{"id":"9398ec4c-3047-4687-9ca9-b0f3199b682b","arxiv_id":"2607.25403","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH, max_{X<q≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)).","lead":"This paper proves a new lower bound on how large certain number-theoretic functions can get, assuming the Generalized Riemann Hypothesis. The result improves a recent bound by Gao for the same family, doubling the leading constant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof applies Lemma 2.2 to c=mn and c=kmn without restricting M to odd integers, violating the lemma's odd-c hypothesis, and (3.3) omits a factor N in S1; these are load-bearing text gaps, though the claims appear repairable by choosing M odd.","rationale":"Read the paper in good faith: the resonance method is standard, the approximate functional equation and GCD-sum lemma are cited from the literature, and the constant 1 follows formally if the external estimates hold. The central claim is probably correct; I do not see a fatal flaw. The most load-bearing soft spot is the hypothesis gap around Lemma 2.2: the lemma is only valid for odd c, but the proof applies it to c=mn and c=kmn without specifying that the extremal set M is odd. Because an even diagonal m=n would contribute a spurious main term, this directly affects both S1 and S2. The same region also contains the inconsistent (3.3) missing factor N. These are substantive enough to justify the reader's CONDITIONAL verdict, but they are mechanical: the known extremal GCD-sum construction can be chosen odd, and the factor N cancels in the ratio. Thus I recommend no change to the verdict. (Additional minor issues: the weight appears to be written as Φ̂(q/X) where Φ(q/X) is presumably intended, and Lemma 2.2 is quoted from an unreviewed preprint; neither changes the assessment.)","tokens_in":5206,"tokens_out":29859,"duration_ms":285252,"concrete_test":"Recompute S1 and S2 using M' = {m: m odd} obtained by stripping the factor 2 from each element of the extremal set in [5, Eq. (1.5)] (or from the authors' chosen M), and correct (3.3) to S1=(1+o(1))Φ̂(1)X|M'|. Verify that (3.7) still gives D ≥ (1+o(1))Φ̂(1)X|M'| exp((2+o(1))√(log|M'|log_3|M'|/log_2|M'|)); if the ratio S2/S1 remains exp((2+o(1))√(...)), the odd-c/even-M gap is repairable and Theorem 1.1 stands. Additionally, a numerical check of Lemma 2.2 for even c=4 (left side zero versus claimed Φ̂(1)X) would confirm the lemma cannot be used without the odd restriction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.2 is stated for positive odd c. For even c the left-hand side vanishes identically (χ_{8q}(c)=0 when q is an odd prime), so the displayed formula is false for even square c. The proof expands R_q^2 and applies Lemma 2.2 with c=mn (S1, line after (3.2)) and c=kmn (S2, after (3.4)). Lemma 2.3 is stated for arbitrary positive square-free integers; the paper never asserts M is odd. If an even m lies in M, the diagonal term m=n has c=m^2 even and is counted as Φ̂(1)X although the actual sum is zero, overcounting S1; the analogous failure affects S2. Independently, (3.3) claims S1 = Φ̂(1)XN+O(...) ≤ (1+o(1))Φ̂(1)X, which is impossible for N=X^{1/4-δ} (the correct upper bound is (1+o(1))Φ̂(1)XN). Since the final lower bound is S2/S1, an overcount of S1 would directly change the ratio. The likely repair is to take M to be a set of odd squarefree integers (the construction in [5] can be chosen this way) and to correct (3.3) to include N; with both repairs the factor N cancels and the claimed exponent still follows from (3.7). As written, however, the main terms in S1 and S2 are not justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves, under GRH, a lower bound of the form max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)), improving the constant 1/2 obtained by Gao for the same family of quadratic Dirichlet L-functions with prime-related moduli. The method is the resonance method: a resonator R_q = Σ_{m∈M} χ_{8q}(m) is constructed from a set M of squarefree integers with large GCD sum (Lemma 2.3), and the ratio S_2/S_1 is bounded below using a GRH-conditional character-sum estimate (Lemma 2.2) and the approximate functional equation (Lemma 2.1). The proof follows the standard pattern of Bondarenko-Seip and de la Bretèche-Tenenbaum adapted to this family.","tokens_in":5622,"tokens_out":3586,"duration_ms":35744,"significance":"If