{"id":"279f53a9-57f4-4734-ba74-7f55a3f9bf22","arxiv_id":"2608.07399","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Under RH, an exceptional sequence of Siegel zeros implies liminf of normalized zero gaps is below 0.4733.","lead":"Assuming the Riemann Hypothesis, the authors show that an infinite family of special L-function zeros called Siegel zeros forces the zeta function to have consecutive zeros closer than 0.4733 times the average spacing. The proof breaks a suspected 1/2-spacing barrier and rules out the strongest versions of the 'Alternative Hypothesis' under these assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (28) mis-states the character orthogonality, so Range III of Proposition 6 is not proved as written.","rationale":"The reader's weakest assumption was the external Chen–Gupta–Li zero-density estimate, but the more immediate obstacle is an internal algebraic error. The orthogonality statement in (28) is false as written: Σ_ψ ψ(u)ψ(k) selects uk≡1, not u≡k. Since χ is quadratic, the confusion is harmless when y=0, but the argument requires y=r\\bar m, which is a general nonzero residue, so the reduction does not represent the original sum. The subsequent bounds in Lemmas 9 and 10 are conjugation-invariant, so replacing ψ(k) by \\bar ψ(k) should repair the proof; this makes the error fixable rather than fatal. I also independently approximated the final ratio in (8) with the stated parameters and found the main term exceeds 1 by only a margin on the order of 10^{-6}, so the opaque computer check should be documented with high-precision arithmetic. Both issues support a CONDITIONAL verdict: the theorem is plausible and likely correct after correction, but the manuscript as written contains a definite error in a key reduction and an insufficiently documented numerical assertion.","tokens_in":16470,"tokens_out":51446,"duration_ms":394043,"concrete_test":"Enumerate the four characters modulo 5 with χ the Legendre symbol, and test (28) with k=2, y=1. Here k inverse is 3, so χ(k^{-1}+y)=χ(4)=1, while the claimed χ(k+y)=χ(3)=-1; the two sides disagree. Then replace ψ(k) by \\bar ψ(k) in (28) and re-derive the subsequent bound (40). If Lemmas 9 and 10 still apply to \\bar ψ with the same constants, the proof is repairable. If the bounds depend essentially on ψ rather than \\bar ψ, Proposition 6 is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Proposition 6, Range III (Section 6.5), the paper defines T_ψ(y)=Σ^*_{u mod q}χ(u+y)ψ(u) and claims, by orthogonality, that (1/φ(q))Σ_{ψ mod q}T_ψ(y)ψ(k)=Σ^*_{u mod q}χ(u+y)1_{u≡k}=χ(k+y). This is incorrect: the orthogonality relation for multiplicative characters gives Σ_ψ ψ(u)ψ(k)=φ(q) if uk≡1 mod q, and zero otherwise. The displayed identity should be χ(k^{-1}+y), not χ(k+y). Since χ is quadratic, the special case y=0 happens to work because χ(k^{-1})=χ(k), but in the application y=r\\bar m is not zero in general, and χ(k^{-1}+r\\bar m) does not equal χ(k+r\\bar m). Thus equation (28), which reduces E(K,M,R) to a sum over ψ of the product of an m,r-sum and Σ_k a_kΛ(k)ψ(k), represents a different character sum than the one in the original off-diagonal expression. The proof would go through if (28) used \\bar ψ(k) instead of ψ(k), and Lemmas 9 and 10 are invariant under conjugation, so the error is probably repairable; but as written the reduction in Range III is invalid. This is a concrete internal inconsistency in a load-bearing step, independent of the correctness of the cited zero-density estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proves a conditional theorem: assuming the Riemann Hypothesis and the existence of an infinite sequence of Siegel zeros whose qualities E_j are bounded below by a sufficiently large absolute constant, the normalized liminf of gaps between consecutive zeros of the zeta function is strictly less than 0.4733. The method follows the Montgomery–Odlyzko ratio I1/I0 but with a resonator of length L = T^{17/14−ε}, well beyond the traditional L ≤ T barrier. The resonator coefficients are χ(n)G_0(log n / log L) for an exceptional quadratic character χ; the Siegel-zero hypothesis enters through Heath-Brown's result that χ(p) = −1 on almost all primes. The diagonal contribution is computed explicitly, and the off-diagonal sum is split into three ranges of the prime variable k: a direct character-sum bound (Lemma 4), a Cauchy–Schwarz/second-moment argument (Lemmas 7 and 8), and a Fourier-orthogonality reduction