{"id":"f5b22966-1abf-46d1-8a47-3be85c7e607c","arxiv_id":"2607.14515","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH plus new pair-correlation hypotheses on Dirichlet L-function zeros, n(q) ≤ (log q)^{1+ε} and p(k) ≤ φ(k) exp(B(log k)^A) with A∈(1/2,1).","lead":"This paper proves that if two strong, unproved conjectures about the spacing of zeros of Dirichlet L-functions are true, then the smallest quadratic non-residue is at most (log q)^{1+ε} and the least prime in a residue class is nearly φ(k)(log k)^{o(1)}. The bounds are conditional on assumptions about zero pair-correlation, so the real content is a web of new implications between conjectures rather than unconditional progress.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's dyadic tail is not controlled for k near the claimed upper range, so Theorem 5's φ(k) bound is unsupported.","rationale":"The reader's verdict focused on Hypothesis 1 and the short-interval PNT step in Lemma 6. Those are legitimate concerns, but the more serious problem is internal to the proof of Theorem 4. The dyadic decomposition in (54) has a tail that is not controlled for moduli near the maximum allowed size. This is not a matter of an unproved conjecture or an extended uniformity range; it is a concrete inequality failure in the stated proof. The tail term x/(2^{J+1}φ(k)) is not bounded by the claimed √x exp(c_1(log x)^{c_2})/φ(k) when J is small. Because Theorem 5 is derived directly from Theorem 4, the headline result for the least prime in an arithmetic progression is not established. I am not objecting to the conditional nature of the paper—conditional results can be valuable—but the proof as written has a demonstrable gap in the range it claims. The same flaw also explains why the φ(k) factor in Theorem 5 is suspect: the correct algebra gives a k factor, not φ(k), unless additional arguments are supplied. This warrants rejection of the paper's central claims in their current form.","tokens_in":18001,"tokens_out":42228,"duration_ms":388304,"concrete_test":"Set B = c_1(log x)^{c_2} and k = x e^{−2B}. In the proof of Theorem 4, take J = 0 and compare the tail x/(2^{J+1}φ(k)) = x/(2φ(k)) ≈ e^{2B}/2 with the claimed error C√x e^B/φ(k) = C e^{3B}/√x. For large x, e^{2B}/2 is much larger than C e^{3B}/√x, so equation (54) fails to establish the theorem's uniformity for this admissible k. This single numerical check settles whether the tail estimate in the central dyadic argument is valid.","verdict_should_be":"REJECT","load_bearing_attack":"The main load-bearing concern is in the proof of Theorem 4, specifically equation (54). The proof defines J by k ≤ (x/2^J) exp(−2c_1(log(x/2^J))^{c_2}) and then bounds the tail ψ(x/2^{J+1}; k, a) − x/(2^{J+1}φ(k)) by x/(2^{J+1}φ(k)), absorbing it into the claimed O(√x exp(c_1(log x)^{c_2})/φ(k)). But this absorption fails exactly at the upper end of the stated k-range. Let B = c_1(log x)^{c_2} and take k = x e^{−2B}. Then J = 0, and the tail is approximately x/(2φ(k)) ≈ e^{2B}/2. The claimed bound is √x e^B/φ(k) ≈ e^{3B}/√x. Since B = o(log x), e^{2B} ≫ e^{3B}/√x, so the tail is not bounded by the claimed error. This invalidates Theorem 4 as stated. The same problem propagates to Theorem 5: the positivity condition x/φ(k) − D√x e^B/φ(k) > 0 gives x ≫ e^{2B}, not x ≫ φ(k)e^{2B}, while the hypothesis range forces x ≫ k e^{2B}; hence the conclusion should at best be p(k) ≪ k exp(B(log k)^A), and the claimed φ(k) version is not justified. Additionally, for k = x e^{−2B}, the asserted error is o(1), which is incompatible with the elementary discreteness of ψ(x;k,a) across residue classes.