the result is correct, it constitutes a genuine improvement over Gao's recent constant 1/2 and matches the constant 1 that is expected from the resonance/GCD-sum method for this family. The argument is not self-contained: it relies on two substantial external inputs, Lemma 2.2 from Gao's preprint and Lemma 2.3 from de la Bretèche-Tenenbaum, but such reliance is normal in this area. The paper's main contribution is the correct adaptation of the GCD-sum method to the moduli 8q, and the result is likely to be of interest to specialists. However, as written, the proof contains two load-bearing technical gaps that prevent the main estimate from being fully justified.","major_comments":[{"comment":"Lemma 2.2 is stated only for positive odd integers c. In the evaluation of S_1, Lemma 2.2 is applied with c=mn, and in S_2 with c=kmn. The set M is taken as in Lemma 2.3, which only requires squarefree integers and does not guarantee that m,n are odd. If an even m∈M exists, then the diagonal term m=n gives c=m^2 even, for which the displayed main term in Lemma 2.2 is false (indeed χ_{8q}(m^2)=0 for odd primes q). Similarly, in S_2 the index k must be restricted to odd integers. This is not a mere formality: the main terms in (3.3) and (3.4) would be incorrect for even c, and the final ratio would change. The likely repair is to choose M to consist of odd squarefree integers, as can be done in the construction of [5], and to sum over odd k in S_2. The authors should state this explicitly and verify the consequences for M's size and GCD-sum bound.","section":"§3, before (3.3) and after (3.4)"},{"comment":"The equation S_1 = Φ̂(1)XN + O(X^{1/2+2ε}N^2) ≤ (1+o(1))Φ̂(1)X is inconsistent: since N=⌊X^{1/4-δ}⌋→∞, the displayed upper bound omits the factor N and is false. The correct bound is S_1 ≤ (1+o(1))Φ̂(1)XN. This error is consequential because if S_1 were only O(X), the ratio S_2/S_1 would gain an extra factor N, artificially changing the final exponent. In the subsequent text the correct normalization appears to be used implicitly, but as printed the proof is self-contradictory. This must be corrected before the claimed lower bound can be accepted.","section":"§3, Eq. (3.3)"},{"comment":"The proof of the lower bound for S_2 depends crucially on Lemma 2.2 being valid uniformly for c=kmn with k up to X^{1/2+ε} and m,n∈M, whose elements may be as large as exp((log N)^{1+o(1)}). The paper imports Lemma 2.2 from [7] without proof and without checking uniformity conditions. In particular, the statement of Lemma 2.2 has an error term X^{1/2+ε} log(c+2), and the number of such c is large. The authors should either quote the precise uniformity range from [7] or prove the needed estimate. This is not a circularity (the external lemma is not fitted to the conclusion), but it is a load-bearing dependency that the manuscript should make explicit and verify.","section":"§3, passage after (3.4)"}],"minor_comments":[{"comment":"\"By Lemma (2.2), ω(ξ) and ω'(ξ) decay exponentially\" should refer to Lemma 2.1, not Lemma 2.2.","section":"§3, after (3.4)"},{"comment":"The introduction says q always represents an odd prime, but later uses both X<q≤2X and X<|q|≤2X. Since q is positive, |q| is redundant; unify the notation.","section":"Notation throughout"},{"comment":"The author list in [10] appears garbled: \"H. Z. Z. Dong, W. Wang and S. Zhao\" should likely list the four authors consistently with the title page. Please correct.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short, technically standard application of the resonance method to a family already considered by Gao. The central claim is plausible and likely repairable, but the two explicit gaps in the text—the parity restriction in Lemma 2.2 and the missing factor N in (3.3)—are load-bearing and must be fixed. Given that the proof relies on two external lemmas, one from a recent preprint, the editor may also wish to ensure that [7] is publicly available and its results accepted. I do not see grounds for rejection, but the present manuscript is not yet in a publishable form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper plausibly proves the sharp lower bound exp((1+o(1))√(log X log_3 X / log_2 X)) for max_{X<q≤2X} |L(1/2, χ_{8q})|, improving Gao's 1/2 constant. The method is the standard resonance method with the de la Bretèche–Tenenbaum GCD-sum bound, which is the right tool and is used in the way you would expect. The result is new: it is not a re-coordinatization of the authors' earlier fundamental-discriminant theorem, since the subfamily {8q} is sparse. The constant 1 is the expected optimal constant, so this is a legitimate step forward.