{"id":"a70d0cf4-6540-4fdb-96ac-b3ae1cc7e486","arxiv_id":"2607.20408","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Assuming GRH, the paper proves the sharp lower bound exp((1+o(1))√(logX log3X/log2X)) for max |L(1/2,χ_d)| with d∼X, improving the constant from 1/2 to 1.","lead":"Under the Generalized Riemann Hypothesis, this paper claims quadratic Dirichlet L-functions can reach exp((1+o(1))√(log X·log log log X / log log X)) near X, doubling the previous best constant. A specialist would read it because the resonance method may have reached the natural limit of this technique.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rankin-trick inequality in (3.7) has the wrong exponent sign; as written, the subtracted term can exceed the main term, so the exponent-1 lower bound is unproved.","rationale":"The reader's weakest assumption identifies exactly the load-bearing defect: the Rankin-trick inequality in (3.7) is backwards. I checked the direction explicitly. For r=[m,n]/(m,n)>X^ε, we have r^{-1/2}=r^{-1/3} r^{-1/6} < X^{-ε/6} r^{-1/3}; therefore the omitted tail is controlled by X^{-ε/6}Σr^{-1/3}, not X^{ε/6}Σr^{-1/3}. With X^{ε/6} (which is >1), the subtracted quantity is larger than the unrestricted GCD sum, so the claimed lower bound has no basis. Theorem 1.1's logarithmic exponent 1 is obtained by substituting (3.7) into (3.8); without (3.7), the proof gives nothing. I also considered whether the y_M bound or the moment estimate for Σr^{-1/3} is a separate fatal gap. It is a real unproved assertion, but the sign error is sufficient for rejection. If the exponent is corrected, the argument appears salvageable: even the crude product bound yields X^{-ε/6}Σr^{-1/3} ≤ N^{1−2ε/3+o(1)}, which is o(N^{1+o(1)}) for fixed ε>0. Thus the paper may be repairable by a minor correction, but the version under review does not contain that correction. The reader's moderate-confidence REJECT is appropriate; my stress test does not change it.","tokens_in":5442,"tokens_out":14751,"duration_ms":114619,"concrete_test":"Re-derive (3.7) using the correct Rankin inequality: for r > X^ε, r^{-1/2} ≤ X^{-ε/6} r^{-1/3}, so the tail is bounded by X^{-ε/6} Σ_{m,n} ((m,n)/[m,n])^{1/3}. Then verify, using the stated bound Σ_{m,n} ((m,n)/[m,n])^{1/3} ≤ N exp(y_M^{2/3}) and y_M ≤ (log N)^{1+o(1)}, whether this tail is o(1) times Σ_{m,n} sqrt((m,n)/[m,n]) as N→∞ with X=N^{4+o(1)} and fixed ε>0. If yes, the proof is repairable; if no, (3.7) fails even after the sign correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 depends on the lower bound (3.7), which is derived by a Rankin-trick truncation at [m,n]/(m,n) ≤ X^ε. The paper asserts\n\nΣ_{m,n: r≤X^ε} r^{-1/2} ≥ Σ_{m,n} r^{-1/2} − X^{ε/6} Σ_{m,n} r^{-1/3},   r = [m,n]/(m,n).\n\nFor r > X^ε the pointwise inequality is r^{-1/2} = r^{-1/3} r^{-1/6} < X^{-ε/6} r^{-1/3}, not X^{ε/6} r^{-1/3}. Thus the tail is bounded above by X^{-ε/6} Σ r^{-1/3}, and the subtracted term in (3.7) should carry X^{-ε/6}, not X^{ε/6}. With the printed exponent, the subtracted term is larger than the unrestricted GCD sum by a factor X^{ε/6}, so the right-hand side can be negative and no lower bound follows. Since the exponential constant 1 in Theorem 1.1 comes entirely from (3.7) through Lemma 2.3, the central claim is not proved as written. If the sign is corrected, the step is plausible: the crude bound Σ r^{-1/3} ≤ N exp(y_M^{2/3}) with y_M ≤ (log N)^{1+o(1)} gives X^{-ε/6} Σ r^{-1/3} ≤ N^{1−2ε/3+o(1)}, which is o(1) times the main term N^{1+o(1)} for fixed ε>0. But that is a repair, not the displayed argument. The set M's y_M bound is also cited rather than proved, but it is secondary to the sign error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Assuming GRH, the paper claims that for sufficiently large X, max_{X<|d|≤2X, d∈F} |L(1/2,χ_d)| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)). This