{"id":"da89d261-81b1-4136-ab57-6791a3e3ebc4","arxiv_id":"2511.06109","paper_version":2,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"An expository report of Young's proof of Levinson's theorem (≥1/3 of zeta zeros on the critical line) and Wu's Dirichlet-L-function bound κ(χ)>0.4172, with the decisive error-term analysis deferred to the source paper.","lead":"This paper is a student report that walks through published proofs about where the zeros of the Riemann zeta function and Dirichlet L-functions lie: at least one third of zeta's nontrivial zeros are on the critical line, and more than two fifths of the zeros of Dirichlet L-functions. It contains no new results; every theorem is attributed to earlier papers by Young, Wu, and Conrey, and the report's own conclusion notes that stronger published bounds already exist.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's 0.4172/0.4074 lower bounds rest on an error-term estimate that the report explicitly defers to [13]; as presented, the proof of Theorem 3.1 is not self-contained and is unsupported internally.","rationale":"The reader's weakest assumption exactly identifies the missing error-term analysis behind Theorem 3.1, and the manuscript's own §III.2 and §III.3 concede the omission. This is the load-bearing concern because all numerical claims depend on Theorem 3.2(B). I agree with the UNVERDICTED verdict: the report contains no new research claim, and its exposition of the Dirichlet case is incomplete. The concrete check would distinguish 'correct by attribution' from 'proved here'; since that distinction is the actual question, the reader's verdict should be maintained. No additional internal inconsistency was found in the zeta-function section beyond the unproved 'Result' in §II.3, which is secondary to the headline generalization.","tokens_in":14621,"tokens_out":4378,"duration_ms":41622,"concrete_test":"Provide a complete derivation of the §III.2 Z-bound from the stated lemmas without referencing [13]. Specifically, expand E_1,E_2 in the g(α,β,w) decomposition, apply Lemma 3.4 and the D(0,...) bound, and show that the displayed Z estimate gives O(T^{-ε0}) uniformly for θ=4/7−ε. If this derivation cannot be reproduced from the report's text alone, the central theorem is not proved in this preprint.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive input for the headline numbers is Theorem 3.2(B): ε0>0 for θ<4/7 only under the special coefficient shape a(n)=μ(n)(F0+F1·(F2*F3))(n). Theorem 3.1 then chooses θ=4/7−ε with a(n)=μ(n)(P1+P2·Σ_{p|n,p≤y^{3/4}}P) and reads off κ>0.4172, κ*>0.4074. But the proof of Theorem 3.2's error term is not supplied. §III.2 states 'After some long calculation using various other previously known bounds, we prove Z≪...' and immediately adds 'The proof of the error term is quite lengthy and is given in more details in [13].' The report also quotes the Levinson inequality and the sieve evaluation Σ(α,β)=q/φ(q)θL M+O(...) in §III.3 without proof. Consequently, if any step in the deferred error analysis—the Z≪y^{7/8}T^{-1/2+11η/2+ε}+y^{7/4}T^{-1+11η/2+ε} bound, or its transition to O(R^{1−ε0}) at θ=4/7−ε—failed, the stated constants would have no support in this manuscript. This is an internal omission, not a dispute with the published literature: stronger bounds are even cited as [14],[15]. The report can at best be certified as a partial digest, not as a proof of Theorem 3.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is an expository report that first sketches Matthew Young's proof of Levinson's theorem for the Riemann zeta function (at least one-third of non-trivial zeros lie on the critical line) and then claims a generalization to Dirichlet L-functions: Theorem 3.1 states that for any Dirichlet character χ and sufficiently large T with log q = o(log T), κ(χ) > 0.4172 and κ*(χ) > 0.4074, where κ and κ* are the proportions of critical zeros and simple critical zeros, respectively. The proof of the Dirichlet part follows Xiaosheng Wu's 2018 paper, using a mollified second moment and a long-mollifier device. The manuscript is not a research announcement of a new theorem; it is a survey based