{"id":"def83d58-3251-4897-b110-7b59da81272d","arxiv_id":"2504.14291","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The first moment of primitive quartic L-functions over F_q(T) with fixed genus g equals q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2}) plus an error of size q^{(3/5+ε)g}.","lead":"A new theorem gives the first asymptotic for the average of quartic Dirichlet L-functions at the critical point over F_q(T), with genus fixed and q≡3 mod 4. The main term has size q^{2g/3} with an error of q^{(3/5+ε)g}, and a corollary gives a positive proportion of nonvanishing values.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Meromorphic continuation of A_4 to S4 relies on bound (3.14) whose proof (Theorem 3.1) contains asserted steps and an unjustified f-independence of coefficients; the S3 derivation in §5.1 also appears to misapply Lemma 3.1's degree exponent.","rationale":"The reader's weakest_assumption matches my own reading: the continuation region S4 is the hinge of the paper. I checked the surrounding details and found two additional concrete danger points: the f-dependence of a_1,a_2 in (3.8)-(3.12) and the degree-exponent mismatch in the q→q^2 substitution in §5.1. Neither is fatal by itself, but together they make the S3 derivation insecure. I also examined Lemma 4.1; its proof is indeed problematic, and χ_D is not defined for all D∈A_q (only for products of inert primes), which threatens the residue computation. However, the continuation is the more load-bearing premise because both the main term and the error term pass through S4. A direct small-q computation of (3.14) would settle the issue. Therefore I do not change the reader's CONDITIONAL verdict.","tokens_in":20492,"tokens_out":21266,"duration_ms":172688,"concrete_test":"For q=3 (or another q≡3 mod 4) and all squarefree f∈A_q with deg f ≤ 4, compute \\tilde Ψ_q(f,u) by direct summation of the Gauss sums in (3.2) at several u with σ=1/2, 3/4, 1, 5/4, 3/2, staying away from |u^4-q^{-5}| and |u^4-q^{-3}|, and compare with |f|^{1/2(3/2-σ)+ε}. If the bound (3.14) fails, S3 and hence S4 are unsupported. Alternatively, re-derive (3.10)-(3.12) from (3.9) while tracking W_{f,i} explicitly; if an |f|^c factor survives, the Phragmén-Lindelöf argument in Theorem 3.1 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 stands or falls on the meromorphic continuation of A_4(u,v) to S4 = {|u|<q^{-1}, |u^{1/2}v|<q^{-5/4}, |u^5 v^8|<q^{-13}} (§5.1). The third convergence region S3 is obtained from Lemma 3.1's bound (3.14), which is proved via Theorem 3.1. That proof is only sketched: equations (3.10)-(3.12) are asserted from (3.9) without derivation, the linear-independence case is handled by fiat, and the key assertion that a_1(s), a_2(s) are 'absolutely bounded ... independently of f' is not demonstrated. These coefficients derive from Hoffstein's functional equation and involve W_{f,i}=τ(χ_4^{2i-1}χ_f); their f-dependence must be controlled before the Phragmén-Lindelöf step is valid. Additionally, §5.1 replaces q by q^2 in Lemma 3.1 and then writes H(ND^2,uv^2) ≪ |u^{1/2}v q^{3/2+ε}|^{deg ND}; since Q=ND^2 has degree deg N + 2 deg D over F_q, the exponent on D should be doubled. This changes the D-summation condition in S3. If either issue is real, the convex hull S4 is not established, the contour shift to |u|=q^{-(9/5-ε)} is unjustified, and the error term O(q^{(3/5+ε)g}) collapses. The paper itself says the proof is 'analogous' and omits the details, so this is the load-bearing technical premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the first moment of central values of primitive quartic L-functions over F_q(T) in the non-Kummer setting q≡3 mod 4, with characters ordered by genus. The main result (Theorem 1.1) states that the sum over primitive quartic characters with χ^2 primitive and genus g equals an explicit main term of size