would improve the previous conditional constant 1/2 of Darbar and Maiti [5] to 1. The proof uses the resonance method: choose a set M of squarefree integers with large GCD sum (Lemma 2.3), set R_d = Σ_{n∈M} χ_d(n), and estimate S1 = Σ R_d^2 and S2 = Σ L(1/2,χ_d) R_d^2 via the approximate functional equation (Lemma 2.1) and a GRH mean-value lemma (Lemma 2.2). The ratio S2/S1 gives the lower bound. The key new ingredient is the Rankin-trick estimate (3.7) for the truncated GCD sum.","tokens_in":5826,"tokens_out":10352,"duration_ms":78119,"significance":"If correct, the theorem is a significant advance: it raises the conditional lower-bound constant for quadratic Dirichlet L-functions from 1/2 to 1, matching the best-known constant for ζ(s) in the Bondarenko–Seip line (though not the √2 of de la Bretèche–Tenenbaum). The method is a direct combination of existing tools, and the proof is short. The main novelty is the application of the large GCD-sum lemma together with the y_M bound. However, the proof as written contains a sign error in a key inequality; after correction, the argument is plausible but needs revision.","major_comments":[{"comment":"The Rankin-trick step is written with the wrong exponent. For r=[m,n]/(m,n)>X^ε, r^{-1/2}=r^{-1/3}r^{-1/6} ≤ X^{-ε/6}r^{-1/3}, not ≥ X^{ε/6}r^{-1/3}. Therefore the tail should be bounded by X^{-ε/6}Σ r^{-1/3}, not X^{ε/6}Σ r^{-1/3}. As printed, X^{ε/6}Σ r^{-1/3} is of size N exp((2ε/3+o(1))log N), while the main term from Lemma 2.3 is only N exp(2√(log N log_3 N / log_2 N)); hence the right-hand side of (3.7) is not a lower bound. This step is load-bearing for Theorem 1.1. With the corrected exponent X^{-ε/6}, the subtracted term is negligible relative to the main term, so the claim appears repairable, but the displayed argument must be fixed and the subsequent estimates rechecked.","section":"§3, derivation of (3.7)"},{"comment":"The proof of Theorem 1.1 uses not only Lemma 2.3 but also the fact that the extremal set M has y_M = max_{m∈M} P^+(m) ≤ (log N)^{1+o(1)}. This property is stated only in an informal note after Lemma 2.3, not as part of the lemma. It is used in the lower bound (3.6) and in the estimate Σ_{n∈M} ((m,n)/[m,n])^{1/3} ≤ exp(y_M^{2/3}). Please state the y_M bound explicitly in Lemma 2.3, with a proof or a precise reference, because without it the lower bound for D in (3.8) is incomplete.","section":"Lemma 2.3 and §3"}],"minor_comments":[{"comment":"The displayed error O(X^{1/2+ε}Σ_{m,n∈M}1) after Lemma 2.2 drops the factors g1(n1), g2(n2); these are not O(1) but exp((log n)^{1-ε}) and exp((log n)^{1/2-ε}). The argument should absorb them into X^ε or display the precise bound.","section":"§3, S1 error term"},{"comment":"There are notational inconsistencies: g2(n1) versus g2(n2), and the phrase 'It is clear for g1(n0)' should refer to n1 consistently.","section":"After Lemma 2.2"},{"comment":"The final 'Taking α→0+' should be formulated as a two-limit argument: for every η>0 choose α small enough, then X large enough. As written it may suggest a single limiting process in which the o(1) also varies.","section":"End of §3"},{"comment":"The phrase 'as d is large' is awkward; use 'as |d|→∞'.","section":"Abstract"},{"comment":"The informal note after Lemma 2.3 would be better integrated into the lemma statement so that the y_M property is a formal hypothesis for later use.","section":"Lemma 2.3 note"}],"recommendation":"major_revision","confidential_remarks":"The sign error in (3.7) is a central flaw, but it is a local typo that can likely be repaired by replacing X^{ε/6} with X^{-ε/6} and rechecking the estimate. I therefore recommend major revision rather than rejection. The paper also needs to make the y_M property a formal part of Lemma 2.3. The result, if repaired, would be a natural and worthwhile improvement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper claims the expected improvement from 1/2 to 1 in the GRH-conditional extreme value problem for quadratic Dirichlet L-functions. The announced bound is new relative to Darbar–Maiti, and the architecture is the standard resonator-plus-GCD-sum method, with a sensible choice of N = X^{1/4−α}. The main term and error estimates follow established lines, and there is no circularity or fitting of constants to the result. If the proof were correct, this would be a clean within-field advance.