combined with the Chen–Gupta–Li zero-density estimate (Lemmas 9 and 10). The final constant c = 0.473275 is verified numerically.","tokens_in":16632,"tokens_out":19778,"duration_ms":159393,"significance":"If completed, the result is surprising and important: it shows that the long-standing expectation that Siegel zeros impose a 1/2 barrier for small gaps is false, at least under RH and the assumed sequence of Siegel zeros. The introduction of long Dirichlet polynomials into the Montgomery–Odlyzko framework is a genuine methodological novelty, and the explicit constant and the careful three-range decomposition are strengths. The paper is also careful to explain why the result does not contradict the Conrey–Iwaniec quantitative bounds. However, the proof depends crucially on a recent preprint (Chen–Gupta–Li) for the zero-density estimate with exponent 7/3, and on a character-orthogonality step that is incorrectly stated; the latter is repairable, but both points must be settled before the result can be accepted.","major_comments":[{"comment":"The character orthogonality is mis-stated. With T_ψ(y)=Σ^*_{u mod q} χ(u+y)ψ(u), the correct identity is (1/φ(q)) Σ_ψ T_ψ(y) \\bar{ψ}(k) = Σ^*_u χ(u+y) 1_{u≡k} = χ(k+y). The displayed identity with ψ(k) instead of \\bar{ψ}(k) gives χ(k^{-1}+y), which is not equal to χ(k+y) when y=r\\bar{m} is nonzero. Since the application in Range III has y=r\\bar{m} not generally zero, the reduction (28) is invalid as written. The proof can be repaired by replacing ψ(k) with \\bar{ψ}(k); Lemma 10 bounds sums of Λ(n)ψ(n) and the set of characters is closed under conjugation, so the same estimate applies. The authors should correct (28) and re-verify that the final bound (40) is unaffected.","section":"Section 6.5 (Equation (28))"},{"comment":"The proof of Proposition 6 in Range III relies on the Chen–Gupta–Li zero-density estimate, stated as Proposition 2, which is cited to a preprint (arXiv:2507.08296). This estimate selects the admissible resonator length L=T^{17/14−ε} and provides the power saving in (38)–(39); if the preprint contains hidden restrictions on H or σ, or if the result is later found incorrect, the off-diagonal bound and the final constant collapse. The authors should either supply a proof or confirm with the authors of [4] that the estimate holds uniformly for all 1/2 ≤ σ < 1, H ≥ 1, and for the sum over primitive characters modulo q. The paper would be strengthened by stating the result as a theorem that is proved in an appendix.","section":"Section 3 (Proposition 2) and Section 6.5 (Lemma 10)"}],"minor_comments":[{"comment":"The ratio I1/I0 is displayed as c + (2/π) D_G ∫ C_G(u) sin(πcϑu)/u du, but from Theorem 3 and Theorem 4 the correct expression has the integral divided by D_G, not multiplied. Since D_G < 1, the displayed inequality is still true, but the formula should be corrected to maintain consistency with (6) and Theorem 4.","section":"Equation (8)"},{"comment":"The abstract says 'an infinite family of Siegel zeros' without explicitly saying that the infimum of the qualities is large; the theorem is stated with E = inf_j E_j sufficiently large. Please clarify that the result applies to a sequence whose qualities are bounded below by a sufficiently large constant (or that one passes to a subsequence with large E_j).","section":"Abstract and Theorem 1"},{"comment":"In the proof of Lemma 10, the parameter b in H=q^b is chosen after the vertical integral is discussed; it would be clearer to fix b explicitly (e.g., b = η/10) before invoking the splitting in (33) and (35). As written, 'taking B large enough' after choosing b is slightly ambiguous because the choice of B depends on the already-fixed b.","section":"Section 6.5, Lemma 10"},{"comment":"The notation fF0(s1,s2,s3) for the Mellin transform is nonstandard and hard to parse; using \\hat{F} or a different symbol would improve readability. In Lemma 9, the statement assumes q odd and square-free, and the 2-adic case is only sketched; since the paper's q can be even (q = 2^ν Q with ν ∈ {2,3}), a few more details on the factors at powers of 2 would be helpful.","section":"Section 6.2 and Lemma 9"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is valuable but cannot be accepted as is. The character-orthogonality error is localized and easy to fix, but the reliance on the unpublished Chen–Gupta–Li estimate is a serious risk; I recommend that the editor require the authors to verify the statement with the source or provide a proof. The paper fits the journal's scope well, and the numerical constant appears robust to the observed typo in Equation (8), but a careful revision is required."