{"id":"21add021-b39b-47d7-9a02-b036e75e7b42","arxiv_id":"2412.15481","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, a positive proportion of zeta zeros begin r consecutive gaps of size at least 2πc/log T, for any fixed r and small enough c.","lead":"This paper asks how often the gaps between consecutive zeros of the Riemann zeta function are all at least moderately large, for several consecutive steps. Assuming standard conjectures about zero correlations, it proves that a positive proportion of zeros start such runs; it also proves an unconditional but weaker statement about intervals containing moderately large gaps, and applies it to bound a logarithmic derivative.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 2.1 uses 'PCC implies almost all zeros simple', but the stated PCC (5) counts only positive separations and does not rule out a positive density of multiple zeros; the bound |S_j|≤(f(c)+o(1))N(T) is unsupported as stated.","rationale":"The reader correctly identifies the Pair Correlation Conjecture as the load-bearing assumption for Theorem 2.1, and the reader's conditional verdict is appropriate. However, I find a more specific gap within that assumption than the reader's stated concern: the paper's proof of Theorem 2.1 requires the number of zeros with γ_{n+1}=γ_n to be o(N(T)), and it attributes this to PCC, but the PCC as stated in (5) only controls pairs with strictly positive separation. The standard, stronger test-function form of PCC does imply the needed simplicity statement, so the theorem is probably correct under the intended conjecture; but the manuscript should either state that stronger form or add an explicit all-but-o(N)-simple assumption. I also agree with the reader that the application section has unsupported steps, notably the separation and sequence assertions in the proof of Theorem 4.1, but those do not affect the main conditional theorem. Because the identified gap is repairable and does not overturn the central result, the reader's CONDITIONAL verdict remains unchanged.","tokens_in":10047,"tokens_out":23564,"duration_ms":213359,"concrete_test":"Construct a model point set satisfying Conjecture (5) for every fixed c>0 while having a positive density of coincident points: take a GUE-type sequence of N points and replace a fixed fraction δN of them by double points, rescaling so the total count remains N; verify that the strict-positive-distance pair count still normalizes to f(c)+o(1) for each c, while the number of repeated ordinates is δN. If such a model exists, Conjecture (5) alone is insufficient to justify the simplicity assertion. Then check whether the proof of Theorem 2.1 can be salvaged by replacing (5) with the full test-function form of PCC (an even Schwartz weight w with w(0)>0); under that stronger hypothesis the repeated-ordinate count is o(N) and the proof goes through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 2.1, after equation (6), the authors bound |S_j(T,c)| by the number of consecutive gaps below 2πc/log T plus the number of repeated ordinates, and they dismiss the latter as o(N(T)) with the sentence 'the Pair Correlation Conjecture implies that almost all zeros are simple.' But Conjecture (5) is an average over pairs with 0<γ_n−γ_m<2πc/log T; it has no term at distance zero. A positive density of multiple zeros contributes exactly zero to every sum in (5), while it contributes ≍N(T) to the repeated-ordinate count. If, for example, a fraction δ>0 of zeros were double zeros, then each of those zeros would have next gap 0, forcing |S_1(T,c)| ≥ δ N(T)+o(N(T)), which for sufficiently small c violates the claimed bound |S_j|≤(f(c)+o(1))N(T) because f(c)→0 as c→0+. Thus the asserted consequence does not follow from the conjecture as written. The standard test-function form of Montgomery's Pair Correlation Conjecture, with an even weight w having w(0)>0, does imply simplicity, so the theorem is likely repairable; but as stated, the central claim's proof has a gap in its load-bearing assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the existence of r consecutive 'moderate' gaps between ordinates of nontrivial zeros of the Riemann zeta function, where a gap is moderate if it is at least 2πc/log T. Under the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, Theorem 2.1 claims that