\n\nThe trouble is in the Rankin-trick step leading to (3.7). The paper wants to lower-bound the restricted GCD sum over r = [m,n]/(m,n) ≤ X^ε by subtracting the tail from the full sum. It asserts the tail is at most X^{ε/6} Σ r^{-1/3}, but for r > X^ε we have r^{-1/2} = r^{-1/3} r^{-1/6} < X^{-ε/6} r^{-1/3}, so the tail is actually bounded by X^{-ε/6} Σ r^{-1/3}. The exponent sign is backwards. With the printed X^{ε/6}, the subtracted term is X^{ε/6} |M| exp(y_M^{2/3}), which for fixed ε>0 is exponentially larger than the main term N exp(2 sqrt(logN log3N / log2N)). Thus (3.7) cannot follow. This is the load-bearing step: Theorem 1.1 is an immediate corollary of (3.7), and without it the result is unproved. The correct sign would make the step plausible, and the rest of the proof would proceed.\n\nThe y_M bound from Lemma 2.3 is cited rather than proved, but that is secondary. The more serious issue is the wrong exponent, which is either a transcription mistake or a genuine gap. Either way, the manuscript in its present form should not be accepted.\n\nI would still send this to a competent referee rather than desk reject, because the error is local and repairable, and the claimed improvement is the natural next step in the literature. If the authors fix the sign, the paper could be solid. As it stands, the central claim is not established.","headline":"The claimed improvement from 1/2 to 1 in the GRH-conditional extreme value bound for quadratic Dirichlet L-functions is undone by a sign error in the Rankin-trick step; as written, Theorem 1.1 does not follow, though the approach looks repairable.","tokens_in":6369,"tokens_out":3785,"would_cite":false,"duration_ms":33345,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming GRH, quadratic Dirichlet L-functions are shown to reach exp((1+o(1))√(log X log_3 X / log_2 X)), doubling the previous best constant from 1/2 to 1.","keywords":["quadratic Dirichlet L-functions","extreme values","resonance method","GCD sums","Generalized Riemann Hypothesis","central values","Rankin trick","lower bounds"],"falsifier":"Check whether the inequality Σ_{r>X^ε} r^{1/2} ≤ X^{ε/6} Σ_r r^{1/3} holds; it does not for any r>X^ε, so the tail term in (3.7) is uncontrolled unless a different argument is supplied.","tokens_in":5274,"feed_emoji":"📈","tokens_out":7790,"duration_ms":56755,"temperature":0.7,"pith_summary":"The paper studies how large quadratic Dirichlet L-functions can be at their central point as the discriminant d grows. Under the Generalized Riemann Hypothesis, it proves that for sufficiently large X some fundamental discriminant d in [X,2X] satisfies |L(1/2, χ_d)| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). This improves the previous conditional constant from 1/2 to 1, matching the leading constant known for the Riemann zeta function. The method combines the resonance construction with large GCD sums and relies on a GRH-based mean-value estimate for quadratic characters.","feed_headline":"Quadratic L-functions double the previous extreme-value exponent","feed_subtitle":"Under GRH, central values reach exp((1+o(1))√(log X log_3 X / log_2 X)).","key_machinery":"The central object is the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]) over a set M of squarefree integers, paired with the resonator R_d = Σ_{n∈M} χ_d(n). Lemma 2.3 