q^{2g/3} times an Euler product, with an error O(q^{(3/5+ε)g}). The method uses a double Dirichlet series A_4(u,v), a Perron-type contour integral, and a meromorphic continuation to a convex hull S4 obtained by combining three convergence regions. The proof follows the strategy of Gao–Zhao and the cubic work of David–Florea–Lalin and Hong et al., and it centers on bounds for the generating function of quartic Gauss sums (Theorem 3.1 and Lemma 3.1). A nonvanishing corollary for quartic characters of genus g is also claimed.","tokens_in":20838,"tokens_out":17974,"duration_ms":147561,"significance":"If Theorem 1.1 is correct, it would be the first power-saving first moment result for primitive quartic L-functions over function fields with fixed genus, with an explicit main term and without the heavy residue computations of earlier approaches. The paper also advertises a clean framework (bounds on Gauss-sum generating functions plus convex hulls) that could apply to higher-order characters. The claimed nonvanishing corollary is a natural consequence. However, the significance is conditional: several load-bearing steps in the proof are either invalid as written (Lemma 4.1) or only sketched (Theorem 3.1), and the derivation of the third convergence region in §5.1 contains apparent degree-exponent and region-logic errors. The central idea is promising, but the manuscript needs substantial technical revision before the result is established.","major_comments":[{"comment":"The proof of Lemma 4.1 is invalid. From σ(Ω(α)) = Ω(α) one may only conclude that Ω(α) lies in F_q, i.e., Ω(α) ∈ {±1}; the nontrivial fourth roots of unity are interchanged by the Frobenius σ. Thus the lemma's conclusion χ_D(F)=1 is not established and is in fact false in general for inert or split primes. This invalidates the simplification used to pass from (4.5) to (4.9): the Euler product should contain characters χ_{P_1}(N) that are quadratic signs, and the factor Y_{P_1|N}(1-|P_1|^{-s}∏(...))^{-1} is not the correct local factor. Since (4.9) underlies the first convergence region and the residue computation at u=q^{-2}, this is a load-bearing error. The argument must either prove that the signs are trivial (which appears not to hold) or track this quadratic twist throughout the subsequent analysis.","section":"§4.2, Lemma 4.1 and equations (4.5)–(4.9)"},{"comment":"The proof of Theorem 3.1 is only a sketch, and the absent details are load-bearing because the bound (3.14) is the key input for the third convergence region in §5.1. Equations (3.10)–(3.12) are asserted without derivation; the dichotomy between the linearly dependent and linearly independent cases of the coefficient pairs is not justified; and the statement that a_1(s) and a_2(s) are absolutely bounded independently of f is not demonstrated, even though W_{f,i} appears in a_2(u) and depends on the Gauss sum of χ_4^{2i-1}χ_f. A complete proof is needed, or the paper should explicitly invoke Corollary 5.4 of [7] and show precisely how the required range 1/2≤σ≤3/2 is obtained.","section":"§3.2, Theorem 3.1 and equations (3.8)–(3.12)"},{"comment":"When Lemma 3.1 is applied over F_q2 to Q=ND^2, the degree of Q over F_q2 is deg N + 2 deg D, not deg ND = deg N + deg D. The bound should read H(ND^2,uv^2) ≪ |u^{1/2}v q^{3/2+ε}|^{deg N + 2 deg D}. With the doubled D-exponent, the D-summation condition changes from |q^{5/2+ε}u^{3/2}v^3|<1 to a different condition, and the claimed region S3 and the convex hull S4 in (5.7) are not established. This is a central technical step: the contour shift in §5.2 and the final error term O(q^{(3/5+ε)g}) both depend on S4.","section":"§5.1, equations (5.4)–(5.6) and the definition of S3"},{"comment":"The region S3 is defined by removing exactly the annulus q^{-3/2} ≤ |u^{1/2}v| ≤ q^{-1/2} (equivalently q^{-3} ≤ |uv^2| ≤ q^{-1}), yet the only upper bound derived in the text, from