a positive proportion of zeros begin r consecutive moderate gaps; Corollary 2.2 gives explicit admissible ranges of c. A similar statement is proved under a weaker Well-Spacing Hypothesis. Section 3 gives an unconditional result: for every fixed r and ε, for large fixed m and h=2πm/log T, a set T⊂(T,2T] of measure >(1-ε)T exists such that each of r consecutive intervals of length h contains a moderate gap. Section 4 applies these ideas to prove, assuming RH, a bound for ξ''/ξ' on horizontal segments along a sequence T_j→∞ with bounded gaps, filling a gap noted in the literature.","tokens_in":10318,"tokens_out":21578,"duration_ms":201116,"significance":"If the proof gaps are repaired, this is a useful contribution. The main conditional result is a clean application of the pair-correlation heuristic, and the paper is transparent about hypotheses: it explicitly conditions on standard conjectures, fits no free parameters, and derives explicit constants recorded in Table 1. Theorem 3.1 is unconditional and its proof via Fujii's second-moment bound is elegant. The application in Section 4 addresses a stated gap in an earlier paper coauthored by the first author. However, as it stands, the proof of the central conditional theorem uses a consequence of the Pair Correlation Conjecture that is not contained in the stated version, and the proof of the application contains unresolved separation and sequence-selection issues. The significance is therefore real but conditional on a careful revision.","major_comments":[{"comment":"The proof claims that the stated Pair Correlation Conjecture implies that almost all zeros are simple, but this does not follow from (5). The sum in (5) is restricted to pairs with 0<γ_n−γ_m<2πc/log T, so a positive density of multiple zeros contributes exactly zero to every such sum for every c>0. If a proportion δ>0 of zeros were double zeros, then each double zero would contribute at least one zero gap of size 0, so |S_1(T,c)|≥δ N(T)+o(N(T)), while the proof's bound would give |S_1(T,c)|≤(f(c)+o(1))N(T); since f(c)→0 as c→0+, these are incompatible for small c. The standard test-function form of the Pair Correlation Conjecture, with an even weight w having w(0)>0, does imply simplicity, so the theorem is likely repairable. Please replace (5) by such a form, or add an explicit hypothesis that all but o(N(T)) zeros are simple.","section":"§2.3, Eq. (5) and proof of Theorem 2.1"},{"comment":"The theorem asserts a sequence T_j→∞ with T_{j+1}−T_j≪1, but the proof does not justify this bounded-gap property. Theorem 3.1 only provides, for each large T, a set T⊂(T,2T] of measure >(1−ε)T. A set of large measure in a long interval need not contain points with bounded gaps: its complement could be a single interval of length εT, which is ≫1 for fixed ε. The proof says only 'there exists a set T ... Now let T_j=γ3+(logγ3)^{-C}', without explaining how the associated heights form a syndetic sequence. This can likely be repaired by showing that the chosen γ3's have positive density in [T,2T], but as written the stated conclusion is unsupported.","section":"§4, Theorem 4.1, sequence selection"},{"comment":"The estimate in Case 2 that the sum over T_j−1≤γ≤γ2 is O((log T)^2) because 'γ−T_j ≫ 1/log T_j for each term' is not justified by the hypotheses of Theorem 3.1. The zero γ2 lies in the first interval (t,t+h], while γ3 lies in the second interval (t+h,t+2h]; nothing prevents γ2 from being extremely close to γ3. If γ3−γ2 is small, then T_j−γ2=(γ3−γ2)+(log γ3)^{-C} can be as small as (log γ3)^{-C}, which is far smaller than 1/log T_j, so the omitted sum is not O((log T)^2) and equation (18) does not follow. The preceding 'without loss of generality' about γ*3 does not fix this, since T_j is fixed to γ3+δ and the alternative γ4−γ*3≥2π/(3log T) is not used in the argument. A separation argument for γ2, γ3, and γ*3, or a different choice of T_j, is needed.","section":"§4, proof of Theorem 4.1, Case 2"}],"minor_comments":[{"comment":"In the proof, the threshold is written as c < (3/π)^{4/3} r^{-1/3}; the calculation f(c)≤(π^2/9)c^3 gives c < (3/π)^{2/3} r^{-1/3}, matching the statement of the corollary. Please correct the exponent.","section":"§2.3, proof of Corollary 2.2"},{"comment":"The statement says each interval contains '< 3/2m ordinates', but the proof