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two conditional improvements under pair-correlation hypotheses for Dirichlet L-functions. Under RH for a quadratic character plus a postulated bound (Hypothesis 1) for F^+_{χ_q}(x,T) in the range x^ε ≤ T < x, Theorem 3 claims n(q) ≪ (log q)^{1+ε}. Under GRH plus a second pair-correlation hypothesis (Hypothesis 2), Theorems 4 and 5 claim a uniform error bound ψ(x;k,a) − x/φ(k) ≪ √x exp(c_1(log x)^{c_2})/φ(k) for k ≤ x exp(−2c_1(log x)^{c_2}), and consequently p(k) ≪ φ(k) exp(2c_1(log k)^{c_2}) with c_2 ∈ (1/2,1). The paper also proves Theorem 1, an upper bound for F^+ in the range x ≤ T ≤ e^x, and Theorem 2, an asymptotic in a restricted range.","tokens_in":18472,"tokens_out":36329,"duration_ms":323360,"significance":"If the conditional results were correct, they would constitute notable improvements over the classical GRH bounds (Ankeny, Bach–Sorenson, Lamzouri–Li–Soundararajan), and the proposed connection between zero pair-correlation and these two classical sieving problems is conceptually attractive. The main conditional theorem for quadratic non-residues (Theorem 3) appears internally coherent and is a genuine conditional implication, although it rests on the very strong, unproved Hypothesis 1. However, the second half of the paper, Theorems 4 and 5, is not merely missing a detail: Theorem 4 contradicts the Friedlander–Granville limitation quoted by the authors themselves, and the proof of Theorem 5 inherits the failure. The advertised bound p(k) ≪ φ(k) exp((log k)^A) is therefore unsupported and likely false as stated. The paper does not provide machine-checked proofs or reproducible code; its strengths are the clear structure and the standard explicit-formula framework.","major_comments":[{"comment":"The tail term ψ(x/2^{J+1};k,a) − x/(2^{J+1}φ(k)) is bounded by (log x)^2 + x/(2^{J+1}φ(k)), and this is then absorbed into the claimed √x exp(c_1(log x)^{c_2})/φ(k). This absorption fails at the upper end of the stated k-range. Let B = c_1(log x)^{c_2} and k = x e^{−2B}. Then J=0 and the tail main-term contribution is at least x/(2φ(k)) ≥ e^{2B}/2, while the claimed error is e^{3B}/√x = o(1). Thus the displayed inequality in (54) is false for this k. This is not a minor technicality: applying the Friedlander–Granville limitation (6) with y=x/2 and the same k gives, for some a, an error ≫ exp(2(1−c_2)B), which also exceeds e^{3B}/√x. Hence Theorem 4 as stated is false.","section":"§5, Theorem 4, Eq. (54)"},{"comment":"The proof of Theorem 5 relies entirely on Theorem 4, so it inherits the failure described above. Independently, the positivity step in the proof is written as if x/φ(k) − D√x e^B/φ(k) > 0 implies x ≫ φ(k)e^{2B}; the direct algebra gives x ≫ e^{2B}. The factor φ(k) has to be imported from the theorem's hypothesis range k ≤ x e^{−2B}. With the correct constraint, the best conclusion one could hope for from the given method is p(k) ≪ k exp(B(log k)^A), not the claimed φ(k) version. Since Theorem 4 is false, Theorem 5 is unsupported.","section":"§5, Theorem 5"},{"comment":"The proof of Lemma 6 bounds E_1 by Tδx using the estimate ψ(n+10^4δx)−ψ(n) ≈ 10^4δx, described as following from the prime number theorem. With δ = √(S(x)/(Tx)) and T ≥ x, the interval length is δx ≪ √(log x). No form of the prime number theorem available under the paper's assumptions gives an asymptotic for ψ in intervals of length o(√n) uniformly for all n ≤ x; the usual error terms, whether unconditional or under RH, are far larger than the interval length. This step is unjustified, so the proofs of Lemma 6, Theorem 1, and Theorem 2 are incomplete. Theorem 3 itself does not use Theorem 1, but this gap affects a stated theorem of the paper.","section":"§3, Lemma 6, Eq. (32)"}],"minor_comments":[{"comment":"The statement of Lemma 11 has a typo: the maximum should