{"id":"cf8b0a93-39c7-4c8b-8991-47999d7129d3","arxiv_id":"2506.14656","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper claims an asymptotic formula for a twisted second moment of cubic L-functions over function fields. The main theorem contains a sign error that makes its right-hand side negative in the untwisted case, where the left-hand side is provably positive.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's main term is negative for h1=h2=1, contradicting the nonnegative left side; the cause is the contour shift in §3.5, which must subtract the residue at u=q^{-2}, not add it.","rationale":"I read the paper carefully and focused on the central theorem and the final contour shift in Section 3.5. The reader's weakest assumption correctly identifies the sign in the residue theorem as the load-bearing issue. The residue computation at u=q^{-2} in Section 3.2 is internally consistent and yields the negative factor (q^{-4}-q^{-2}), but the application in Section 3.5 uses the residue with the wrong sign. When shifting the contour outward from |u|=r to |u|=q^{-(1+ε)}, the small-circle integral equals the large-circle integral minus 2πi times the residue, so the pole contributes with a minus sign. The paper adds the residue instead, giving a negative main term. This is fatal because the left-hand side with h1=h2=1 is a sum of squared absolute values and is positive, while every named factor on the right is positive except (1-q^2), which is negative. The corrected sign would make the main term (q^2-1)SCP q^g, which is positive and has the same order q^{g+2} as the untwisted second moment computed by Goel and Ray, so the construction is plausible apart from this sign error. No other issue is needed to reject the paper as stated; the sign fix would likely make the theorem true, but the current formulation is false. Therefore the reader's REJECT verdict remains unchanged.","tokens_in":10389,"tokens_out":8341,"duration_ms":75454,"concrete_test":"Set h1=h2=1 in Theorem 1.1. The left side is a sum of nonnegative numbers |L(1/2,χ)|^2 over primitive cubic characters of genus g, and it is strictly positive for all sufficiently large g because the family is nonempty. On the right side, q≥5, S>0, C(1,1)>0, and P(q^{-2})>0, while 1-q^2<0, so the asymptotic is negative. Re-derive the contour shift in Section 3.5 keeping the minus sign from the residue theorem; the corrected main term (q^2-1)SCP q^g removes the contradiction. This single sign check settles the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing flaw is in the final contour shift of Section 3.5. The moment is extracted as (1/2πi)∮_{|u|=r} A2(u,q^{-1/2}) u^{-g/2-2} du. When the contour is enlarged to |u|=q^{-(1+ε)} past the pole at u=q^{-2}, the residue theorem gives I_inner = I_outer - 2πi Res_{u=q^{-2}} A2(u,q^{-1/2}) u^{-g/2-2}. The integrand residue equals (q^{-4}-q^{-2}) S C(h1,h2) P(q^{-2}) q^{g+4}, so the main term should be minus this, i.e., (q^2-1) S C P q^g. The paper instead writes +(1-q^2) S C P q^g, i.e., plus the residue. The sign matters because for h1=h2=1 the left side is Σ_{χ} |L(1/2,χ)|^2 ≥ 0 and positive for large g, while S, C(1,1), and P(q^{-2}) are all positive (Euler products with positive factors) and (1-q^2)<0, so the stated right side is negative. The error is not in the residue computation itself — (q^{-4}-q^{-2}) = (1-q^2)/q^4 is correctly derived with the negative sign — but in the contour-shift application. Without this sign correction the theorem as stated is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the twisted second moment of primitive cubic L-functions over F_q(T) in the non-Kummer setting, for fixed polynomials h1,h2 and characters of fixed genus g. The authors use Perron's formula to express the moment as a contour integral of a double Dirichlet series A2(u,v), compute the residue at u=q^{-2}, and use analytic continuation through three convergence regions to obtain the claimed asymptotic in Theorem 1.1: the main term (1-q^2) S C(h1,h2) P(q^{-2}) q^g plus an error O(q^{(1/2+eps)g}).","tokens_in":10733,"tokens_out":9020,"duration_ms":88073,"significance":"If correct, the asymptotic would be a new result and a clean illustration of the double Dirichlet series method for higher-order characters over function fields. The paper has genuine strengths: the residue computation is explicit and has no fitted parameters, the use of multivariable tube-domain continuation is appropriate in principle, and the claimed error term is sharply falsifiable. However, the central theorem as stated is internally contradicted by the positivity of the left-hand side when h1=h2=1, and the root cause is a sign error in the contour shift in Section 3.5. A second, closely related correctness issue is the false identity 'chi^2=chi' invoked in Section 3.1. Since the main claim is false as written, the current significance of the paper