\n\nThe paper is honest about its reliance on two external inputs: Gao's GRH character-sum estimate (Lemma 2.2) and the GCD-sum theorem. Self-citation [10] is contextual, not load-bearing. No constants are fitted and no circularity beyond the standard conditional framework.\n\nSoft spots, in order of seriousness. First, the line after (3.3) displays S1 = Φ̂(1)XN + O(...) ≤ (1+o(1))Φ̂(1)X. That cannot be right: N = X^{1/4−δ} → ∞, so the displayed inequality is false as written. The proof only needs S1 to be O(Φ̂(1)XN), and the ratio S2/S1 would cancel the N, so this is a mechanical gap, but it is exactly the kind of thing that will confuse a referee. Second, Lemma 2.2 is stated for odd c only. The proof applies it with c=mn and c=kmn without ever saying M is a set of odd squarefree integers. If any m is even, the diagonal m=n has even c=m², the character sum is actually zero, and the main term claimed is wrong. The construction in [5] can certainly be restricted to odd integers, so this is repairable, but it is load-bearing in the sense that as written the main terms in S1 and S2 are not justified. The error-term estimate for S2 appears routine, and the arithmetic leading to (3.5)–(3.7) is standard.\n\nOverall: the central argument holds up modulo the odd-M fix and the missing N in (3.3). The paper deserves a serious referee, but the authors should be asked to correct these two points before publication. I would send it to a standard analytic number theory referee.","headline":"The main theorem is likely true and the constant 1 is a genuine improvement over Gao's 1/2, but the written proof needs two small repairs before it is referee-ready.","tokens_in":6108,"tokens_out":3372,"would_cite":true,"duration_ms":31983,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11M06","11N56"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming GRH, some prime q in (X,2X] makes |L(1/2, χ_{8q})| as large as exp((1+o(1))√(log X log log log X / log log X)).","keywords":["quadratic Dirichlet L-functions","extreme values","resonance method","character sums","GCD sums","Generalized Riemann Hypothesis","prime moduli","central values"],"falsifier":"Evaluate the character sum in Lemma 2.2 for c = 4: since χ_{8q}(4) = 0 for every odd prime q, the sum is 0, whereas the lemma's asserted main term would be Φ̂(1)X (because 4 is a square). This shows the lemma cannot be applied to even c, and thus the resonator set M in the proof must be restricted to odd integers for the diagonal main term S1 ≈ Φ̂(1)XN to be correct.","tokens_in":5110,"feed_emoji":"📈","tokens_out":17822,"duration_ms":140862,"temperature":0.7,"pith_summary":"This paper proves a new lower bound for the extreme values of quadratic Dirichlet L-functions with prime-related moduli, specifically the characters χ_{8q} where q is an odd prime. Under the Generalized Riemann Hypothesis, it shows that for all large X there is a prime q in (X,2X] with |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log log log X / log log X)). This improves the previous constant 1/2 for this family to 1, matching the constant obtained for the full family of all fundamental discriminants. The proof uses the resonance method: a carefully chosen set of square-free integers M defines a resonator R_q, and the ratio of two character-sum averages is bounded below by a GCD sum that supplies the exponential growth.","feed_headline":"Constant 1 achieved for prime-related quadratic L-function extremes","feed_subtitle":"Under GRH, |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log log log X / log log X)) for some prime q in (X,2X].","key_machinery":"The machinery is the resonance method with a GCD-sum amplifier. The central object is the resonator R_q = Σ_{m∈M} χ_{8q}(m), where M is a set of square-free integers of size X^{1/4-δ} maximizing the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). Two estimates carry the proof: Lemma 2.2, a GRH-conditional bound ∑_{q}^{*}(log q)χ_{8q}(c)Φ̂(q/X) = 1_{c=□}Φ̂(1)X + O(X^{1/2+ε} log(c+2)) for odd c, which