Lemma 3.1, is valid precisely in that annulus. Hence the proof does not establish holomorphy of A_4(u,v) on the stated S3; the logic appears to be backwards. The relation between the region where the estimate applies and the region where holomorphy is claimed must be corrected.","section":"§5.1, definition of S3 and its proof"},{"comment":"For v=q^{-1/2}, a point with |u|=q^{-9/5+ε} does not satisfy the third defining inequality of S4, |u^5v^8|<q^{-13}, because |u^5v^8|=q^{-13+5ε}. The contour should be at |u|=q^{-9/5-ε} in order to lie inside S4. As written, the integration contour is outside the region of meromorphic continuation, so the proof of Theorem 1.1 is incomplete at this step; if this is a sign typo, it must be corrected and the effect on the ε in the error term checked.","section":"§5.2, contour shift to |u|=q^{-9/5+ε}"}],"minor_comments":[{"comment":"In the theorem statement the condition is |u^4-q^{-5}|>δ, but equation (3.9) and Lemma 3.1 also require |u^4-q^{-3}|>δ; the relationship between these exclusions and the poles at u^4=q^{-5}, q^{-3} should be clarified.","section":"§3.2, Theorem 3.1 statement"},{"comment":"The Perron formula is written with a denominator (1-u)u^{N+1}; the text should make explicit the contour orientation and the required radius r for the integral to be valid.","section":"§4.1, equation (4.2)"},{"comment":"There are typographical errors: 'gievs' should be 'gives' and 'Menawhile' should be 'Meanwhile'. More importantly, the final lower bound ≫ q^{2g/3}q^{-ε(g/3+1)} follows only if the implicit constants in the preceding inequalities are independent of g, which should be stated.","section":"§5.3, proof of Corollary 1.1"},{"comment":"The set H_{q^2,g/3+1} is described as monic square-free polynomials of degree g/3+1, which implicitly assumes 3 divides g; the paper should state what happens for other g, even if the moment is then trivially zero or the degree parameter is adjusted.","section":"§2.1, Lemma 2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript cites the author's own work [13] only as motivation for the cubic analogue, and the quartic proof does not appear to depend on the results of [13], so there is no circularity concern from self-citation. The main problems are technical: the proof of Lemma 4.1 is wrong in a way that affects the main Euler product derivation, and the third convergence region in §5.1 has several apparent errors (degree substitution, region logic, contour radius). These may be fixable, but a resubmission should contain a complete proof of Theorem 3.1 and a corrected §5.1, and the authors should verify the convex hull computation with correct degree exponents. The paper's core strategy is reasonable and the claimed main term is explicit, but the current version is not yet publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first fixed-genus first moment for primitive quartic L-functions over F_q(T), and that is a genuine step forward in that subfield. The main term is explicit, there are no fitted parameters, and the debts to Hoffstein, Patterson, Gao–Zhao, and David–Florea–Lalin are acknowledged honestly. The self-citation [13] is used only as motivation. So the result is important if true, and I would not dismiss it.\n\nThat said, the proof as written is not sound. Lemma 4.1 is the clearest problem. The Frobenius argument shows only that σ(Ω(α)) = Ω(α), which forces α ∈ F_q; since q ≡ 3 mod 4, the quartic roots in F_q are {±1}, not just {1}. The conclusion χ_D(F) = 1 does not follow and is likely false. This matters because the lemma is what removes χ_D(N) from the D-sum in (4.5)–(4.6). If χ_D(N) can be −1, the Euler product and the residue formula for the main term change. That is load-bearing, not cosmetic.