{"id":"6145c305-cfd6-48ca-863e-83b88a6d393e","arxiv_id":"2607.04632","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Guth–Maynard’s 2024 zero-density estimate improves Ingham’s 1940 bound for σ ≤ 3/4 and implies the prime-number theorem holds in intervals of length x^{17/30}.","lead":"This expository paper surveys the history of zero-density estimates for the Riemann zeta function and explains the 2024 Guth–Maynard improvement, the first progress near the critical line since Ingham in 1940. It shows how the new bound yields the prime-number theorem in shorter intervals than previously known.","discovery_kind":"review","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly identifies that the survey’s only non-classical ingredient is the Guth–Maynard large-values estimate, which is already published in a top journal and is merely sketched here. No internal inconsistency, hidden assumption, or unsupported numerical claim appears in the manuscript. The concrete verification step above is therefore only a routine citation check, not a potential falsifier. Consequently the reader’s ACCEPT / high-confidence assessment stands without modification.","tokens_in":18252,"tokens_out":383,"duration_ms":3621,"concrete_test":"Cross-check the three displayed exponents in Theorem 7.4 and the resulting density exponent 15(1-σ)/(3+5σ) against the corresponding statements in Guth–Maynard, Ann. of Math. (2) 203 (2026), 623–675 (or the arXiv version [GM24]); if they match exactly, the survey’s central claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper is an accurate, carefully written survey of classical zero-density estimates and of the already-published Guth–Maynard large-values theorem. Its strongest claim is simply a correct restatement of that theorem and of the standard implication (Theorem 4.4) that yields primes in short intervals of length x^{17/30}. The only potential soft spot is the high-level sketch of the new large-values estimate in §7.3, but the paper never pretends to supply a self-contained proof; it explicitly defers to the Annals paper [GM26]. Because the survey’s own claims are modest, well-cited, and free of original technical assertions that could fail, there is no load-bearing concern that would alter the reader’s ACCEPT verdict.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"This expository paper surveys the history of zero-density estimates for the Riemann zeta-function and their consequences for primes in short intervals, culminating in the 2024/2026 Guth–Maynard theorem. After recalling the explicit formula and the classical zero-free regions, it explains why zero-density estimates are needed for short-interval prime-number theorems, reconstructs the Bohr–Landau, Ingham, Montgomery and Huxley arguments (via Littlewood’s lemma and via zero-detecting Dirichlet polynomials), and states the new large-values estimate of Guth–Maynard. The resulting bound N(σ,T)≪T^{15(1-σ)/(3+5σ)+ε} improves Ingham’s 1940 exponent for σ≤3/4 and, via the standard implication recorded as Theorem 4.4, yields the prime-number theorem in intervals of length x^{17/30} (and almost all intervals of length x^{2/15}).","tokens_in":18399,"tokens_out":744,"duration_ms":6316,"significance":"The paper supplies a clear, carefully referenced account of an 80-year-old barrier that has just been broken. By placing the Guth–Maynard large-values estimate in the classical lineage of zero-density methods and by spelling out the immediate arithmetic consequences, it makes a major recent advance accessible to a broad analytic-number-theory audience. The historical reconstructions are standard and accurate; the only original technical content is the high-level sketch of the new large-values argument, which correctly identifies the role of additive energy and the range of N for which the new estimate is decisive. No machine-checked proofs or code are claimed, but the exposition itself is a valuable service.","major_comments":[],"minor_comments":[{"comment":"In the abstract and introduction the announcement is dated 2024 and the published version 2026; a single consistent citation style (e.g., always [GM26] after first mention) would avoid any momentary confusion for readers who have not yet seen the Annals paper.","section":null},{"comment":"Section 7.3 sketches the Guth–Maynard large-values estimate and the resulting zero-density bound, but the precise range of N for which the new estimate is applied is given only in prose. Adding a short displayed inequality (as the authors do for the classical Montgomery–Halász–Huxley estimate) would make the comparison with earlier work more immediate.","section":null},{"comment":"Figures 2 and 3 are helpful, yet the vertical axis label “upper bound on A(σ)” is slightly ambiguous; a parenthetical “(i.e., the exponent of T^{A(σ)(1-σ)})” would remove any residual doubt.","section":null},{"comment":"A few minor typographical inconsistencies appear (e.g., “Korbov” for Korobov in the introduction, occasional missing spaces around ≪). These are easily corrected in proof.