actually establishes the two-sided bound m/2 < M_j < 3m/2. The lower bound is needed for the average-spacing argument and should be included in the statement for clarity.","section":"§3, Theorem 3.1"},{"comment":"The intervals are written as '(t + (j − 1)h), t + jh]' with mismatched parentheses; they should be '(t+(j−1)h, t+jh]'.","section":"§3, proof of Theorem 3.1"},{"comment":"The sentence 'We prove the first assertion only as the proof of the second is similar' does not correspond to the theorem as stated, which contains a single assertion; this appears to be a vestigial remark from an earlier version and should be removed or corrected.","section":"§4, proof of Theorem 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central conditional theorem is likely correct once the Pair Correlation Conjecture is formulated in a way that controls the diagonal, and the unconditional Section 3 is solid. The main obstacles are the unsupported simplicity claim in Theorem 2.1 and the incomplete justification of the bounded-gap sequence and the separation estimates in Theorem 4.1. Each of these seems repairable within the paper's scope, so I do not recommend rejection; however, the current text is not publishable without addressing them. The bounded-gap issue in Theorem 4.1 may be fixable by a density argument on the selected γ3's, but it needs to be written out. I would also ask the authors to verify whether the bounded-gap conclusion is actually needed for the intended application, since weakening it would reduce the burden of proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"We read the Gonek–Sahay note. The short version: the main conditional theorem is a nice idea but the proof as written has a gap that is real and repairable; the unconditional result in Section 3 is the most solid part; the application in Section 4 is the least polished.\n\nWhat's new: the question of r consecutive moderate gaps hasn't been addressed before, and the decomposition N_r(T,c)=N(T)−sum|S_j| is exactly the right way to see it. The union bound over first-failure positions is simple and, once you have the right pair-correlation input, gives a positive proportion. The unconditional Theorem 3.1 is clean: using Fujii's mean-square bound, the authors show that on a set of t of density 1, each of r consecutive intervals of length 2πm/log T contains about m zeros, and therefore contains a gap of size ≫1/log T. That is a solid, self-contained contribution.\n\nThe problem: the proof of Theorem 2.1 claims that the Pair Correlation Conjecture (5) implies almost all zeros are simple. The stress-test note is correct. Equation (5) sums over pairs with 0<γ_n−γ_m<2πc/log T; a positive density of double zeros contributes nothing to that sum, yet each double zero has next gap zero and forces |S_1| to be a positive fraction of N(T). The standard Montgomery form with an even test function whose Fourier transform is positive does imply simplicity, so the theorem can be repaired by quoting the right form of the conjecture; but as written the proof's load-bearing step does not follow from the displayed conjecture. This is a genuine soft spot, not a nitpick, but it is fixable.\n\nThe application section has its own soft spots. The choice of T_j = γ3 + (log γ3)^{−C} and the claim that the zero of ξ' in the middle interval is at distance ≥2π/(3 log T) from an endpoint are asserted without proof. The estimates in Case 2 are sketched and some of the sums look like they need more care than the text gives. This is the least trustworthy part of the paper.\n\nOverall: the paper deserves a serious referee. The main result is likely correct under the standard form of PCC; the unconditional result is genuinely useful. The referee should push for a rewritten proof of Theorem 2.1 using a pair-correlation statement that actually gives simplicity, and a fuller justification of Theorem 4.1. I'd send it out.","headline":"The consecutive-moderate-gaps counting argument is new and mostly right, but the proof of Theorem 2.1 has a fixable gap in deducing simplicity from the stated Pair Correlation Conjecture; the unconditional theorem is the cleanest part.","tokens_in":10837,"tokens_out":2564,"would_cite":true,"duration_ms":23563,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06","11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under