{"id":"83d6f321-a073-4e65-b5ae-cad89e1cdb65","arxiv_id":"2505.12015","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For q ≡ 2 mod 3, it claims the second moment of primitive cubic L-functions of genus g is (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).","lead":"This paper claims an asymptotic formula for the second moment of cubic Dirichlet L-functions over the rational function field F_q(T) when q is 2 modulo 3. The proof relies on a lemma stating that the relevant cubic characters are trivial on polynomials over F_q, and that lemma appears to be false.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No load-bearing mathematical error: Lemma 2.5 is valid for the F_{q^2}[T]-character used in (3.13); the reader's counterexample evaluates a different §2.5 symbol.","rationale":"The reader's verdict of REJECT rests on the claim that Lemma 2.5 is false, supported by the example q=5, P=T^2+T+2, χ_P(T)≠1. This example is correct for the F_q[T] cubic residue symbol defined in Section 2.5, but Lemma 2.5 is about the character associated to F over F_{q^2}[T]. Under that intended definition, a split prime P contributes χ_Qχ_{\\tilde Q} to χ_F, and the proof correctly shows this product is 1 on F_q[T] because χ_{\\tilde Q}(f)=χ_Q(f)^q=χ_Q(f)^{-1}. The Möbius inversion in Section 3.3 uses χ_D(f) for D∈F_q[T] viewed as an element of F_{q^2}[T], exactly the object for which the triviality statement is true. The proof's notation is admittedly confusing, since χ_P was already used for a different symbol in Section 2.5, and the lemma omits the necessary assumption (f,F)=1. These are expository flaws that should be corrected, not a mathematical gap that invalidates the central asymptotic. The honest stress-test finding is that the single most load-bearing concern raised by the reader does not hold; the paper should not be rejected on that basis, though the ambiguity in Lemma 2.5 warrants a conditional acceptance rather than outright acceptance as written.","tokens_in":24206,"tokens_out":44922,"duration_ms":401111,"concrete_test":"Explicit computation for q=5, P=T^2+T+2: factor P=Q\\tilde Q over F_{25}[T], with Q=T-β and \\tilde Q=T-β^5. Compute χ_Q(T) by evaluating T^8 mod Q = β^8; this is a nontrivial cube root ω in F_{25}^×. Compute χ_{\\tilde Q}(T) by β^{40}=ω^5=ω^{-1}. Then χ_Qχ_{\\tilde Q}(T)=1, confirming Lemma 2.5. Separately compute the §2.5 F_5[T] symbol χ_P(T) via T^8 mod P; verify it equals the same ω (not 1). This demonstrates that the reader's counterexample targets the wrong character and that the lemma's intended statement is true.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader's attack on Lemma 2.5 does not land. Lemma 2.5 is explicitly about the character χ_F associated to F over F_{q^2}[T]. For a prime P∈F_q[T] of even degree, this character has factor χ_Q χ_{\\tilde Q}, where Q and \\tilde Q are the two primes over P in F_{q^2}[T]. The proof shows χ_{\\tilde Q}(f)=χ_Q(f)^q=χ_Q(f)^{-1} for f∈F_q[T] because q≡2 mod 3, so the product is 1. The reader's q=5 computation evaluates the §2.5 F_q[T] cubic residue symbol χ_P, defined by exponent (q^{deg P}-1)/3, which is not the F_{q^2}[T] character used in Lemma 2.5. For P=T^2+T+2 over F_5, χ_P(T) is a nontrivial cube root, but χ_Qχ_{\\tilde Q}(T)=1. The application at (3.13) requires exactly the F_{q^2}[T] character for the divisor D∈F_q[T], because D viewed in F_{q^2}[T] contains both conjugate primes for each split factor; thus χ_D(f)=1 is correct. The lemma's statement should add the coprimality hypothesis (f,F)=1, since χ_F(f)=0 when a prime divides both; this hypothesis is satisfied at the application site where (D,f)=1 is imposed. No correctness issue for Theorem A arises from this lemma.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24511,"tokens_out":45224,"duration_ms":426879,"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":[{"comment":"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.","section":"§3.1, Eqs. (1.3), (3.1)–(3.2), Prop. 2.3"},{"comment":"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.","section":"§3.5, proof of Theorem A"},{"comment":"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.","section":"§3.4, Eq. (3.26) to Mt"}],"minor_comments":[{"comment":"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.","section":"Lemma 2.5"},{"comment":"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.","section":"Eq. (3.3)"},{"comment":"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.","section":"Notation for χ_F"}],"recommendation":"major_revision","confidential_remarks":"The first reader's rejection rested mainly on Lemma 2.5. I have checked that the lemma is actually correct for its intended use: the F_{q^2}[T] character is the product over the two conjugate primes, and the product is 1 on F_q[T] because q≡2 mod 3. The real problems are the dual-term index inconsistency and the g-factors in the error term; these are load-bearing but appear fixable within the scope of a major revision. The paper is likely salvageable, and the central asymptotic