be over t, i.e. max_{U≤t≤T} F^+_k(x,t), not F^+_k(x,T).","section":"§1.2, Lemma 11"},{"comment":"The line 'Let k > xε' should read 'Let k > x^ε'.","section":"§4, Lemma 9"},{"comment":"Remark 4 refers to Lemma 13, which is introduced later in Section 5. This is a minor organizational issue.","section":"§4, Remark 4"},{"comment":"The abstract and introduction state that the paper 'surpasses' classical GRH bounds. Since the results are conditional on additional unproved Hypotheses 1 and 2, the wording should be qualified (e.g., 'conditional on a pair-correlation hypothesis').","section":"Abstract / Introduction"}],"recommendation":"reject","confidential_remarks":"The most serious issue is that Theorem 4 contradicts the paper's own citation of the Friedlander–Granville limitation (6). The authors appear to have checked the k-uniformity range against (6) but not the size of the error term. The false step in (54) is load-bearing for the second half of the paper. Theorem 3 is a valid conditional result, but it rests on Hypothesis 1, which is an unproved extension of Theorem 1 into T < x. Overall, the manuscript's main new claims for primes in arithmetic progressions cannot stand as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Conditional improvements are plausible in spirit, but the proof of the least-prime theorem has a real gap in the dyadic tail, and the base pair-correlation bound leans on an unjustified short-interval prime estimate. I would not take Theorems 4 and 5 as established as stated.\n\nThe genuinely new part is the application of Montgomery-type pair-correlation to the least quadratic non-residue, and the clean conditional implications: GRH plus Hypothesis 1 gives n(q) << (log q)^{1+ε}, which is the best known under that kind of assumption. Theorem 3's derivation through explicit formulas and dyadic decomposition is coherent, and the paper carefully acknowledges Granville's obstruction about the limits of the k-uniformity range. That honesty is appreciated.\n\nThe soft spots are not cosmetic. First, in Lemma 6, the estimate in Eq. (32) claims a short-interval PNT bound ψ(n+10^4 δx)-ψ(n) ≪ δx. But for the ranges used, δ is about √(log x)/√(T x), and since T ≥ x, the interval length 10^4 δx is of order √(log x). That is far too short for the ordinary PNT to give the claimed main term; under RH the error √x log^2 x dominates. The step is unjustified and it undercuts the proof of Theorem 1, the base result behind Hypothesis 1.\n\nSecond, the dyadic tail in Theorem 4 is not controlled. In (54), the final term ψ(x/2^{J+1}) - x/(2^{J+1}φ(k)) is absorbed into the claimed √x exp(...)/φ(k) error. At the top of the k-range, J=0, and that tail is roughly x/(2k) ≈ exp(2c1(log x)^c2), while the claimed error is about exp(3c1(log x)^c2)/√x, which is smaller. So the absorption fails. This invalidates Theorem 4 as stated, and Theorem 5's φ(k) bound inherits the problem. At best the method seems to give p(k) ≪ k exp(B(log k)^A), not the sharper φ(k) form.\n\nWhat is solid: the conditional implications are logically structured, Lemma 9 and 13 use standard dyadic and Cauchy-Schwarz machinery, and I don't see circularity in the main hypotheses. But the two technical gaps are load-bearing. The paper is worth reading for the conceptual link and for the explicit statements, but it needs substantial revision before the main claims can be trusted. I would send it to a serious referee rather than desk reject, since the flaws are concrete and the approach is worth engaging.","headline":"Original conditional results, but the proof of the main theorems has two concrete gaps that need fixing before the claims can be