\n\nThe analytic continuation is also thinner than the paper suggests. Theorem 3.1's Phragmén–Lindelöf proof skips the derivation of (3.10)–(3.12), handles the linear-independence case by fiat, and asserts that a_1(s), a_2(s) are bounded independently of f. Those claims may be true, but they are not demonstrated. Then in §5.1 the bound from Lemma 3.1 is transplanted to H(ND^2, uv^2) with the exponent written as deg ND. Since Q = ND^2 has degree deg N + 2 deg D, the D-summation exponent should reflect that. If it is a typo, the S3 region and the convex hull S4 still need to be recomputed; if it is not a typo, the contour shift to |u| = q^{-9/5+ε} is unjustified.\n\nOn the positive side, the family size calculation, the residue computation at u = q^{-2}, and the nonvanishing corollary are sensible. The paper is clearly written within the right framework, and I see no circularity or invented structure. The problems are technical gaps that a careful revision could close, but they need to be closed before the theorem is asserted.\n\nBottom line: the central claim is plausible and the paper would be a useful addition to the function-field L-function literature. A serious referee should ask for a correct Lemma 4.1, a complete proof of the continuation bound, and a careful redo of the exponent in §5.1. Send it out for peer review, but expect major revision.","headline":"A plausible first-moment asymptotic for quartic L-functions over function fields, but the written proof has a false lemma and a sketched continuation argument; worth serious refereeing, not acceptance as is.","tokens_in":21418,"tokens_out":6781,"would_cite":false,"duration_ms":60981,"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":"For q≡3 mod 4, the first moment of primitive quartic L-functions over F_q(T) equals an explicit main term plus an error of size q^{(3/5+ε)g}, and many characters in the family are nonzero at the central point.","keywords":["central values","quartic L-functions","Gauss sums","function field","double Dirichlet series","first moment","nonvanishing","fixed genus"],"falsifier":"A direct check would be to compute, for a small prime power $q\\equiv3\\pmod4$ such as $q=3$ or $q=7$, the left-hand side of Theorem 1.1 for a few small genera $g$ by enumerating the primitive quartic characters and evaluating $L_q(\\tfrac12,\\chi)$ via the functional equation; any difference from the claimed main term that grows faster than $q^{(3/5+\\varepsilon)g}$ would disprove the theorem. A cheaper test targets Lemma 3.1 directly: fix a square-free $f\\in\\mathbb{F}_q[T]$, evaluate $\\Psi_q(f,u)$ at a point with $|u^4-q^{-5}|>\\delta$ and $\\sigma\\in[\\tfrac12,\\tfrac32]$, and see whether the asserted bound $|f|^{\\frac12(\\frac32-\\sigma)+\\varepsilon}$ holds.","tokens_in":20227,"feed_emoji":"🧮","tokens_out":19511,"duration_ms":155927,"temperature":0.7,"pith_summary":"This paper claims to pin down the average value of primitive quartic Dirichlet L-functions over the rational function field $\\mathbb{F}_q(T)$, taken over characters of a fixed genus $g$ in the case $q\\equiv 3\\pmod 4$. The claimed formula gives a main term $q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2})$ plus an error of size $q^{(3/5+\\varepsilon)g}$, which is genuinely smaller than the main term. Because the main term is nonzero, the formula forces many of these L-functions to be nonzero at the central point $s=\\tfrac12$, a nonvanishing statement for quartic L-functions in a function-field family where none was known. The proof routes the whole computation through a double Dirichlet series $A_4(u,v)$, using an upper bound for the generating function of quartic Gauss sums to continue the series to a convex hull and read the answer off a residue.","feed_headline":"Quartic L-functions: first moment has a power-saving error","feed_subtitle":"The average matches an explicit main term, forcing many nonvanishing central values.","key_machinery":"The load-bearing object is the double Dirichlet series $$A_4(u,v)=\\sum_{\\substack{F\\in H_{$q^{2}$}\\\\ P\\mid F\\Rightarrow P\\notin\\mathbb{F}_q[T]}} L_q(w,\\chi_F)\\,$u^{{\\deg F}}$,\\qquad v=$q^{{-w}}$,$$ where $H_{q^2}$ is the set of monic square-free polynomials in $\\mathbb{F}_{q^2}[T]$ whose prime divisors do not come from $\\mathbb{F}_q[T]$, together with the quartic Gauss-sum generating function $\\Psi_q(f,u)=\\sum_{F\\in\\mathbb{F}_q[T]}G_q(f,F)u^{\\deg F}$. Perron's formula rewrites the desired family sum as a contour integral of $A_4(u,\\tfrac12)\\,u^{-g/3-2}$, and the main term is the residue at the pole $u=q^{-2}$. The estimate that makes the whole argument possible is the bound (3.14), $\\widetilde\\Psi_q(f,u)\\ll |f|^{\\frac12(\\frac32-\\sigma)+\\varepsilon}$ in the strip $\\tfrac12\\le\\sigma\\le\\tfrac32$; this comes from a functional equation for $\\Psi_q$ and a Phragmén–Lindelöf argument. With that bound in hand, the paper continues $(u-q^{-2})A_4(u,v)$ holomorphically to the convex hull $$S_4=\\{(u,v): |u|<$q^{{-1}}$,\\ |$u^{{1/2}}$v|<$q^{{-5/4}}$,\\ |$u^{5}$$v^{8}$|<$q^{{-13}}$\\},$$ and shifting the contour to $|u|=q^{-9/5+\\varepsilon}$ produces the error term $q^{(3/5+\\varepsilon)g}$.","core_discovery":"For $q\\equiv 3\\pmod 4$ and every $\\varepsilon>0$, the paper claims that $$\\sum_{\\substack{\\chi\\text{ primitive quartic}\\\\ \\$chi^{2}$\\text{ primitive}\\\\ \\mathrm{genus}(\\chi)=g}} L_q(\\tfrac12,\\chi) = $q^{{2g/3}}$(1-$q^{2}$)P($q^{{-2}}$)Z($q^{{-2}}$,$q^{{-1/2}}$) + O($q^{{(3/5+\\varepsilon)g}}$),$$ where $P(u)$ and $Z(u,v)$ are explicit Euler products introduced in Section 4.3. It then derives the lower bound $\\#\\{\\chi:L_q(\\tfrac12,\\chi)\\neq 0\\}\\gg q^{2g/3-\\varepsilon(g/3+1)}$. The author presents this as the quartic analogue of the cubic first moment over function fields, obtained without approximating the functional equation or computing individual Gauss-sum residues; all of the arithmetic input is concentrated in an estimate for the quartic Gauss-sum generating function.","pith_inferences":["The same convex-hull scheme should carry over to the case $q\\equiv1\\pmod4$ with minor modifications, which the paper notes but does not carry out; spelling out the Gauss-sum bound there would complete the quartic picture.","A second moment of the same family would convert the nonvanishing lower bound into an asymptotic count of nonvanishing characters; the first-moment main term here is the natural starting point for such a calculation.","Since the error exponent is controlled by the Phragmén–Lindelöf width in Lemma 3.1, a sharper analytic treatment of $\\Psi_q$ would directly shrink the error in Theorem 1.1.","The same strategy may apply to sextic or higher-order characters over function fields, with the convex hull growing more complicated as the order increases; the bottleneck would be the analogous Gauss-sum generating function bound."],"forward_implications":["The first moment of the quartic family is asymptotic to $q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2})$, with an error $q^{(3/5+\\varepsilon)g}$ that is smaller than the main term.","At least $q^{2g/3-\\varepsilon(g/3+1)}$ primitive quartic characters of genus $g$ have $L_q(\\tfrac12,\\chi)\\neq0$.","The average of $L_q(\\tfrac12,\\chi)$ over the family is bounded, so the main term has the same order as the family size $q^{2g/3}$; a typical character contributes a bounded amount at the central point.","The method shows that explicit residue computations for Gauss sums can be replaced by convex-hull continuation plus a single bound on the Gauss-sum generating function, reducing the computational burden of such moment problems.","If the bound (3.14) were improved, the error exponent $3/5$ in Theorem 1.1 would improve accordingly."],"supporting_citations":[{"why":"Supplies the functional equation for the quartic Gauss-sum generating function that underlies the key bound (3.14).","marker":"[12]"},{"why":"Provides the cubic-function-field first-moment computation and the Gauss-sum lemmas that the quartic case extends.","marker":"[6]"},{"why":"Constructs the primitive quartic characters over F_q(T) and the correspondence between genus and conductor degree used to set up the family sum.","marker":"[14]"},{"why":"Supplies the tube-domain continuation theorem underlying the convex-hull step.","marker":"[4]"},{"why":"Supplies the multivariable convex-hull continuation proposition used to bound the continued double Dirichlet series.","marker":"[5]"},{"why":"Introduces the multiple-Dirichlet-series treatment of first moments for fixed-order characters that the paper adapts to function fields.","marker":"[11]"},{"why":"Gives a range of the key Gauss-sum bound that the paper extends to the full strip, serving as a baseline for Lemma 3.1.","marker":"[7]"}],"fun_headline_variants":["Quartic L-function mean gets power-saving error","Explicit main term for quartic L-function average","Nonvanishing quartic L-values from exact mean","Quartic L-function first moment: error term tamed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on one estimate: that the generating function of quartic Gauss sums, averaged over polynomials coprime to a fixed index, grows at most like $|f|^{(\\frac32-\\sigma)/2+\\varepsilon}$ in the strip $\\tfrac12\\le\\sigma\\le\\tfrac32$; if this bound fails, the meromorphic continuation to the convex hull fails and the power-saving error term collapses.","fun_headline_variants_meta":{"raw":{"variants":["Quartic L-function mean gets power-saving error","Explicit main term for quartic L-function average","Nonvanishing quartic L-values from exact mean","Quartic L-function first moment: error term tamed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000199,"raw_usage":{"total_tokens":1331,"prompt_tokens":862,"completion_tokens":469,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":406}},"tokens_in":478,"tokens_out":469,"duration_ms":5286,"temperature":1.0,"reasoning_tokens":406,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:54:50.183785+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check would be to compute, for a small prime power $q\\equiv3\\pmod4$ such as $q=3$ or $q=7$, the left-hand side of Theorem 1.1 for a few small genera $g$ by enumerating the primitive quartic characters and evaluating $L_q(\\tfrac12,\\chi)$ via the functional equation; any difference from the claimed main term that grows faster than $q^{(3/5+\\varepsilon)g}$ would disprove the theorem. A cheaper test targets Lemma 3.1 directly: fix a square-free $f\\in\\mathbb{F}_q[T]$, evaluate $\\Psi_q(f,u)$ at a point with $|u^4-q^{-5}|>\\delta$ and $\\sigma\\in[\\tfrac12,\\tfrac32]$, and see whether the asserted bound $|f|^{\\frac12(\\frac32-\\sigma)+\\varepsilon}$ holds.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tube-domain continuation theorem underlying the convex-hull step."},{"cited_title":"Hoffstein","cited_arxiv_id":null,"evidence_quote":"Supplies the functional equation for the quartic Gauss-sum generating function that underlies the key bound (3.14)."},{"cited_title":"David, A","cited_arxiv_id":null,"evidence_quote":"Provides the cubic-function-field first-moment computation and the Gauss-sum lemmas that the quartic case extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the primitive quartic characters over F_q(T) and the correspondence between genus and conductor degree used to set up the family sum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the multivariable convex-hull continuation proposition used to bound the continued double Dirichlet series."},{"cited_title":"Gao and L","cited_arxiv_id":null,"evidence_quote":"Introduces the multiple-Dirichlet-series treatment of first moments for fixed-order characters that the paper adapts to function fields."}],"review_version":1}