REVIEW 2 major objections 4 minor 22 references
Siegel zeros and small gaps between zeros of the Riemann zeta function
T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Assuming RH, an infinite family of Siegel zeros makes some normalized gaps between zeta zeros smaller than 0.4733.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 6.5 (Equation (28))] 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 3 (Proposition 2) and Section 6.5 (Lemma 10)] 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.
minor comments (4)
- [Equation (8)] 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.
- [Abstract and Theorem 1] 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 6.5, Lemma 10] 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 6.2 and Lemma 9] 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.
Circularity Check
No circularity: the small-gap bound is derived from external Siegel-zero and zero-density inputs, with the constant c verified numerically rather than fitted.
full rationale
No circular step found. Theorem 1 takes RH and the existence of an exceptional sequence of Siegel zeros as hypotheses, while the target quantity mu is defined independently of the construction. The resonator coefficients are built from the exceptional character, and the key claim that chi(p) = -1 on almost all primes (Proposition 3) is imported from Heath-Brown [14], an external result whose statement does not involve mu. The zero-density estimate of Chen-Gupta-Li (Proposition 2) is likewise external and is used to bound character sums; it does not encode the small-gap conclusion. The constant c = 0.473275 and the weight G0 are used in an explicit numerical verification of the sufficient condition I1/I0 > 1; c is not fitted to the zeros being predicted but is a free parameter in an inequality. The correlation sums and off-diagonal estimates are derived from first principles in Lemmas 4, 7, 8, 9, and 10, and no derivation step reduces to its own input. The reviewer's concern about the orthogonality identity in equation (28), if sustained, would be a correctness gap in Range III, not a circularity, and therefore does not affect this circularity score.
Assumptions & free parameters
free parameters (3)
- c =
0.473275
- delta =
10^-6
- G0 polynomial coefficient 63/125 =
63/125
assumptions (5)
- domain assumption Riemann Hypothesis for the Riemann zeta function
- domain assumption Existence of an exceptional sequence of Siegel zeros with quality E sufficiently large
- domain assumption Chen-Gupta-Li zero-density estimate (Proposition 2): sum over primitive characters of N(sigma,H,psi) is O((qH)^(7(1-sigma)/3+epsilon))
- domain assumption Heath-Brown exceptional character distribution lemma (Proposition 3): sum over p <= q^500 with chi(p)=1 of log p/p is O(log q / sqrt(log E))
- standard math Guinand-Weil explicit formula for compactly supported test functions
Cite this review
Pith. "Pith review of Siegel zeros and small gaps between zeros of the Riemann zeta function." pith.science (2026). https://pith.science/paper/GNQWSVS3
@misc{pith2026260807399,
author = {Pith},
title = {Pith review of: Siegel zeros and small gaps between zeros of the Riemann zeta function},
year = {2026},
howpublished = {\url{https://pith.science/paper/GNQWSVS3}},
note = {Machine review of arXiv:2608.07399}
}
abstract
On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/2\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\varepsilon}$ into the Montgomery--Odlyzko method.
Reference graph
Works this paper leans on
-
[4]
B. Chen, V. Gupta, and Y. C. Li,Large value estimates for Dirichlet polynomials, and the density of zeros of DirichletL-functions, preprint, arXiv:2507.08296
-
[1]
H. Bui, D. Goldston, M. Milinovich, and H. L. Montgomery,Small gaps and small spacings between zeta zeros, Acta Arith.210(2023), 133–153
work page 2023
-
[2]
H. M. Bui and M. B. Milinovich,Gaps between zeros of the Riemann zeta-function, Quart. J. Math.69(2018), 403–423
work page 2018
-
[3]
H. M. Bui, M. B. Milinovich, and N. Ng,A note on the gaps between consecutive zeros of the Riemann zeta function, Proc. Amer. Math. Soc.138(2010), 4167–4175
work page 2010
-
[5]
Chinis,Siegel zeros and Sarnak’s conjecture, J
J. Chinis,Siegel zeros and Sarnak’s conjecture, J. London Math. Soc.113(2026)
work page 2026
- [6]
-
[7]
J. B. Conrey,The Riemann hypothesis, Notices Amer. Math. Soc.50no. 3 (2003), 341–353
work page 2003
-
[8]
J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown,On the distribution of gaps between zeros of the zeta-function, Quart. J. Math. Oxford Ser. (2)36(1985), 43–51
work page 1985
Show all 22 references
-
[9]
J. B. Conrey, A. Ghosh, and S. M. Gonek,A note on gaps between zeros of the zeta function, Bull. London Math. Soc.16(1984), 421–424
1984
-
[10]
J. B. Conrey and H. Iwaniec,Spacing of zeroes of HeckeL-functions and the class number problem, Acta Arith.103(2002), 259–312
2002
-
[11]
Feng and X
S. Feng and X. Wu,On gaps between zeros of the Riemann zeta-function, J. Number Theory 132(2012), 1385–1397
2012
-
[12]
D. A. Goldston, T. S. Trudgian, and C. L. Turnage-Butterbaugh,On the Montgomery–Odlyzko method regarding gaps between zeros of the zeta-function, J. Math. Anal. Appl.527no. 2 (2023), Paper No. 127548
2023
-
[13]
Guth and J
L. Guth and J. Maynard,New large value estimates for Dirichlet polynomials, Ann. of Math. 203no. 2 (2026), 623–675. SIEGEL ZEROS AND SMALL GAPS 27
2026
-
[14]
D. R. Heath-Brown,Prime twins and Siegel zeros, Proc. London Math. Soc.47no. 2 (1983), 193–224
1983
-
[15]
Riemann Hypothesis
D. R. Heath-Brown,Small class numbers and the pair correlation of zeros, talk given at AIM conference “Riemann Hypothesis”, 1996. Video available at Vimeo
1996
-
[16]
M. N. Huxley,Large values of Dirichlet polynomials, III, Acta Arith.26(1975), 435–444
1975
-
[17]
Inoue,Small gaps between consecutive zeros of the Riemann zeta function, preprint, arXiv:2604.05733
S. Inoue,Small gaps between consecutive zeros of the Riemann zeta function, preprint, arXiv:2604.05733
-
[18]
Inoue, H
S. Inoue, H. Kobayashi, and Y. Toma,Explicit extreme values of the argument of the Riemann zeta function, Math. Proc. Cam. Phil. Soc., online ahead of print, (2026), 1–15
2026
-
[19]
Kerr,Bounds of multiplicative character sums over shifted primes, Proc
B. Kerr,Bounds of multiplicative character sums over shifted primes, Proc. Steklov Inst. Math. 314no. 1 (2021), 64–89
2021
-
[20]
H. L. Montgomery and A. M. Odlyzko,Gaps between zeros of the zeta function, inTopics in Classical Number Theory(Budapest, 1981), Colloq. Math. Soc. J´ anos Bolyai, vol. 34, North- Holland, Amsterdam, 1984, pp. 1079–1106
1981
-
[21]
Preobrazhenskiˇ ı,A small improvement in the gaps between consecutive zeros of the Riemann zeta-function, Res
S. Preobrazhenskiˇ ı,A small improvement in the gaps between consecutive zeros of the Riemann zeta-function, Res. Number Theory2(2016), Art. 28
2016
-
[22]
Tao and J
T. Tao and J. Ter¨ av¨ ainen,The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero, J. London Math. Soc.106no. 4 (2022), 3317–3378. Department of Mathematical Sciences, Norwegian University of Science and Tech- nology (NTNU), 7491 Trondheim, Norway. Email add...
2022
Reviewed August 15, 2026 · model on record in the stance chip above.
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