is plausible, but the current text does not prove Theorem A as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper claims the first second-moment asymptotic for cubic Dirichlet L-functions in the non-Kummer function-field family, with an explicit Euler-product constant. That is a genuinely new computation, a natural extension of David–Florea–Lalín's first moment, and the machinery is mostly borrowed faithfully. The reader's rejection rests on Lemma 2.5, and I think that objection misses the mark. The lemma is about the F_{q^2}[T]-character χ_F, not the §2.5 F_q[T] cubic residue symbol. For a split prime P=Q\\tilde Q, the factor of χ_F is χ_Qχ_{\\tilde Q}; for f∈F_q[T] these local values are inverse cube roots, because χ_{\\tilde Q}(f)=χ_Q(f)^q=χ_Q(f)^{-1} when q≡2 mod 3. The counterexample with q=5 evaluates χ_P(T) from §2.5, which is one factor, not the product. So the lemma is true in the sense used at (3.13).\n\nThat said, the paper deserves blame for the confusion. The proof writes 'χ_P' for the product χ_Qχ_{\\tilde Q} after §2.5 defined χ_P as the single-symbol cubic residue character. That is an overloading waiting to trip a reader, and the lemma as stated omits the necessary coprimality assumption (f,F)=1 (otherwise χ_F(f)=0). Both are presentation fixes, not mathematical holes. I also noticed the Möbius step in the dual term displays D∈M_{q^2,g/2+1} where it should be D∈F_q[T]; the later generating functions use the correct range, so it looks like a typo.\n\nThe rest of the argument is a workmanlike adaptation of DFL22 and Florea: Perron twice, residues at z=1/q^2 and u=q^{-1} and q^{-3/2}, and the dual term via Gauss-sum averages from DFL22. I did not verify every line of the Euler products, but nothing leaps out as circular or fitted. The main term is O(g^2 q^g) with error O(q^g), so the asymptote is meaningful.\n\nWho should read this: anyone working on moments of cubic L-functions over function fields. It is a serious contribution and deserves refereeing. The referee's main job will be checking the dual-term Gauss-sum average stage and asking for the Lemma 2.5 rewrite.\n\nRecommendation: do not desk reject. Send it to a referee; expect a minor-revision outcome if the technical checks pass.","headline":"The main theorem is probably right and the reader's Lemma 2.5 objection misfires, but the paper needs a notation cleanup before publication.","tokens_in":25047,"tokens_out":24177,"would_cite":true,"duration_ms":206897,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R59","11M38","11R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $O(q^g)$.","keywords":["cubic Dirichlet L-functions","function fields","second moment","genus","Gauss sums","approximate functional equation","Perron formula","non-Kummer case"],"falsifier":"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.","tokens_in":23947,"feed_emoji":"🧮","tokens_out":10639,"duration_ms":92269,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cubic L-function second moment has one explicit main term","feed_subtitle":"When q ≡ 2 mod 3, the genus-g second moment is a constant times g^2 q^(g+2), up to an error of size q^g.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the first-moment asymptotic that this paper extends, along with the Gauss-sum average propositions (2.7, 2.8) and the functional-equation lemma used here.","marker":"[DFL22]"},{"why":"Provides the principal/dual decomposition template and the error-term strategy adapted to the even-character functional equation.","marker":"[Flo17b]"},{"why":"Establishes the Perron-integral and approximate-functional-equation method for first moments over function fields.","marker":"[AK12]"},{"why":"Gives the original Eisenstein-series approach to first moments, motivating the shape of the main term.","marker":"[HR92]"},{"why":"Supplies the cubic reciprocity and Gauss sum structure underlying the character parametrisation.","marker":"[Kub69]"},{"why":"Provides the metaplectic theta-function estimates that the Gauss-sum average results rely on.","marker":"[Hof92]"}],"fun_headline_variants":["Explicit second moment for cubic L-functions over F_q(T)","Cubic L-function second moment: explicit asymptotic main term","Function-field cubic L-functions: second moment main term is explicit","Second moment of cubic L-functions: one explicit main term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit second moment for cubic L-functions over F_q(T)","Cubic L-function second moment: explicit asymptotic main term","Function-field cubic L-functions: second moment main term is explicit","Second moment of cubic L-functions: one explicit main term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00104,"raw_usage":{"total_tokens":4346,"prompt_tokens":884,"completion_tokens":3462,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":3390}},"tokens_in":500,"tokens_out":3462,"duration_ms":26442,"temperature":1.0,"reasoning_tokens":3390,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:46:56.151729+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}