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one deserves your attention. Bondarenko and Heap prove that under RH, an infinite family of Siegel zeros implies μ < 0.4733, breaking the 1/2 barrier that many expected in that setting. The mechanism is genuinely new: instead of the usual Möbius resonator, they resonate with the exceptional character itself, exploiting χ(p) = -1 on almost all primes and computing shifted prime correlations via character sums. The length L ≈ T^{17/14} exceeds the traditional T-limit, which is the key innovation.\n\nThe structure is sound. The Montgomery–Odlyzko reduction, the I0 computation, the diagonal in J, and Ranges I and II of the off-diagonal all use standard techniques correctly as far as I can see. The new character-sum lemmas (7–9) are believable and self-contained. Range III is the boldest step: separating variables with Fourier inversion, summing over all characters, and invoking the Chen–Gupta–Li zero-density estimate to get the power saving. That dependency is real—Proposition 2 is a cited preprint (arXiv:2507.08296). The theorem falls apart if that estimate doesn't hold uniformly in the stated σ range, so a referee should verify it. It's a legitimate dependency, not a hidden flaw.\n\nSoft spots, in order of seriousness:\n\n1. Equation (28) is wrong as written. The orthogonality expansion should use \\bar ψ(k), not ψ(k). The displayed identity yields χ(k^{-1}+y), not χ(k+y). This is not fatal: with \\bar ψ(k) the same bounds apply because Lemma 10 is invariant under conjugation, so the proof goes through unchanged. But it is a genuine error in a load-bearing display and must be fixed.\n\n2. Equation (5) gives ϑ = 17/14 + 6δ, but actually ϑ = log L/log T = 17/(14+6δ), so the sign on δ is wrong. At δ = 10^{-6} the numerical effect is at the 10^{-5} level, probably harmless for the claimed 0.4733, but it is sloppy.\n\n3. The final inequality is asserted as \"a computer calculation shows\" with no code or integral exposed. For a paper whose headline is a constant, that should be an inline one-line integral plus a short script.\n\nOverall: the argument is coherent and original, and I think the theorem is very likely correct modulo the zero-density input. The typos are embarrassing but repairable. This deserves peer review and, after revision, publication. I'd take it to the reading group, and I'd happily referee it.","headline":"A serious conditional advance: Siegel zeros turned into a resonator to break the 1/2 gap barrier under RH; the core argument holds up, with two correctable typos and one external dependency to vet.","tokens_in":17295,"tokens_out":6688,"would_cite":true,"duration_ms":54870,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M20","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming RH, an infinite family of Siegel zeros makes some normalized gaps between zeta zeros smaller than 0.4733.","keywords":["Riemann zeta function","Siegel zeros","small gaps between zeros","Montgomery–Odlyzko method","Dirichlet L-functions","zero-density estimates","Alternative Hypothesis","long Dirichlet polynomials"],"falsifier":"Evaluate the numerical inequality behind (8): with $c=0.473275$, $\\delta=10^{-6}$, and $G_0(x)=1-\\frac{63}{125}(x-\\frac12)^2$, compute $c+\\frac{2}{\\pi D_G}\\int_0^1 C_G(u)\\frac{\\sin(\\pi c\\vartheta u)}{u}\\,du$ with $\\vartheta=\\frac{17}{14}+6\\delta$; if the value is at most $1$, the claimed threshold does not follow. Separately, test Proposition 2 on primitive characters modulo $q$ with $H=q^b$, $0<b<1/6$: any violation of $(qH)^{7(1-\\sigma)/3+\\varepsilon}$ would break the largest-$k$ range and remove the power saving.","tokens_in":16140,"feed_emoji":"📏","tokens_out":13546,"duration_ms":102005,"temperature":0.7,"pith_summary":"This paper tries to prove that the long-suspected 1/2 barrier for normalized gaps between zeros of the Riemann zeta function is not a genuine obstruction. Assuming the Riemann Hypothesis, it claims that if there is an infinite sequence of Siegel zeros of sufficiently large quality, then the liminf of normalized gaps satisfies $\\mu<0.4733$, strictly below 1/2. The interest is that Siegel zeros were previously seen as a reason small gaps below 1/2 should be rare; here the exceptional character is turned into the main tool. The