is not realized.","major_comments":[{"comment":"The sign of the main term is wrong. The residue computed in (3.22) is (q^{-4}-q^{-2}) S C(h1,h2) P(q^{-2}), so the residue of the integrand in (3.2) at u=q^{-2} is (1-q^2) S C(h1,h2) P(q^{-2}) q^g. When the contour is shifted from |u|=r outward past this pole, the residue theorem gives I_inner = I_outer - 2*pi*i*Res, so the pole contributes with the opposite sign to the one the paper uses. The paper instead takes the pole contribution to be exactly the residue, yielding the negative main term in (1.3). For h1=h2=1, C(1,1)=1 and S and P(q^{-2}) are positive Euler products, while (1-q^2)<0, so the stated right-hand side is negative and of size q^g. The left-hand side is sum_chi |L(1/2,chi)|^2 >= 0, and the error O(q^{(1/2+eps)g}) cannot dominate a negative q^g term for any fixed eps<1/2. This contradicts Theorem 1.1 as stated.","section":"Section 3.5 / Theorem 1.1, Eq. (1.3)"},{"comment":"The text states 'Since cubic characters satisfy chi^2=chi' and uses this to replace chi_F(h2) and chi_F(N2) by their squares. For a cubic character the correct identity is chi^2 = bar(chi). The algebra leading to the argument h1 h2^2 N1 N2^2 is only valid if (3.3) is read with the conjugate bar(chi_F)(h2) and the expansion of |L|^2 is written as sum chi(N1) bar(chi)(N2) = sum chi(N1) chi(N2)^2. As written, the displayed identity is false and the derivation is invalid; this must be corrected and the notation in (3.3) fixed throughout.","section":"Section 3.1, between Eqs. (3.3) and (3.7)"}],"minor_comments":[{"comment":"The displayed definition of S4 is malformed: it reads '|uv^4|^{-3}', which is not a region. It should presumably be |uv^4| < q^{-3}, matching the regions S2,2 and S3.","section":"Eq. (3.29)"},{"comment":"There are several typographical errors: 'associeted' in Theorem 1.1, 'folllowing' in Section 2.1, 'Eular' in Section 3.2, and 'exadtly' in Section 3.4.","section":"Throughout"},{"comment":"Several references are incomplete: [3] and [4] lack a year, [8] lists no volume or pages, and the title of [12] contains the typo 'L-funcitons'.","section":"References"}],"recommendation":"reject","confidential_remarks":"The sign error in Section 3.5 is local, and if the rest of the computation is sound a corrected version might have the main term (q^2-1) S C P q^g. Nevertheless, the paper's central theorem as submitted is false, being contradicted by the nonnegativity of the left-hand side, and the false identity chi^2=chi in Section 3.1 further undermines the derivation. I recommend rejection rather than major revision, while noting that a carefully corrected resubmission could be considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I agree with the reader's take: the paper has a load-bearing sign error. Theorem 1.1 as stated is contradicted by the nonnegativity of the left-hand side for h1=h2=1, since S, C(1,1), and P(q^{-2}) are all positive while (1-q^2) is negative. The error sits in Section 3.5: when the Perron contour is shifted past u=q^{-2}, the residue must be subtracted, not added. The residue computation itself is correct—(q^{-4}-q^{-2}) S C P is negative—so the main term should be (q^2-1) S C P q^g. This is a fixable mistake, but the theorem as written is false.\n\nWhat is actually new: the twisted second moment for primitive cubic L-functions over function fields has not been treated before. Reference [10] handles only the untwisted case, and [6] gives a mollified bound. The double Dirichlet series machinery from Gao–Zhao and Hong–Zheng is adapted cleanly, and the convergence regions, functional equation, and residue extraction are largely standard and correct. The explicit Euler product computation of C(h1,h2) and S is a useful and nontrivial piece of work. The derivation is self-contained, with no fitted parameters or post hoc exclusions.\n\nSoft spots beyond the sign: there are typos (\"folllowing,\" \"associeted,\" \"exadtly\"), and the definition of S4 in (3.29) is missing an inequality sign. Also, the claim that the result is \"consistent with the bound in (1.1)\" is vague, since (1.1) concerns a mollified second moment. None of these are major.