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a pure survey of already-published theorems; its claims are modest and correctly attributed. I see no novelty or priority issues. The journal’s usual standards for high-quality expositions are met, and I recommend acceptance with only the lightest copy-editing."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is pure exposition. Turnage-Butterbaugh does not claim a new bound or method; she walks through the classical zero-density story (Bohr–Landau, Ingham, Montgomery–Huxley) and then places the 2024/2026 Guth–Maynard large-values estimate for Dirichlet polynomials in that line. The payoff is correctly stated: N(σ,T) ≪ T^{15(1−σ)/(3+5σ)+ε} improves Ingham for σ ≤ 3/4 and, via the standard implication, gives the prime-number theorem in intervals of length x^{17/30} (and almost all of length x^{2/15}).\n\nWhat the paper does well is the scaffolding. The sketches of Littlewood’s lemma, the zero-detecting polynomials, and the passage from large values of Dirichlet polynomials to Type-I zeros are standard but carefully referenced and readable. The two figures that track the successive upper bounds on A(σ) make the 84-year gap and the new improvement visually immediate. Citation pattern is clean; the Annals paper is the sole source for the new estimate, and the author is explicit that §7.3 is only a high-level sketch.\n\nThe only soft spot is exactly that sketch: it correctly identifies the additive-energy idea and the matrix-norm reformulation, but anyone who wants the actual proof still has to open Guth–Maynard. That is not a flaw for an exposition; it is the genre. No circularity, no invented claims, no free parameters.\n\nThis is for analytic number theorists who want a single, reliable account of why the short-interval exponent moved after half a century, and for graduate students who need the historical thread before diving into the Annals paper. It deserves a serious referee for an expository journal or a proceedings volume. I would bring it to reading group if we are covering recent advances in zero-density estimates, and I would cite the survey for the clean statement of the implication to primes in short intervals.","headline":"Clean, accurate survey of the Guth–Maynard zero-density advance; no new theorems, but a useful map of the landscape after 80 years of stasis.","tokens_in":18982,"tokens_out":515,"would_cite":true,"duration_ms":5347,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N05"],"pacs":[],"model":"grok-4.5","headline":"Guth and Maynard improve Ingham's 1940 zero-density bound for the first time in over 80 years, yielding the prime-number theorem in shorter intervals.","keywords":["zero-density estimates","Riemann zeta-function","Dirichlet polynomials","large values","primes in short intervals","prime-number theorem","critical strip"],"falsifier":"An independent verification (or counter-example) of the large-values inequality R ≪ T^{o(1)}(N^{2}V^{-2} + N^{18/5}V^{-4} + T N^{12/5}V^{-4}) for Dirichlet polynomials of length N with coefficients of size at most 1, evaluated at well-spaced points up to height T.","tokens_in":19159,"feed_emoji":"π","tokens_out":1107,"duration_ms":13347,"temperature":0.7,"pith_summary":"This expository paper explains that a zero-density estimate bounds how many zeros of the Riemann zeta-function can lie away from the critical line, giving quantitative evidence toward the Riemann Hypothesis and controlling primes in short intervals. It places the 2024 Guth–Maynard theorem in the long history of such estimates, showing that their new large-values bound for Dirichlet polynomials is the first improvement on Ingham's 1940 exponent in the range of real parts up to 3/4. The resulting density theorem N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} is