asserts the maximal GCD sum for |M|=N is N exp((2+o(1))√(log N log_3 N/log_2 N)). The proof also uses the approximate functional equation (Lemma 2.1) and a GRH-conditional mean-value estimate (Lemma 2.2) to compute the first and second moments of L(1/2,χ_d) R_d^2. The mechanism is that the second moment inherits the GCD sum of M while the first moment is essentially |M|, so their ratio produces the extreme value.","core_discovery":"The paper's central claim is Theorem 1.1: under GRH, for large X, max_{X<|d|≤2X, d∈F} |L(1/2, χ_d)| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a set M of squarefree integers with near-maximal GCD sums and defines a resonator R_d = Σ_{n∈M} χ_d(n). Expanding the weighted first and second moments, S_1 and S_2, and applying a GRH-conditional mean-value theorem (Lemma 2.2), the ratio S_2/S_1 is shown to be at least the GCD sum of M. With the optimal GCD sum from Lemma 2.3, this yields the stated lower bound.","pith_inferences":["If the flagged tail-bound inequality is repaired, the method could plausibly push the constant beyond 1, since the GCD sum construction is already at the conjectured optimum; the next bottleneck would be the moment error terms.","The dependence on GRH enters only through the mean-value estimate; an unconditional substitute for that estimate would immediately yield an unconditional (but weaker) extreme-value lower bound.","The technique might extend to higher moments or derivatives of L-functions, where the same GCD sum machinery could yield new lower bounds for central values and beyond.","A natural stress test is whether the same exponent appears for non-quadratic characters or for L-functions of higher rank, where the resonance form changes but the GCD sum may still dominate."],"forward_implications":["The extreme-value exponent for quadratic Dirichlet L-functions now matches the best known for the Riemann zeta function, indicating a common threshold across L-function families.","Under GRH, this is the sharpest lower bound known for |L(1/2,χ_d)| in terms of the leading constant.","The resonance plus GCD sum argument extends naturally to other families whose character sums admit similar mean-value theorems.","The proof gives a template for converting GCD-sum maxima into central-value lower bounds for real primitive characters."],"fun_headline_variants":["Quadratic L-functions set new GRH extreme record","Extreme exponent for quadratic L-functions doubled","Central values of quadratic L-functions reach new peak","GRH boosts quadratic L-function extreme values","Quadratic Dirichlet L-functions: extreme values spike"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof relies on a step asserting that pairs with very large least common multiple contribute negligibly to the GCD sum; the inequality used for that step runs the wrong way, so the asserted bound is not actually derived.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic L-functions set new GRH extreme record","Extreme exponent for quadratic L-functions doubled","Central values of quadratic L-functions reach new peak","GRH boosts quadratic L-function extreme values","Quadratic Dirichlet L-functions: extreme values spike"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000108,"raw_usage":{"total_tokens":805,"prompt_tokens":590,"completion_tokens":215,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":334,"completion_tokens_details":{"reasoning_tokens":140}},"tokens_in":334,"tokens_out":215,"duration_ms":2969,"temperature":1.0,"reasoning_tokens":140,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:57:19.604784+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the inequality Σ_{r>X^ε} r^{1/2} ≤ X^{ε/6} Σ_r r^{1/3} holds; it does not for any r>X^ε, so the tail term in (3.7) is uncontrolled unless a different argument is supplied.","supporting_citations":[],"review_version":1}