trusted.","tokens_in":18949,"tokens_out":8554,"would_cite":false,"duration_ms":81808,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N13","11L40","11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming a vertical zero-spacing hypothesis for Dirichlet L-functions, the paper pushes the least quadratic non-residue down to (log q)^{1+ε} and the least prime in an arithmetic progression down to φ(k) times a subexponential factor.","keywords":["least quadratic non-residue","primes in arithmetic progressions","pair correlation","Dirichlet L-functions","generalized Riemann hypothesis","zero spacing","Linnik's constant","Montgomery pair-correlation conjecture"],"falsifier":"Compute F^+_{χ_□}(x,T) for a fixed small prime modulus (say q ≈ 10^6) at heights in the gap between x^ε and x; if for some T in that range F^+/(T log(qx)) exceeds a constant, then Hypothesis 1 fails and Theorem 3 has no content. Also, deriving a contradiction from the short-interval prime number theorem used at Eq. (32) would invalidate Theorem 1's uniformity range.","tokens_in":17850,"feed_emoji":"🔢","tokens_out":5944,"duration_ms":48098,"temperature":0.7,"pith_summary":"The paper asks whether the vertical spacing—not just the horizontal location—of the nontrivial zeros of Dirichlet L-functions can improve the classical conditional bounds for two famous problems. Its central claim is that if a natural pair-correlation bound F^+ ≪ T log(kx) holds down to heights T as small as x^ε (Hypothesis 1), then the least quadratic non-residue modulo a prime q satisfies n(q) ≪ (log q)^{1+ε}, beating the long-standing GRH bound (log q)^2. Symmetrically, if a related pair-correlation estimate holds (Hypothesis 2), then the least prime in an arithmetic progression satisfies p(k) ≪ φ(k) exp(B (log k)^A) for some A ∈ (1/2,1), improving on the previous k^{1+ε} under the same framework. Both results are conditional on the Generalized Riemann Hypothesis and on pair-correlation hypotheses that the paper formulates but does not prove; the unconditional part of the paper is a Montgomery-type bound for F^+ in the range x ≤ T ≤ e^x. A sympathetic reader will see the paper as establishing a bridge: vertical zero repulsion, if confirmed, converts a spectral regularity into sharp arithmetical bounds.","feed_headline":"Pair correlation trims least quadratic non-residue to (log q)^(1+ε)","feed_subtitle":"Under RH plus a zero-spacing hypothesis, the classical (log q)^2 bound drops below (log q)^(1+ε).","key_machinery":"The load-bearing object is the pair-correlation sum F^+_{χ_□}(x,T) (and its full-sum analogue F_k(x,T) for characters mod k), which measures how zeros repel: it is essentially the Fourier transform of a quadratic character's zero-count, weighted by W(u)=4/(4+u²). Montgomery's 1973 argument gives a bound in the range T ≥ x; the paper's innovation is to treat the short-T range as a hypothesis and push it through an explicit-formula dyadic argument, converting the spectral input into bounds for n(q) and p(k). The Sobolev–Gallagher inequality and a Goldston–Montgomery Fourier-series lemma are the technical tools that let the authors pass from the pair-correlation bound to estimates on weighted r","core_discovery":"On the paper's own terms, the central discovery is that the pair-correlation function F^+_{χ_□}(x,T), which sums x^{i(γ_1−γ_2)} over pairs of zeros of the quadratic character L-function weighted by W(γ_1−γ_2)=4/(4+(γ_1−γ_2)^2), satisfies F^+ ≪ T log(kx) for x ≤ T ≤ e^x (Theorem 1) and, under GRH, has the Montgomery-type asymptotic F^+ ∼ (T/2π) log x (Theorem 2). The paper then postulates (Hypothesis 1) that the same bound persists for T as small as x^ε, and uses that extension—via a dyadic decomposition of the explicit formula for θ(x,χ_□)—to prove n(q) ≪ (log q)^{1+ε} (Theorem 3). For primes in