on the cited works [7], [10], and [13], with explicit choices of polynomials and numerical parameters. The central technical claim is Theorem 3.2, an asymptotic for the mollified second moment I_R(Q,χ), and Section III.3 uses it together with quoted Levinson-type inequalities and a sieve estimate to obtain the numerical bounds.","tokens_in":1742,"tokens_out":2039,"duration_ms":69998,"significance":"If the exposition were fully self-contained and accurate, the paper would be a useful pedagogical survey of the Levinson method, bringing together Young's streamlined proof for ζ(s) and Wu's twisted-moment treatment for Dirichlet L-functions. However, the manuscript contains no new mathematical result. Its main value lies in assembling known results and numerical constants. The strength of the manuscript is its honest identification of sources: the proof of Levinson's theorem is explicitly attributed to Young [7], and the Dirichlet generalization is attributed to Wu [13]. The stated numerical constants are consistent with those in the cited literature. On the other hand, the exposition is not self-contained: several load-bearing estimates are asserted without proof and explicitly deferred to [13]. In particular, the error-term analysis behind Theorem 3.2 is not carried out in the manuscript, and the quoted Levinson inequality and sieve evaluation in Section III.3 are used as black boxes. Thus the paper cannot serve as a proof of Theorem 3.1 as written, though the underlying theorem is true by the cited work. The significance of the manuscript depends on the venue: as an expository","major_comments":[{"comment":"The proof of the decisive error term is not supplied. After stating \"After some long calculation using various other previously known bounds, we prove Z ≪ ...\" and \"This completes the proof of the proposition,\" the text immediately adds: \"The proof of the error term is quite lengthy and is given in more details in [13].\" This is a citation, not a proof. Since Theorem 3.1's constants 0.4172 and 0.4074 are read off from Theorem 3.2 with θ=4/7−ε and the special coefficient shape a(n)=μ(n)(P1+P2·Σ_{p|n,p≤y^{3/4}}P), the claimed lower bound has no support in this manuscript unless the bound on Z and the transition to the stated error term are proved here. Moreover, Theorem 3.2's error term is written as O(R^{1−ε0}); with R=1.3 this is O(1), which is meaningless as an asymptotic error term. Compare the proposition's O(T^{−ε0})—this is very likely a typo for O(T^{1−ε0}), and must be corrected.","section":"III.2, Theorem 3.2"},{"comment":"The final derivation of κ(χ)>0.4172 and κ*(χ)>0.4074 depends on two quoted results that are not proved: the Levinson-type inequality κ(χ) ≥ 1 − (1/R) log(T^{−1} I_R(Q,χ)) + o(1) (and the analogous statement for simple zeros when Q is linear), and the sieve estimate Σ(α,β) = q/φ(q) θ L M + O((log log)^7 log^{−2} y). These are essential to convert the asymptotic of Theorem 3.2 into the stated numerical bounds. The manuscript gives no derivation of either, so the proof of Theorem 3.1 is incomplete at its final step. Either the proofs should be included, or the paper should be explicitly reframed as a survey relying on [13] for these ingredients.","section":"III.3"},{"comment":"Even in the Levinson/Young part, the exposition is a sketch rather than a proof. Lemma 2.3 is asserted to follow from a contour manipulation and an unproved 'Result' that supplies the main term of J_{α,β}(M). The statement 'The proof is completed using the following result, qed' is not a proof of that result. Similarly, the assertion after contour shifting that 'the new contour of integration gives O(T^{1−ε})' is not substantiated. If the paper's goal is to present a self-contained proof of Levinson's theorem, these gaps must be filled; if the goal is a survey, the text should say so explicitly.","section":"II.3, Lemma 2.3 and the 'Result'"}],"minor_comments":[{"comment":"The displayed analytic continuation formula is garbled: 'Γ((1−2)/2)' should presumably be 'Γ((1−s)/2)', and the notation around the incomplete gamma factor is unclear. There are also typos ('cosnider', 'its