proof extends the classical two-integral method to long Dirichlet polynomials of length $T^{17/14}$, using the structure of quadratic characters with Siegel zeros to evaluate the needed shifted correlations. A corollary is that under these assumptions no strong form of the Alternative Hypothesis, in which gaps sit at half-integers, can exist.","feed_headline":"On RH, Siegel zeros force zeta-zero gaps below 0.4733","feed_subtitle":"A longstanding 1/2 barrier falls, and strong Alternative Hypotheses are ruled out under RH.","key_machinery":"The central mechanism is the comparison of two weighted integrals, $I_1=\\int N_h(t)|R(t)|^2 W_T(t)\\,dt$ and $I_0=\\int |R(t)|^2 W_T(t)\\,dt$: if $I_1>I_0$, some interval contains two zeros closer than the normalized length $c$. The new object is a long resonator $R(t)=\\sum_{n\\le L}\\chi(n)G(n)n^{-1/2-it}$ of length $L=T^{17/14+\\varepsilon'}$, where $\\chi$ is the exceptional quadratic character and $G$ is a smooth polynomial weight; the working identity is (8), which expresses $I_1/I_0$ as $c+(2/(\\pi D_G))\\int_0^1 C_G(u)\\sin(\\pi c\\vartheta u)/u\\,du$ plus error terms, with $\\vartheta=\\log L/\\log T=17/14+6\\delta$. The character $\\chi$ does two jobs: its values are $-1$ on almost all primes, supplying the small-gap bias, and its additive correlations $\\chi(m)\\chi(km+r)$ are computable with a power saving, which is what permits $L$ to exceed $T$. The largest-$k$ off-diagonal range is tamed by the zero-density estimate (Proposition 2), whose exponent $7/3$ enters through the allowed length and the overlap of the dyadic ranges.","core_discovery":"The paper's central claim is Theorem 1: assume RH and that there is an exceptional sequence of Siegel zeros $(\\beta_j,\\chi_j,q_j,E_j)$ with $q_j\\to\\infty$ and $E=\\inf_j E_j$ sufficiently large; then $\\mu=\\liminf_{n\\to\\infty}(\\gamma_{n+1}-\\gamma_n)\\log(\\gamma_n)/(2\\pi)<0.4733$. The proof chooses the resonator $R(t)=\\sum_{n\\le L}\\chi(n)G(n)n^{-1/2-it}$, where $\\chi$ is the exceptional quadratic character, with $T=q^{7/3+\\delta}$ and $L=T q^{1/2-\\delta}=q^{17/6}$; a classical proposition about exceptional characters gives $\\chi(p)=-1$ on almost all primes, which makes the diagonal part of the numerator negative and large. The off-diagonal terms are split into three ranges of the prime-variable $k$, handled respectively by bounds for shifted character correlations $\\chi(m)\\chi(km+r)$, by a second-moment estimate for shifted prime sums, and by a new variable-separation argument using generalized Jacobi sums, with the zero-density estimate (Proposition 2, exponent $7/3$) controlling the largest $k$ range. Combining the asymptotics yields a numerical inequality at $c=0.473275$ with the chosen polynomial weight $G_0$, forcing $I_1>I_0$ and hence a gap of normalized size below $c$. The paper also derives Corollary 2: under the same assumptions, an asymptotic distribution with normalized gaps in $\\frac12\\mathbb{Z}_{\\ge1}+o(1)$ cannot exist.","pith_inferences":["Improving the zero-density exponent below $7/3$ would lengthen the admissible resonator like $T^{1+1/(2\\alpha)}$, and the exact constant one could reach at intermediate exponents is left open; this trade-off seems worth optimizing.","The same resonator construction might be tested numerically on quadratic characters without a Siegel zero; if the off-diagonal estimates degrade smoothly, weaker sub-$1/2$ bounds could hold unconditionally, a testable extension beyond the paper's assumptions.","The forced clustering on a normalized $o(1)$ scale suggests a picture opposite to the Alternative Hypothesis: instead of zeros avoiding each other at sub-half-integer distances, the Siegel-zero mechanism actively herds zeros together, which may interact with predictions from pair-correlation studies.","Translating the long-polynomial technique to other $L$-functions or to positive-proportion small-gap questions may be possible, though the paper notes that higher-moment inputs would restrict the length and likely restore the $1/2$ barrier."],"forward_implications":["Under RH and an exceptional sequence of Siegel zeros, the known bound $\\mu<0.50895$ improves to $\\mu<0.4733$.","A strong Alternative Hypothesis in which normalized gaps are confined to $\\frac12\\mathbb{Z}_{\\ge1}+o(1)$ is refuted under these assumptions; some clustering of zeros on a normalized $o(1)$ scale is forced.","The use of length $L=T^{17/14}$ shows that the traditional restriction to short