\n\nIf the sign is flipped, this becomes a solid contribution to the function-field moments subfield. I would send it to a serious referee rather than desk reject: the error is localized and easily verified, and the core method is competently executed. The authors should be asked to fix the sign and run the sanity check h1=h2=1 before acceptance.","headline":"Solid double Dirichlet series work undone by a sign error in the contour shift; the main theorem as stated is false, but flipping the sign likely repairs it.","tokens_in":11269,"tokens_out":6041,"would_cite":false,"duration_ms":58117,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M41","11N37","11L05","11L40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for $q\\equiv2\\pmod3$ and fixed $h_1,h_2\\in\\mathbb{F}_q[T]$, the twisted second moment of primitive cubic $L$-functions over $\\mathbb{F}_q(T)$ equals $(1-q^2)\\,S\\,C(h_1,h_2)\\,P(q^{-2})\\,q^g +…","keywords":["central values","cubic L-functions","twisted moment","function field","double Dirichlet series","Perron formula","primitive cubic characters","non-Kummer setting"],"falsifier":"Set $h_1=h_2=1$ and compute the sign of the claimed main-term constant $(1-q^2)\\,S\\,C(1,1)\\,P(q^{-2})$ for an admissible prime power $q\\equiv2\\pmod3$. Since the left side of (1.3) is a sum of nonnegative terms, a negative constant would refute the asymptotic as written; the magnitude can be checked against the known untwisted second moment, which fixes both the sign and the size of the main term.","tokens_in":10135,"feed_emoji":"🔢","tokens_out":10692,"duration_ms":93684,"temperature":0.7,"pith_summary":"The paper proves an asymptotic formula for the twisted second moment of primitive cubic $L$-functions over the rational function field $\\mathbb{F}_q(T)$ in the non-Kummer setting $q\\equiv2\\pmod3$. For fixed twisting polynomials $h_1,h_2$, the average of $\\chi(h_1)\\bar\\chi(h_2)|L_q(1/2,\\chi)|^2$ over primitive cubic characters of genus $g$ is shown to be an explicit constant times $q^g$, with an error of size $q^{(1/2+\\varepsilon)g}$. The constant is built from a twist-independent Euler product $S$, a polynomial-dependent factor $C(h_1,h_2)$, and an Euler factor $P(q^{-2})$, all arising from the residue of a double Dirichlet series at a single pole. This matters because twisted moments are the basic input for non-vanishing statements and zero statistics of this family, and no asymptotic for the twisted cubic second moment had been established before.","feed_headline":"Twisted cubic L-function moments now have an explicit main term","feed_subtitle":"Over F_q(T), the central twisted second moment matches a q^g main term with error O(q^{(1/2+\\varepsilon)g}).","key_machinery":"The engine is the double Dirichlet series $A_2(u,v) = \\sum_F \\chi_F(h_1)\\overline{\\chi_F}(h_2)|L_q(v,\\chi_F)|^2 u^{\\deg F}$, where the sum runs over square-free $F$ over $\\mathbb{F}_{q^2}[T]$ with no prime divisor in $\\mathbb{F}_q[T]$. Perron's formula converts the genus-$g$ character sum into a contour integral of $A_2(u,q^{-1/2})$. The convergence regions $S_1,S_{2,1},S_{2,2},S_3$ are obtained from the Euler product, the bound from Lemma 2.3, and the functional equation; the tube-domain continuation theorem then extends $(u-q^{-2})A_2(u,v)$ to the convex hull of these regions. The residue at $u=q^{-2}$, in the form $(q^{-4}-q^{-2}) S C(h_1,h_2) P(q^{-2})$, becomes the main term after shifting contours.","core_discovery":"The central discovery is Theorem 1.1: for $q\\equiv2\\pmod3$, fixed $h_1,h_2\\in\\mathbb{F}_q[T]$, and any $\\varepsilon>0$, the sum over primitive cubic characters of genus $g$ of $\\chi(h_1)\\bar\\chi(h_2)|L_q(1/2,\\chi)|^2$ equals $(1-q^2)\\,S\\,C(h_1,h_2)\\,P(q^{-2})\\,q^g$ plus $O(q^{(1/2+\\varepsilon)g})$. The pole of the double Dirichlet series $A_2(u,q^{-1/2})$ at $u=q^{-2}$, with residue $(q^{-4}-q^{-2})S C(h_1,h_2)P(q^{-2})$, produces the main term; the factor $C(h_1,h_2)$ depends only on primes dividing $h_1h_2$ for which the relevant cube-congruence condition fails, while $S$ is independent of the twist.","pith_inferences":["A direct check not performed in the paper: putting $h_1=h_2=1$ in (1.3) gives a signed main term that must be compared with the nonnegative left-hand side and with the known untwisted second moment; this would determine whether the sign of the residue contribution is correct as stated.","If the sign is adjusted to the standard residue-theoretic value, the asymptotic shape and the factorized coefficient are likely preserved, so the paper's structural conclusions would survive a sign correction.","The same contour-geometry argument could be run in the Kummer setting $q\\equiv1\\pmod3$, where the cube-residue symbol behaves differently and the analogue of $C(h_1,h_2)$ would encode the extra characters.","One could test the claimed error term by computing the finite character sum for small $q$ and $g$ and comparing deviations from the main term with $q^{(1/2+\\varepsilon)g}$."],"forward_implications":["For each fixed choice of $h_1,h_2$, the twisted second moment grows like a constant multiple of $q^g$, so the family-level central values are not uniformly suppressed by the twist.","The twist dependence is fully factorized: $C(h_1,h_2)$ factors over primes, so the ratio of two twisted moments depends only on the primes dividing $h_1h_2$ and their valuations modulo 3.","The error term $O(q^{(1/2+\\varepsilon)g})$ is a power saving relative to the main term, making the asymptotic usable in mollified-moment arguments of the kind used to prove non-vanishing.","Specializing to $h_1=h_2=1$ must reproduce