beneficial').","section":"I.1, Theorem 1.1"},{"comment":"The text says 'The non-trivial zeroes of ζ(s) lie in the strip 0≤Re(s)≤1' in the Dirichlet L-function section; this should refer to L(s,χ), not ζ(s).","section":"I.7"},{"comment":"The name 'Balasubramaian' should be 'Balasubramanian'. The displayed asymptotic for the zeta-moment includes the phrase 'where χ is a primitive character', but no character appears in that formula.","section":"II.4"},{"comment":"In the proof of Theorem 3.2, the notation Z is introduced for 'the term in the expression of I_i involving E_i' but the reader is not told what E_i is; the decomposition I_i = M_i + R_i + E_i is only explained by reference to Lemma 3.4. Please spell out the decomposition.","section":"III.2"},{"comment":"The notation e(x) is used (e.g., 'e(−xv/q)') without definition; standard is e(x)=e^{2πix}. Also 'zamd' should be 'z and'.","section":"III.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript reads as a student report or expository summary rather than a research article. It has no new results, and its own proof claims are not self-contained: the decisive error-term analysis is deferred to [13], and the final numerical extraction relies on quoted inequalities and sieve estimates. If the journal is a research journal, the appropriate standard would be reject, since the central claim 'we present a proof' is not met. I recommend major_revision rather than reject only because the underlying results are true and the paper could be reshaped into an honest, clearly attributed survey after substantial revision, with the proof claims either completed or removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis is not a research paper. It is a supervised expository report (November 2025) that walks through Young's short proof of Levinson's theorem and Wu's generalization for Dirichlet L-functions. There is zero new mathematics: the theorems are cited as [7], [10], [13], and the report itself concedes stronger bounds (0.5865 and 0.60261) in its own conclusion. Don't send it to a research venue expecting a new result.\n\nWhat it does well: it is honest and readable. The structure is sensible—preliminaries, Young's proof, then Wu's theorem and the parameter choices that yield κ>0.4172 and κ*>0.4074. The author clearly understands the shape of the method, and he attributes every substantive theorem to its source. That attribution is not perfunctory; it is the backbone of the report.\n\nThe soft spots are exactly where you'd expect. The decisive input for Theorem 3.1 is Theorem 3.2(B)—the twisted-moment asymptotic with θ<4/7 under the special coefficient shape. But the error term is not proved. Section III.2 says 'After some long calculation... we prove Z≪ ...' and then immediately says the proof is lengthy and given in [13]. So the report's proof of Theorem 3.1 is not self-contained. The Levinson inequality in §III.3 and the sieve evaluation of Σ(α,β) are also quoted without proof. The same pattern appears in Section 2, where the key 'Result' behind Lemma 2.3 is stated without proof. None of this disputes the published results; it just means this manuscript cannot be certified as a proof of the theorems it presents. The stress-test note lands.\n\nWho gets value from it? A student or a seminar group wanting a digest of Young and Wu, with the caution that the hard analytic parts are skipped and referenced rather than derived. For that purpose it is a reasonable starting point. But it is not a paper to cite, and it should not go through research peer review as a claim of new work.