,V) appearing in the sums lies within the stated hypotheses, the transfer is legitimate; if Florea's proof requires, say, d(V) <= n or n in a different range, identify the first failing term and check whether its contribution is still bounded by q^{g/2(1+epsilon)}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.1, the entire control of S(V != square), rests on the estimate |delta_{V;n}(u)| << q^{n/2(1+epsilon)} imported from Florea's odd-degree paper. The manuscript states this bound in (6.6) with only a citation to [9, Section 7], and then applies it in Section 6 to all ranges of n and V arising after the Perron/C-sum manipulations. No verification is given that those ranges, which include even-degree f with n up to floor(g/2) and non-square V of degree roughly up to 2n, satisfy the hypotheses of Florea's proof. The definition of delta_{V;n} itself has no D-dependence, but that does not automatically make the bound valid for every n and V; the proof could use properties such as d(V) <= n or a particular summation order that need not hold here. This is load-bearing because the claimed error term O(q^{g/2(1+epsilon)}) is exactly what separates the new secondary main terms from the older Jung error term. Additionally, the paper's own cross-reference to 'Lemma 6.3' for bounding the tail integrals C_{k,ell} in Lemmas 5.4 and 5.5 points to a lemma that does not appear in the manuscript, so the error control is not fully self-contained even for the V-square terms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":42195,"tokens_out":8672,"duration_ms":95430,"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":[{"comment":"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.","section":"Section 6, Proposition 6.1 and Eq. (6.6)"},{"comment":"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.","section":"Sections 5.1 and 5.2, after Eqs. (5.12) and (5.18)"}],"minor_comments":[{"comment":"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":"Section 3, p. 7"},{"comment":"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.","section":"Section 6.1, displayed formulas for tilde{S}^e and hat{S}^e"},{"comment":"There are several grammatical slips (e.g., 'extra technical difficulties that doesn't appear'); these should be corrected in a final revision.","section":"Abstract and Introduction"},{"comment":"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.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript is a technical adaptation of Florea's method, and the announcement of secondary main terms is plausible. I am not recommending rejection because the gaps are omissions of verification rather than demonstrated contradictions. However, the missing Lemma 6.3 and the unverified transfer of the Gauss-sum bound (6.6) are exactly the points on which the claimed error term depends, so they must be supplied before the paper can be accepted. If the authors provide those arguments and clarify the notation in Section 6.1, I would be willing to support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nShort 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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":42708,"tokens_out":3880,"would_cite":false,"duration_ms":34957,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M38","11M06","11G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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)}).","keywords":["first moment","quadratic Dirichlet L-functions","function fields","real quadratic","hyperelliptic ensemble","trivial zero","Poisson summation","asymptotic expansion"],"falsifier":"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.","tokens_in":41725,"feed_emoji":"🧮","tokens_out":9955,"duration_ms":93309,"temperature":0.7,"pith_summary":"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.","feed_headline":"First moment of quadratic L-functions gains three new main terms","feed_subtitle":"Average of L(1/2,χ) over real quadratic function fields is now known to order q^{g/2(1+ε)}.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Poisson summation formula, the Dirichlet series B(z,w) and its Euler product factorization, and the character-sum estimate used for the error term.","marker":"[9]"},{"why":"Gives the approximate functional equation for even-degree D and the earlier asymptotic that this theorem sharpens.","marker":"[13]"},{"why":"Provides the corrected approximate functional equation used as Lemma 2.4.","marker":"[20]"},{"why":"Provides the power-of-q identity used to cancel residue terms in the proof of Lemma 5.6.","marker":"[14]"},{"why":"Supplies the background facts on zeta functions, characters, and the functional equation of L(u, χ_D).","marker":"[19]"},{"why":"Gives the fact that L(u, χ_D) has a trivial zero exactly when deg D is even, which drives the even-degree structure.","marker":"[21]"}],"fun_headline_variants":["Quadratic L-function first moment gains three main terms","Extra main terms in real quadratic function field L-function average","First moment of quadratic L-functions: three new terms","Even-degree quadratic fields get refined L-function first moment","Three terms beyond error refine quadratic L-function moment"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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($q^{{g/2(1+ε)}}$).","fun_headline_variants_meta":{"raw":{"variants":["Quadratic L-function first moment gains three main terms","Extra main terms in real quadratic function field L-function average","First moment of quadratic L-functions: three new terms","Even-degree quadratic fields get refined L-function first moment","Three terms beyond error refine quadratic L-function moment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000701,"raw_usage":{"total_tokens":3157,"prompt_tokens":933,"completion_tokens":2224,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":2147}},"tokens_in":549,"tokens_out":2224,"duration_ms":22375,"temperature":1.0,"reasoning_tokens":2147,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:52:21.847656+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson summation formula, the Dirichlet series B(z,w) and its Euler product factorization, and the character-sum estimate used for the error term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the approximate functional equation for even-degree D and the earlier asymptotic that this theorem sharpens."},{"cited_title":"Rubinstein and K","cited_arxiv_id":null,"evidence_quote":"Provides the corrected approximate functional equation used as Lemma 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the power-of-q identity used to cancel residue terms in the proof of Lemma 5.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the background facts on zeta functions, characters, and the functional equation of L(u, χ_D)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the fact that L(u, χ_D) has a trivial zero exactly when deg D is even, which drives the even-degree structure."}],"review_version":1}