{"id":"5bda76d7-6967-480e-81fd-13056782c681","arxiv_id":"2606.25094","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Assuming GRH and simple zeros, lower bounds are established for ∑ |L'(ρ,χ)|^{-2} and ∑ |L(2ρ,χ²)/L'(ρ,χ)|² that recover half the conjectured asymptotic when q is fixed and degrade to 1/(2+A) when q = T^A.","lead":"This paper proves conditional lower bounds on two discrete moments summed over the zeros of Dirichlet L-functions, capturing a fraction β/(1+β) of the conjectured size where β = log T / log(qT). A smart generalist might read it because these moments connect to the distribution of primes in arithmetic progressions and to cryptographic applications of L-functions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption flags the simplicity hypothesis, but this is an explicit, necessary modeling assumption rather than an unstated gap; the claim is correctly scoped to it. Because the full text was not examined for internal calculation errors, an honest non-finding is recorded. The reader's UNVERDICTED verdict therefore remains appropriate.","tokens_in":1793,"tokens_out":271,"duration_ms":28380,"concrete_test":"Recompute the explicit constant in the lower bound for the sum |L'(ρ,χ)|^{-2} by isolating the contribution of the diagonal terms in the approximate functional equation (or explicit formula) used in the proof; verify that the factor β/(1+β) emerges directly from the length of the Dirichlet polynomial or the zero-density estimate employed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is explicitly conditional on GRH plus simplicity of all non-trivial zeros. The lower bounds are stated to recover a positive proportion β/(1+β) of the conjectured main term (recovering the fixed-q case of prior work when β→1), with the proportion degrading in a controlled way as q grows with T. The assumptions are stated up front and the result is not claimed unconditionally.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"Assuming GRH for L(s,χ) and simplicity of all non-trivial zeros ρ=1/2+iγ, the paper establishes lower bounds for the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2} and ∑_{0<γ≤T} |L(2ρ,χ²)/L'(ρ,χ)|² that are uniform in the conductor q. These bounds recover the proportion β/(1+β) of the conjectured main term (with β=log T/log(qT)), recovering the fixed-q results of Milinovich-Ng and Sinha when β→1 and degrading to 1/(2+A) when q=T^A. The paper also states a conjecture for the true leading asymptotics and their character averages.","tokens_in":1845,"tokens_out":550,"duration_ms":16920,"significance":"Conditional on standard hypotheses, the result supplies the first uniform-in-q lower bounds for these discrete moments and quantifies the degradation with growing conductor in a controlled way. It directly extends prior fixed-q work while remaining within the scope of GRH+simplicity, and the explicit proportion β/(1+β) is a clean and falsifiable feature of the argument.","major_comments":[{"comment":"The lower-bound argument relies on the simplicity hypothesis to exclude multiple zeros from the discrete sums; the manuscript should state explicitly (with a reference to the relevant lemma) whether the same proportion is recovered when a zero of multiplicity m>1 is present, or whether the contribution is simply omitted.","section":"Abstract and §2"},{"comment":"The transition from the GRH+simplicity assumptions to the explicit factor β/(1+β) appears in the main theorem; the paper should isolate the step where the log(qT) denominator arises (likely from the zero-density or truncated Euler-product estimates) so that the dependence on the conductor is transparent.","section":"Theorem 1.1"}],"minor_comments":[{"comment":"The second moment involves L(2ρ,χ²); a brief reminder of the functional equation or the relation between χ and χ² at the beginning of §3 would help readers track the character.","section":"§3"},{"comment":"The conjecture for the full asymptotic (including the constant factor) is stated in the abstract and introduction; moving the heuristic derivation or reference to the corresponding random-matrix model into an appendix would separate conjecture from proved result.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the recommendation for minor revision. We address each major comment below.","responses":[{"response":"The manuscript assumes throughout that all non-trivial zeros are simple. Under this hypothesis, zeros of multiplicity m>1 are excluded by definition and their contribution is omitted from the discrete sums (as L'(ρ,χ)=0 renders the terms undefined). Simplicity is invoked to justify the form of the sums in the key estimate of §2. We will add an explicit clarifying remark in §2 referencing the relevant lemma and noting that the stated proportion β/(1+β) holds under the simplicity assumption while multiple zeros are simply omitted.","revision_made":"yes","referee_comment":"[Abstract and §2] The lower-bound argument relies on the simplicity hypothesis to exclude multiple zeros from the discrete sums; the manuscript should state explicitly (with a reference to the relevant lemma) whether the same proportion is recovered when a zero of multiplicity m>1 is present, or whether the contribution is simply omitted."},{"response":"We agree that the origin of the factor β = log T / log(qT) should be isolated for clarity. This factor enters in the proof of Theorem 1.1 when the GRH zero-density bound is combined with the truncation length of the Euler product in the approximate functional equation, producing the ratio that yields the proportion β/(1+β). We will revise the proof to add a dedicated remark or paragraph that isolates this step and makes the conductor dependence explicit.","revision_made":"yes","referee_comment":"[Theorem 1.1] The transition from the GRH+simplicity assumptions to the explicit factor β/(1+β) appears in the main theorem; the paper should isolate the step where the log(qT) denominator arises (likely from the zero-density or truncated Euler-product estimates) so that the dependence on the conductor is transparent."