arithmetic progressions, the analogous Hypothesis 2 yields, under GRH, a Chebyshev-type error ψ(x","pith_inferences":["The real bottleneck is Hypothesis 1: Theorem 1 only proves F^+ ≪ T log(kx) for T ≥ x, and the entire improvement from (log q)^2 to (log q)^{1+ε} rests on the unproved extension into T ≥ x^ε. A numerical check of F^+ in that short-T range for a small prime modulus would be a cheap way to test whether the hypothesis is plausible.","A subtle gap sits inside the proof of Theorem 1: the estimate at Eq. (32), which uses a short-interval prime number theorem, is not derived from the stated RH assumption. If that step fails, the uniformity range of the base pair-correlation bound—and hence the input to Hypothesis 1—would need revisiting.","If these hypotheses are ever proved, the method would likely yield similar sharpening for other GRH-limited problems, such as the Pólya–Vinogradov bound on character sums, by the same 'vertical repulsion' mechanism.","The Friedlander–Granville limitations quoted in the paper show that the uniformity range in Theorem 4 is essentially maximal; the subexponential term cannot be replaced by a power of log x, so the paper's bounds are near the boundary of what is possible."],"forward_implications":["Under RH plus Hypothesis 1, n(q) ≤ (log q)^{1+ε} for large primes q, nearly attaining Vinogradov's conjecture n(q) ≪ q^ε.","If the T-range in Hypothesis 1 can be extended to (log x)^2, the bound refines to n(q) ≪ (log q)(log log q)^2.","Under GRH plus Hypothesis 2, p(k) ≪ φ(k) exp(B (log k)^A) with A < 1, a subexponential bound that improves on p(k) < k^{1+ε}; it remains far from Heath-Brown's conjectured k(log k)^2.","The same hypothesis yields uniform Chebyshev-error equidistribution: ψ(x;k,a) − x/φ(k) ≪ √x/φ(k) exp(c_1 (log x)^{c_2}) for k up to x exp(−2c_1(log x)^{c_2}).","Theorem 2 gives the Montgomery-type asymptotic F^+ ∼ (T/2π) log x under GRH in the range x ≤ T ≤ e^x, evidence that quadratic-character zeros behave like the zeta zeros."],"fun_headline_variants":["Pair-correlation hypothesis cuts bound for least non-residue and prime","Zero-pair spacing sharpens GRH bounds for least prime and non-residue","Pair-correlation lowers least non-residue to (log q)^(1+ε)","New pair-correlation tool beats GRH for prime and non-residue bounds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole improvement for the least quadratic non-residue rests on Hypothesis 1—that the pair-correlation bound F^+ ≪ T log(kx) continues to hold when the height T drops from x down to x^ε—while the proof only establishes this bound for T ≥ x.","fun_headline_variants_meta":{"raw":{"variants":["Pair-correlation hypothesis cuts bound for least non-residue and prime","Zero-pair spacing sharpens GRH bounds for least prime and non-residue","Pair-correlation lowers least non-residue to (log q)^(1+ε)","New pair-correlation tool beats GRH for prime and non-residue bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001436,"raw_usage":{"total_tokens":5619,"prompt_tokens":728,"completion_tokens":4891,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":4803}},"tokens_in":472,"tokens_out":4891,"duration_ms":37742,"temperature":1.0,"reasoning_tokens":4803,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T01:54:05.335734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute F^+_{χ_□}(x,T) for a fixed small prime modulus (say q ≈ 10^6) at heights in the gap between x^ε and x; if for some T in that range F^+/(T log(qx)) exceeds a constant, then Hypothesis 1 fails and Theorem 3 has no content. Also, deriving a contradiction from the short-interval prime number theorem used at Eq. (32) would invalidate Theorem 1's uniformity range.","supporting_citations":[],"review_version":1}