RH and pair correlation, a positive proportion of zeta zeros start r consecutive moderate gaps.","keywords":["Riemann zeta function","zeros","gaps","pair correlation conjecture","Riemann Hypothesis","moderate gaps","Xi-function","short intervals"],"falsifier":"Compute the empirical pair-counting function on high zeros: for a fixed c ≤ c_r/2 and a large height T, form the ratio N_r(T,c)/N(T) and compare it with 1 - r f(c). If for some fixed r and c it is eventually below 1 - r f(c) by a positive amount, or if the raw pair count in (5) deviates by a fixed proportion from f(c)N(T), then Theorem 2.1's conclusion is not derivable from the stated hypotheses and the conjecture behind it must be revisited.","tokens_in":9835,"feed_emoji":"","tokens_out":4800,"duration_ms":41350,"temperature":0.7,"pith_summary":"The paper asks whether the ordinates of Riemann zeta zeros can be followed by arbitrarily many consecutive gaps all of size at least a fixed multiple of the average gap. Its main conditional theorem says yes: assuming the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, for every fixed r ≥ 1 there is a c > 0 such that a positive proportion of zeros γ_n ≤ T have r consecutive gaps ≥ 2πc/log T. The proof is a short counting argument: the zeros whose first 'small' gap occurs j steps later are each bounded by f(c)N(T), where f(c) is the pair-correlation function. The paper also proves an unconditional result that, in most short intervals of length 2πm/log T, each of r consecutive subintervals contains a moderate gap, and applies this to bound ξ''/ξ' on the critical strip.","feed_headline":"Positive share of zeta zeros begin r moderate gaps","feed_subtitle":"Under two standard conjectures, fixed runs of moderately large gaps occur with positive frequency.","key_machinery":"The counting device is the partition of all zeros into sets S_j(T,c): zeros whose current and next j-1 gaps are moderate but whose (j+1)-st gap is small. These sets are disjoint, yielding the exact identity N_r(T,c) = N(T) - Σ_{j=1}^r |S_j(T,c)|. Under the Pair Correlation Conjecture each |S_j(T,c)| is at most f(c)N(T) plus an error, because every such zero has some nearby follower within 2πc/log T; the function f(c) is the conjectured limiting count of close pairs. The unconditional theorem instead uses the classical formula N(t+h)-N(t) = (h/2π) log t + S(t+h)-S(t) together with Fujii's mean-square bound on S(t+h)-S(t) to show that in most short intervals the zero count is close to its expected value, forcing a moderate gap among each subinterval's zeros.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.1: under RH and the Pair Correlation Conjecture, N_r(T,c), the number of ordinates γ_n ≤ T whose next r gaps are all at least 2πc/log T, satisfies N_r(T,c) ≥ (1 - r f(c) + o(1)) N(T), where f(c) = ∫_0^c (1 - (sin πu/(πu))^2) du. Because f(c) → 0 as c → 0+, this exceeds (1+o(1))N(T) times a positive constant for small c, so a positive proportion of zeros begin r consecutive moderate gaps. The paper gives explicit c_r values for r up to 1000, and shows unconditionally that every sufficiently long interval of the form (T,2T] contains a large set of heights t for which r consecutive subintervals of length 2πm/log T each contain a moderate gap.","pith_inferences":["If the GUE Hypothesis predicts the joint distribution of r consecutive gaps, it should give the exact limiting density of zeros with r moderate gaps; the paper's f(c) bound is only a first-order lower bound obtained from pair data alone.","The disjoint decomposition N_r = N - Σ|S_j| is very soft; it would transfer verbatim to any sequence of points whose close-pair counting function reflects a similar level repulsion, so the positive-proportion conclusion should hold for other point processes with pair-correlation control.","A numerical check on high zeros would compare the empirical proportion of zeros with r runs of moderate gaps at c = c_r/2 against 1 - r f(c); systematic agreement would support the conjecture, while a serious deficit would indicate that pair correlations alone do not capture consecutive-gap structure."],"forward_implications":["For every fixed r ≥ 1, Theorem 2.1 implies that for all sufficiently small c a positive proportion of zeros up to T are starting points of r consecutive moderate gaps.","The