\n\nMy recommendation: desk-reject as a research preprint; if the author wants it to serve as an expository note, the gaps should be clearly labeled as 'proof sketches, details in [13]' and the venue should be a teaching-oriented outlet, not a research journal.","headline":"An honest, clearly written student report that reproduces known theorems by Young and Wu, with no new mathematics and a load-bearing error-term estimate deferred to Wu's paper.","tokens_in":15635,"tokens_out":2394,"would_cite":false,"duration_ms":23407,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"At least one-third of zeta's non-trivial zeros, and more than 41.72% of Dirichlet L-function zeros, lie on the critical line; more than 40.74% of non-trivial zeros are simple and on the line.","keywords":["Levinson's theorem","Levinson's method","Riemann zeta function","Dirichlet L-functions","mollified second moment","simple zeros","critical line","twisted mean square"],"falsifier":"Two checks settle it. (1) Reproduce the deferred estimate of §III.2 — Z ≪_ε y^(7/8)T^(−1/2+11η/2+ε) + y^(7/4)T^(−1+11η/2+ε) — and confirm it holds uniformly for θ < 4/7 with a(n) = μ(n)(F0 + F1·(F2*F3))(n); any θ ≥ 4/7 failure would break Theorem 3.1. (2) Recompute the numerical step: insert the listed polynomials P1, P2, P, Q into the formula for c(P, Q, R, θ) and verify κ ≥ 1 − (1/R) log c gives 0.4172 (and 0.4074 for the linear Q with R = 1.116); a smaller result would refute the constants.","tokens_in":14379,"feed_emoji":"🎯","tokens_out":32568,"duration_ms":227413,"temperature":0.7,"pith_summary":"This report sets out to show that Levinson's method — converting a mean value of an L-function against a short Dirichlet polynomial, called a mollifier, into a lower-bound count of zeros on the critical line — proves two positive-proportion theorems: at least one-third of the non-trivial zeros (those in the critical strip) of the Riemann zeta function lie on Re(s) = 1/2, and for any Dirichlet L-function with conductor growing slowly relative to height, more than 41.72% of non-trivial zeros lie on the critical line, with more than 40.74% of them both simple and on the line. The gain over the one-third bound comes from lengthening the mollifier to T^(4/7−ε), the longest range for which the needed error-term estimate is claimed, then optimizing auxiliary polynomials; longer mollifiers detect more zeros. These are unconditional statements toward the Riemann and generalized Riemann hypotheses — no unproved hypothesis is assumed. The report follows a 2010 short proof of Levinson's theorem and a 2018 generalization for Dirichlet L-functions, with the heaviest estimates cited from the literature rather than re-derived.","feed_headline":"Over 41% of Dirichlet L-function zeros lie on the critical line","feed_subtitle":"The same Levinson method puts a third of zeta's zeros on the line — an unconditional step toward the Riemann hypothesis.","key_machinery":"The load-bearing object is the mollified second moment I_R(Q, χ) = ∫_T^(2T) |V(1/2 + it + R/L_χ, χ)|² |B(1/2 + it, χ)|² dt, where B(s, χ) = Σ_(n≤y) χ(n)a(n)n^(−s) is the mollifier (a short Dirichlet polynomial of length y = T^θ) and V is the L-function acted on by Q(−(1/L_χ)d/ds). Levinson's inequality, κ(χ) ≥ 1 − (1/R) log(T^(−1)I_R(Q, χ)) + o(1), converts an asymptotic for this moment into a zero-proportion bound. The essential input is Theorem 3.2, which evaluates I_R(Q, χ) up to an error T^(1−ε0), ε0 > 0, for θ < 4/7 when the coefficients have the form a(n) = μ(n)(F0 + F1·(F2*F3))(n) with separable factors F_i; this longer-mollifier regime is what pushes the proportion past two-fifths, a","core_discovery":"Theorem 3.1 asserts that for any Dirichlet character χ, with log q = o(log T), the proportion κ(χ) of non-trivial zeros of L(s, χ) on the critical line exceeds 0.4172 for large T, and the proportion κ*(χ) of zeros both on the line and simple exceeds 0.4074. This generalizes the 1989 two-fifths result for zeta, and the report also proves Levinson's theorem: at least one-third of zeta's non-trivial zeros lie on the line. Both rest on one mechanism: Theorem 3.2, an asymptotic for the mollified second moment with error T^(1−ε0) (ε0 > 0) valid for mollifier length θ = 4/7 − ε under a special coefficient shape, converted by Levinson's inequality into a zero count. The longer mollifier — beyond Lev","pith_inferences":["Editorial inference: the proof as written is a condensation — §III.2 states that the decisive error estimate 'is quite lengthy and is given in more details in [13]', and the sieve evaluation of Σ(α, β) is quoted without proof — so a reader seeking the constants 0.4172/0.4074 fully verified from this report alone must supply those deferred arguments.","Editorial inference: the report's own conclusion cites later results pushing the Dirichlet bounds well above two-fifths (roughly 0.59 for critical zeros, over 0.60 for simple critical zeros), so the report is best understood as an expository reconstruction of the longer-mollifier mechanism and its numerical optimization rather than a new record.","Editorial inference (testable extension): the constants depend on a numerical optimization of P1, P2, P, Q at fixed θ = 4/7 − ε; re-running that optimization with higher-degree polynomials or a swept range of R would reveal how much headroom the same error-term range still contains, without any new analytic input.","Editorial inference: because the report's preliminaries set up the method for general L-functions (approximate functional equation, completed L-function), the same two-step recipe — a twisted-moment asymptotic plus Levinson's inequality — would give an analogous zero-proportion statement for any other L-function family whose moment asymptotic can be established."],"forward_implications":["Unconditional positive proportions: at least one-third of zeta's non-trivial zeros, more than 41.72% of Dirichlet L-function zeros, and more than 40.74% of zeros simple and on the line — all without assuming the Riemann hypothesis or any unproved input.","The same template transfers: each historical extension of the admissible mollifier length (the report surveys θ < 17/33 and θ < 6/11, each raising the zeta proportion) goes through the same Theorem-3.2-plus-Levinson-inequality route, so any future error-term improvement directly upgrades the constants.","The numerical step is explicit and reproducible: with θ = 4/7 − ε, R = 1.3 and the listed Q, P1, P2, P, the formula for c(P, Q, R, θ) gives κ > 0.4172; with the linear Q and R = 1.116 it gives κ* > 0.4074.","Simplicity comes from the same inequality: when Q is a linear polynomial, Levinson's inequality bounds the simple-and-critical proportion κ*(χ), so the simplicity statement is a by-product of the same moment evaluation, not a separate argument."],"fun_headline_variants":["Dirichlet L-function zeros: >41% on the critical line","Levinson's method: >41% of Dirichlet L-function zeros on line","Generalized Levinson beats two-fifths for Dirichlet L-functions","Simple zeros included: >40% of Dirichlet L-zeros on critical line","Same technique, new bound: >41% for Dirichlet L-function zeros"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the error term in the mollified second-moment asymptotic (Theorem 3.2) is genuinely small — T^(1−ε0) with ε0 > 0 — all the way to mollifier length θ = 4/7 − ε for the special coefficient shape a(n) = μ(n)(F0 + F1·(F2*F3))(n); the report defers this estimate to the cited literature, and if that bound's range or exponent is wrong, the constants 0.4172 and 0.4074 do not follow from anything shown here.","fun_headline_variants_meta":{"raw":{"variants":["Dirichlet L-function zeros: >41% on the critical line","Levinson's method: >41% of Dirichlet L-function zeros on line","Generalized Levinson beats two-fifths for Dirichlet L-functions","Simple zeros included: >40% of Dirichlet L-zeros on critical line","Same technique, new bound: >41% for Dirichlet L-function zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000321,"raw_usage":{"total_tokens":1650,"prompt_tokens":754,"completion_tokens":896,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":795}},"tokens_in":498,"tokens_out":896,"duration_ms":7205,"temperature":1.0,"reasoning_tokens":795,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:22:25.521511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Two checks settle it. (1) Reproduce the deferred estimate of §III.2 — Z ≪_ε y^(7/8)T^(−1/2+11η/2+ε) + y^(7/4)T^(−1+11η/2+ε) — and confirm it holds uniformly for θ < 4/7 with a(n) = μ(n)(F0 + F1·(F2*F3))(n); any θ ≥ 4/7 failure would break Theorem 3.1. (2) Recompute the numerical step: insert the listed polynomials P1, P2, P, Q into the formula for c(P, Q, R, θ) and verify κ ≥ 1 − (1/R) log c gives 0.4172 (and 0.4074 for the linear Q with R = 1.116); a smaller result would refute the constants.","supporting_citations":[],"review_version":1}