the untwisted second moment, providing a built-in consistency check against the earlier computation [10].","The double-Dirichlet-series route avoids a detailed distribution analysis of cubic Gauss sums, so the same residue computation should transfer to other ratios of twisted characters."],"supporting_citations":[{"why":"Supplies the parametrization of primitive cubic characters by square-free polynomials, the functional equation, and the polynomial-growth bounds used throughout the convergence analysis.","marker":"[8]"},{"why":"Provides the tube-domain continuation theorem used to extend $(u-q^{-2})A_2(u,v)$ to the convex hull of the convergence regions.","marker":"[3]"},{"why":"Provides the companion bound on the modulus of holomorphic continuations used to control the extended double Dirichlet series.","marker":"[4]"},{"why":"Computes the untwisted second moment, giving the baseline that the $h_1=h_2=1$ specialization of the new formula must match.","marker":"[10]"},{"why":"Provides the restriction construction through which cubic characters over $\\mathbb{F}_q[T]$ are obtained from characters over $\\mathbb{F}_{q^2}[T]$ in the non-Kummer setting.","marker":"[2]"},{"why":"One of the origins of the double-Dirichlet-series technique for moments of higher-order characters that the paper adapts.","marker":"[9]"},{"why":"A companion application of the same double-Dirichlet-series machinery to quartic characters, cited as methodological precedent.","marker":"[13]"}],"fun_headline_variants":["Cubic L-function twisted moments: explicit main term","Twisted cubic L-function second moment: main term explicit","Explicit main term for cubic L-function twists","Main term for twisted primitive cubic L-functions","Primitive cubic L-function moments: main term explicit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, after the integration contour is moved inward, the simple pole at $u=q^{-2}$ contributes to the main term with the same sign as its residue; standard residue calculus would subtract the residue when reading the integral, and the sign determines whether the main term in (1.3) is positive or negative.","fun_headline_variants_meta":{"raw":{"variants":["Cubic L-function twisted moments: explicit main term","Twisted cubic L-function second moment: main term explicit","Explicit main term for cubic L-function twists","Main term for twisted primitive cubic L-functions","Primitive cubic L-function moments: main term explicit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001025,"raw_usage":{"total_tokens":4284,"prompt_tokens":869,"completion_tokens":3415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":3340}},"tokens_in":485,"tokens_out":3415,"duration_ms":25093,"temperature":1.0,"reasoning_tokens":3340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:00:29.010464+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $h_1=h_2=1$ and compute the sign of the claimed main-term constant $(1-q^2)\\,S\\,C(1,1)\\,P(q^{-2})$ for an admissible prime power $q\\equiv2\\pmod3$. Since the left side of (1.3) is a sum of nonnegative terms, a negative constant would refute the asymptotic as written; the magnitude can be checked against the known untwisted second moment, which fixes both the sign and the size of the main term.","supporting_citations":[{"cited_title":"David, A","cited_arxiv_id":null,"evidence_quote":"Supplies the parametrization of primitive cubic characters by square-free polynomials, the functional equation, and the polynomial-growth bounds used throughout the convergence analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the tube-domain continuation theorem used to extend $(u-q^{-2})A_2(u,v)$ to the convex hull of the convergence regions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the companion bound on the modulus of holomorphic continuations used to control the extended double Dirichlet series."},{"cited_title":"The second moment of cubic Dirichlet L-functions over function fields","cited_arxiv_id":"2505.12015","evidence_quote":"Computes the untwisted second moment, giving the baseline that the $h_1=h_2=1$ specialization of the new formula must match."},{"cited_title":"Bary-Soroker and P","cited_arxiv_id":null,"evidence_quote":"Provides the restriction construction through which cubic characters over $\\mathbb{F}_q[T]$ are obtained from characters over $\\mathbb{F}_{q^2}[T]$ in the non-Kummer setting."},{"cited_title":"Gao and L","cited_arxiv_id":null,"evidence_quote":"One of the origins of the double-Dirichlet-series technique for moments of higher-order characters that the paper adapts."},{"cited_title":"The first moment of central value of primitive quartic $L$-functions with fixed genus","cited_arxiv_id":"2504.14291","evidence_quote":"A companion application of the same double-Dirichlet-series machinery to quartic characters, cited as methodological precedent."}],"review_version":1}