then combined with classical zero-free regions to prove that the prime-number theorem holds in every interval of length x^{17/30} and in almost every interval of length x^{2/15}. A sympathetic reader cares because these are the first substantial advances on both fronts in half a century or more, turning an 84-year-old analytic bottleneck into concrete shorter intervals that contain the expected number of primes.","feed_headline":"First 80-year advance on zero-density bounds shortens prime intervals","feed_subtitle":"Guth–Maynard cut the short-interval length for the prime-number theorem to x^{17/30}","key_machinery":"The Guth–Maynard large-values estimate for Dirichlet polynomials: if a length-N polynomial takes values at least V at R well-spaced frequencies, then R is bounded by T^{o(1)}(N^{2}V^{-2} + N^{18/5}V^{-4} + T N^{12/5}V^{-4}); this replaces older Montgomery–Halász–Huxley bounds and directly controls the number of Type-I zeros.","core_discovery":"Guth and Maynard prove a new large-values estimate for Dirichlet polynomials that yields the zero-density bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} uniformly for 1/2 ≤ σ ≤ 1; this is the first improvement on Ingham's 1940 estimate throughout the range σ ≤ 3/4 and, when combined with earlier results, gives A(σ) < 30/13, which in turn implies the prime-number theorem in short intervals of length x^{17/30} and almost all intervals of length x^{2/15}.","pith_inferences":["The same large-values method is likely to give parallel improvements for Dirichlet L-functions and hence for primes in arithmetic progressions of short length.","Once the additive-energy analysis is fully optimized, the critical exponent 15/(3+5σ) may be lowered further without new ideas about the zeta function itself.","The result supplies a concrete numerical target: any future zero-density theorem that beats 15(1-σ)/(3+5σ) in the range 1/2 ≤ σ ≤ 3/4 would immediately shorten the 17/30 exponent for short-interval primes."],"forward_implications":["The prime-number theorem holds for every sufficiently large x in the interval (x, x + x^{17/30}].","The prime-number theorem holds for almost every x in intervals of length x^{2/15}.","The best uniform exponent A(σ) drops below 30/13 ≈ 2.308 across the whole critical strip.","Zero-density estimates for σ near 7/10 become strong enough to improve several classical applications that previously relied on Huxley's 1972 bound."],"fun_headline_variants":["Guth–Maynard first improve Ingham zero-density bounds in 80 years","New zero-density estimate shortens prime intervals to x^{17/30}","80-year advance on zero-density yields primes in x^{17/30} intervals","Guth-Maynard bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} for primes short intervals","Zero-density progress after 1940 cuts almost-all prime intervals to x^{2/15}"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The survey's claimed density exponent rests on accepting that Guth and Maynard's new large-values bound for Dirichlet polynomials holds with the stated powers; the paper only sketches the argument and refers the full proof elsewhere.","fun_headline_variants_meta":{"raw":{"variants":["Guth–Maynard first improve Ingham zero-density bounds in 80 years","New zero-density estimate shortens prime intervals to x^{17/30}","80-year advance on zero-density yields primes in x^{17/30} intervals","Guth-Maynard bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} for primes short intervals","Zero-density progress after 1940 cuts almost-all prime intervals to x^{2/15}"]},"model":"grok-4.5","effort":"low","cost_usd":0.004606,"raw_usage":{"total_tokens":1315,"prompt_tokens":725,"num_sources_used":0,"completion_tokens":132,"cost_in_usd_ticks":46060000,"prompt_tokens_details":{"text_tokens":725,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":458,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":725,"tokens_out":132,"duration_ms":3975,"temperature":1.0,"reasoning_tokens":458,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T16:04:41.560853+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An independent verification (or counter-example) of the large-values inequality R ≪ T^{o(1)}(N^{2}V^{-2} + N^{18/5}V^{-4} + T N^{12/5}V^{-4}) for Dirichlet polynomials of length N with coefficients of size at most 1, evaluated at well-spaced points up to height T.","supporting_citations":[],"review_version":1}