explicit bound f(c) ≤ (π/3)^2 c^3 for c ≤ 1/π yields a positive proportion whenever c < (3/π)^{4/3} r^{-1/3}; the table lists c_r for r = 1,...,1000.","Under the weaker Well-Spacing Hypothesis, the same argument gives N_r(T,c) ≥ (1 - r M c^δ + o(1))N(T), with a positive proportion whenever c is small enough relative to r, M, and δ.","Unconditionally, Theorem 3.1 gives heights t in a set of measure > (1-ε)T in (T,2T] such that each of r consecutive intervals (t+(j-1)h, t+jh] contains at least one pair of consecutive zeros with gap ≥ 4π/(3 log T).","Assuming RH, Theorem 4.1 supplies a sequence T_j → ∞ with T_{j+1}-T_j ≪ 1 on which ξ''/ξ'(σ+iT_j) ≪ (log T_j)^{C+1} uniformly for -1 ≤ σ ≤ 2, supporting estimates used in work on the derivative of ξ."],"supporting_citations":[{"why":"Montgomery's Pair Correlation Conjecture is the hypothesis from which the f(c) count and the theorem's lower bound are obtained.","marker":"[9]"},{"why":"Titchmarsh's N(t) formula supplies the explicit counting identity used in Theorem 3.1 and in the ξ'/ξ estimates.","marker":"[14]"},{"why":"Fujii's mean-square bound for S(t+h)-S(t) controls deviations of zero counts in short intervals, the core of Theorem 3.1.","marker":"[7]"},{"why":"Davenport's ζ'/ζ expansion is used to relate ξ'/ξ to sums over nearby zeros in Theorem 4.1.","marker":"[5]"},{"why":"Farmer-Gonek-Lee is the originating context that motivates and uses the bound on ξ''/ξ' obtained in Theorem 4.1.","marker":"[8]"},{"why":"Baluyot's theorem under the Alternative Hypothesis is the source of the stronger conclusion in Section 2.1.","marker":"[2]"}],"fun_headline_variants":["Zeta zeros: positive share start r moderate gaps","RH yields positive density of r-gap runs","Many zeta zeros begin consecutive moderate gaps","Positive fraction of zeros with r moderate gaps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Montgomery's Pair Correlation Conjecture, which asserts that the number of pairs of zeros within 2πc/log T of each other is asymptotically f(c)N(T); if this count differs, the proof's bound on each |S_j(T,c)| collapses.","fun_headline_variants_meta":{"raw":{"variants":["Zeta zeros: positive share start r moderate gaps","RH yields positive density of r-gap runs","Many zeta zeros begin consecutive moderate gaps","Positive fraction of zeros with r moderate gaps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1173,"prompt_tokens":875,"completion_tokens":298,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":240}},"tokens_in":491,"tokens_out":298,"duration_ms":2539,"temperature":1.0,"reasoning_tokens":240,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:25:25.843475+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the empirical pair-counting function on high zeros: for a fixed c ≤ c_r/2 and a large height T, form the ratio N_r(T,c)/N(T) and compare it with 1 - r f(c). If for some fixed r and c it is eventually below 1 - r f(c) by a positive amount, or if the raw pair count in (5) deviates by a fixed proportion from f(c)N(T), then Theorem 2.1's conclusion is not derivable from the stated hypotheses and the conjecture behind it must be revisited.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Montgomery's Pair Correlation Conjecture is the hypothesis from which the f(c) count and the theorem's lower bound are obtained."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Titchmarsh's N(t) formula supplies the explicit counting identity used in Theorem 3.1 and in the ξ'/ξ estimates."},{"cited_title":"Fujii, On the distribution of the zeros of the Riemann Zeta function i n short intervals , Bull","cited_arxiv_id":null,"evidence_quote":"Fujii's mean-square bound for S(t+h)-S(t) controls deviations of zero counts in short intervals, the core of Theorem 3.1."},{"cited_title":"Davenport, Multiplicative Number Theory , Graduate Texts in Math","cited_arxiv_id":null,"evidence_quote":"Davenport's ζ'/ζ expansion is used to relate ξ'/ξ to sums over nearby zeros in Theorem 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Farmer-Gonek-Lee is the originating context that motivates and uses the bound on ξ''/ξ' obtained in Theorem 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Baluyot's theorem under the Alternative Hypothesis is the source of the stronger conclusion in Section 2.1."}],"review_version":1}