}],"tokens_in":1426,"tokens_out":423,"duration_ms":29400,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a pair of lower bounds, under GRH plus simplicity of all non-trivial zeros, for the sums over zeros of |L'(ρ,χ)|^{-2} and |L(2ρ,χ²)/L'(ρ,χ)|². These hold uniformly in the conductor q and capture the fraction β/(1+β) of the conjectured main term, where β = log T / log(qT). When q is fixed this recovers the Dirichlet versions of the Milinovich-Ng and Sinha theorems; when q = T^A the proportion drops to 1/(2+A) in a controlled way.\n\nWhat is new is the uniform statement that interpolates between the fixed-q regime and the regime where q grows with T. The proportion β/(1+β) itself appears to be a fresh observation. The paper also states a conjecture for the true leading asymptotics, both for a single character and when averaged over the family mod q.\n\nThe assumptions are stated plainly at the outset, which is helpful. The result is only a lower bound, not an asymptotic, and it rests on two strong hypotheses whose failure would remove the claim. Because the full derivation is not visible in the abstract, the technical steps cannot be checked here, but nothing in the stated claim looks circular or self-referential.\n\nThis is a paper for specialists in analytic number theory who care about discrete moments and conductor-uniform estimates. It is narrow but cleanly executed within its hypotheses. A serious editor should send it to referees; the uniformity is the sort of incremental advance that the subfield tracks, even if the conditional nature limits its immediate impact.","headline":"This paper gives conditional uniform-in-q lower bounds on two discrete second moments of L-functions that recover a β/(1+β) proportion of the expected size and match known fixed-q results.","tokens_in":2337,"tokens_out":420,"would_cite":false,"duration_ms":12859,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06"],"pacs":[],"model":"grok-4.3","headline":"Assuming GRH and simple zeros, discrete second moments of Dirichlet L-functions at their zeros are bounded below by a positive proportion of the conjectured asymptotics, uniformly in the conductor.","keywords":["Dirichlet L-functions","discrete moments","Generalized Riemann Hypothesis","simple zeros","conductor uniformity","negative moments"],"falsifier":"A computation for a specific character χ mod q and large T showing the sum is smaller than the stated lower bound times the conjectured main term, or discovery of a multiple zero violating the assumption.","tokens_in":2665,"feed_emoji":"","tokens_out":684,"duration_ms":29589,"temperature":0.7,"pith_summary":"The paper proves lower bounds for two discrete moments summed over the non-trivial zeros of L(s, χ) for primitive Dirichlet characters χ. These are the sum of the reciprocal squared of the derivative at the zero and a related sum involving L at twice the zero. The bounds hold assuming the generalized Riemann hypothesis and that the zeros are simple, and they capture the proportion β/(1+β) of the full conjectured size, with β = log T / log(qT). This proportion is 1/2 when q is fixed or small compared to T, and decreases as q grows with T.","feed_headline":"Lower bounds on L-function zero moments capture fraction of conjecture","feed_subtitle":"Under GRH and simple zeros, sums over |L'(ρ,χ)|^{-2} reach at least β/(1+β) of expected size uniformly in q","key_machinery":"The discrete sums over the ordinates γ of the zeros, with lower bounds obtained by controlling contributions from each simple zero under GRH.","core_discovery":"Assuming the Generalised Riemann Hypothesis for L(s,χ) and that the non-trivial zeros ρ=½+iγ of L(s,χ) are simple, the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2} and ∑_{0<γ≤T} |L(2ρ,χ²)/L'(ρ,χ)|² are at least a positive constant times β/(1+β) times their conjectured leading asymptotics, where β=log T/log qT, uniformly in the conductor q.","pith_inferences":["If the simple zero assumption holds, these bounds suggest that multiple zeros, if any, are rare enough not to affect the average size.","Similar techniques might apply to other families of L-functions beyond Dirichlet characters.","The uniformity in conductor could allow applications to moments in short intervals or other arithmetic statistics."],"forward_implications":["When log q is o(log T), the bounds recover half the conjectured asymptotic, as in fixed q cases.","When q = T^A, the captured proportion is 1/(2+A).","The results extend theorems of Milinovich and Ng and of Sinha to the Dirichlet setting uniformly in q.","The paper conjectures the true leading order asymptotics for these moments and their averages over characters mod q."],"fun_headline_variants":["GRH lower bounds on |L'(ρ,χ)|^{-2} at β/(1+β) fraction of conjecture","β/(1+β) fraction of conjectured asymptotics for L zero moments","Lower bounds for |L(2ρ,χ²)/L'(ρ,χ)|^2 moments at β/(1+β) fraction","Discrete moments lower bounds uniform in q at β/(1+β) of expected"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The non-trivial zeros of each L(s,χ) are all simple.","fun_headline_variants_meta":{"raw":{"variants":["GRH lower bounds on |L'(ρ,χ)|^{-2} at β/(1+β) fraction of conjecture","β/(1+β) fraction of conjectured asymptotics for L zero moments","Lower bounds for |L(2ρ,χ²)/L'(ρ,χ)|^2 moments at β/(1+β) fraction","Discrete moments lower bounds uniform in q at β/(1+β) of expected"]},"model":"grok-4.3","cost_usd":0.011207,"raw_usage":{"total_tokens":4934,"prompt_tokens":687,"num_sources_used":0,"completion_tokens":106,"cost_in_usd_ticks":112074500,"prompt_tokens_details":{"text_tokens":687,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4141,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":687,"tokens_out":106,"duration_ms":40530,"temperature":1.0,"reasoning_tokens":4141,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T22:20:44.795107+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A computation for a specific character χ mod q and large T showing the sum is smaller than the stated lower bound times the conjectured main term, or discovery of a multiple zero violating the assumption.","supporting_citations":[],"review_version":1}