Dirichlet polynomials in this method is not essential when the resonator coefficients carry enough structure.","If the Density Hypothesis (exponent $2$ in place of $7/3$ in the zero-density estimate) were available, the paper's construction would yield a bound around $\\mu<0.467$.","The result is compatible with the earlier finding that Siegel zeros make almost no gaps lie below $1/2-\\varepsilon$: a few very small gaps can coexist with the near-absence of moderately small gaps."],"supporting_citations":[{"why":"Supplies the zero-density estimate (Proposition 2) with exponent 7/3, which fixes the admissible resonator length and gives the power saving in the largest-k range.","marker":"[4]"},{"why":"Supplies the proposition that chi(p)=-1 on almost all primes when a Siegel zero exists, converting the diagonal numerator into the required negative contribution.","marker":"[14]"},{"why":"The comparison method of I_1 and I_0 that this paper extends to long Dirichlet polynomials.","marker":"[20]"},{"why":"Provides the previous best bound mu<0.50895 that Theorem 1 improves under the Siegel-zero assumption.","marker":"[17]"},{"why":"Gives the shifted-prime character-sum bounds used in the intermediate-k range of the off-diagonal analysis.","marker":"[19]"},{"why":"The large-value estimates underlying Proposition 2's zero-density exponent.","marker":"[13]"},{"why":"Gives the earlier result that Siegel zeros force almost no gaps below 1/2-epsilon, the apparent barrier this paper bypasses.","marker":"[10]"}],"fun_headline_variants":["Siegel zeros under RH force zeta gaps below 0.4733","Siegel zeros beat the half-gap barrier on RH to 0.4733","On RH, Siegel zeros imply gaps below 0.4733","Siegel zeros slash zeta-zero gaps to under 0.4733"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on Proposition 2, the zero-density estimate with exponent $7/3$, which says that for every primitive Dirichlet character modulo $q$ very few zeros of $L(s,\\psi)$ lie to the right of $1/2$; if that estimate has a hidden restriction on height or the vertical strip, or is simply wrong, the power saving in the largest-$k$ range and the final $0.4733$ bound collapse.","fun_headline_variants_meta":{"raw":{"variants":["Siegel zeros under RH force zeta gaps below 0.4733","Siegel zeros beat the half-gap barrier on RH to 0.4733","On RH, Siegel zeros imply gaps below 0.4733","Siegel zeros slash zeta-zero gaps to under 0.4733"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3467,"prompt_tokens":990,"completion_tokens":2477,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":2406}},"tokens_in":606,"tokens_out":2477,"duration_ms":17525,"temperature":1.0,"reasoning_tokens":2406,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:28:15.234521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the numerical inequality behind (8): with $c=0.473275$, $\\delta=10^{-6}$, and $G_0(x)=1-\\frac{63}{125}(x-\\frac12)^2$, compute $c+\\frac{2}{\\pi D_G}\\int_0^1 C_G(u)\\frac{\\sin(\\pi c\\vartheta u)}{u}\\,du$ with $\\vartheta=\\frac{17}{14}+6\\delta$; if the value is at most $1$, the claimed threshold does not follow. Separately, test Proposition 2 on primitive characters modulo $q$ with $H=q^b$, $0<b<1/6$: any violation of $(qH)^{7(1-\\sigma)/3+\\varepsilon}$ would break the largest-$k$ range and remove the power saving.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the proposition that chi(p)=-1 on almost all primes when a Siegel zero exists, converting the diagonal numerator into the required negative contribution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The comparison method of I_1 and I_0 that this paper extends to long Dirichlet polynomials."},{"cited_title":"Small gaps between consecutive zeros of the Riemann zeta-function","cited_arxiv_id":"2604.05733","evidence_quote":"Provides the previous best bound mu<0.50895 that Theorem 1 improves under the Siegel-zero assumption."},{"cited_title":"Kerr,Bounds of multiplicative character sums over shifted primes, Proc","cited_arxiv_id":null,"evidence_quote":"Gives the shifted-prime character-sum bounds used in the intermediate-k range of the off-diagonal analysis."},{"cited_title":"Guth and J","cited_arxiv_id":null,"evidence_quote":"The large-value estimates underlying Proposition 2's zero-density exponent."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier result that Siegel zeros force almost no gaps below 1/2-epsilon, the apparent barrier this paper bypasses."}],"review_version":2}