isolates the diagonal main terms; and Lemma 2.3, a lower bound for GCD sums of square-free integers, which provides the exponential growth. The restriction [m,n]/(m,n) ≤ X^ε, enforced by Rankin's trick, lets the smooth weight ω be treated as 1.","core_discovery":"Theorem 1.1 states that, assuming GRH, for sufficiently large X, max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a resonator R_q = Σ_{m∈M} χ_{8q}(m), with M a set of N = X^{1/4-δ} square-free integers chosen to maximize the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). The core of the argument is the ratio S2/S1: S1 counts the diagonal terms m=n via a conditional character-sum estimate (Lemma 2.2), and S2 is bounded below using the same estimate plus the observation that the smooth weight ω is close to 1 when [m,n]/(m,n) ≤ X^ε. The GCD-sum bound of Lemma 2.3 gives the exponential factor, and choosing δ → 0 yields the constant 1.","pith_inferences":["The proof applies Lemma 2.2 to c = mn and c = kmn, but the lemma is stated only for odd c; since χ_{8q}(even) = 0, applying it to even c would give a wrong main term. The set M must therefore be chosen to contain only odd integers, a condition the paper does not explicitly state — a repairable but real gap in the written proof.","The same construction could be tested for moduli of the form d·q with fixed squarefree d; the character-sum estimate would need adjustment, but the GCD-sum part should transfer unchanged.","Because the error bounds require log(c+2) ≪ X^ε, the method implicitly constrains M to elements of size exp((log N)^{1+o(1)}); a resonator with larger elements would break the uniformity of Lemma 2.2.","A direct computation of the character sum in Lemma 2.2 at c = 4 would confirm that the main term is zero, not Φ̂(1)X, illustrating why the oddness of M is necessary."],"forward_implications":["Under GRH, the family {χ_{8q}} of prime-related moduli has extreme values of the same asymptotic size as the full family of quadratic Dirichlet L-functions, with constant 1 instead of 1/2.","The proven lower bound grows faster than any fixed power of log X, so |L(1/2, χ_{8q})| can be exceptionally large even when the conductor is restricted to 8 times a prime.","The resonance method remains effective on a thin subfamily of conductors, provided the conditional character-sum estimate holds uniformly for all odd c up to exp((log N)^{1+o(1)}).","The GCD-sum bound is the sole source of the exponential factor; improvements to that bound would directly improve the constant in the lower bound."],"fun_headline_variants":["GRH: optimal exponent for prime-moduli L-functions","Prime-moduli quadratic L-functions: extreme values sharpened","Constant 1 in exponent for L-function extremes at primes","New lower bound for quadratic L-functions at prime moduli","Optimal growth for L-functions over prime moduli"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof's load-bearing premise is that a GRH-conditional estimate for sums of χ_{8q}(c) over primes (Lemma 2.2) holds uniformly for every odd c up to exp((log N)^{1+o(1)}); the paper uses it for c = mn and c = kmn without expressly requiring M to be odd, and for even c the estimate's main term is false.","fun_headline_variants_meta":{"raw":{"variants":["GRH: optimal exponent for prime-moduli L-functions","Prime-moduli quadratic L-functions: extreme values sharpened","Constant 1 in exponent for L-function extremes at primes","New lower bound for quadratic L-functions at prime moduli","Optimal growth for L-functions over prime moduli"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000307,"raw_usage":{"total_tokens":1555,"prompt_tokens":667,"completion_tokens":888,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":804}},"tokens_in":411,"tokens_out":888,"duration_ms":8668,"temperature":1.0,"reasoning_tokens":804,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:32:37.538564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the character sum in Lemma 2.2 for c = 4: since χ_{8q}(4) = 0 for every odd prime q, the sum is 0, whereas the lemma's asserted main term would be Φ̂(1)X (because 4 is a square). This shows the lemma cannot be applied to even c, and thus the resonator set M in the proof must be restricted to odd integers for the diagonal main term S1 ≈ Φ̂(1)XN to be correct.","supporting_citations":[],"review_version":1}