Unconditional Bounds for Two, Three and Four Consecutive Moderate Gaps between Zeros of the Riemann Zeta Function

Boyang Zhao; C\'esar Ang\'on; Erick Ross; Ze Sen Tang

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Unconditional Bounds for Two, Three and Four Consecutive Moderate Gaps between Zeros of the Riemann Zeta Function

César Angón, Erick Ross, Ze Sen Tang, Boyang Zhao

Number Theory · Submitted 16 Oct 2026

Pith's verdict: Unconditional positive-proportion bounds for consecutive moderate gaps – a solid, novel result whose central lemma relies on an unreviewed preprint but is otherwise rigorous and clearly presented.

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arXiv ID2610.22927 (version 2)

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CitationBoyang Zhao, C\'esar Ang\'on, Erick Ross, Ze Sen Tang. "Unconditional Bounds for Two, Three and Four Consecutive Moderate Gaps between Zeros of the Riemann Zeta Function." arXiv:2610.22927 (2026). https://pith.science/paper/ENVCBC4J

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Editor's note

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The paper does what it says: it proves unconditional lower bounds for two, three, and four consecutive moderate gaps between zeros of the Riemann zeta function. This was previously only known conditionally (RH + pair correlation), so it is a genuine advance. The numerical bounds are explicit and the constant C_MT is classical. The method—combining Lamzouri’s Hilbert-space framework with a cluster-counting argument and spectral estimates—is clever and carefully executed. The proofs of the new lemmas (e.g., the spectral estimates for blocks and the cluster-counting proposition) are detailed and look correct.

The one soft spot is the stress-test concern: Lemma 1, the key analytic estimate giving the second moment and the constant C_MT, is taken directly from Lamzouri’s arXiv preprint [7] and not independently derived here. The paper sketches the construction of η_δ and states that “the argument in the proof of [7, Lemma 3.2]” yields (12), but it does not verify the convergence C_{η_δ}→C_MT or the error estimate. So the whole numerical lower bound rests on an unreviewed source. That said, the dependence is transparent, the cited preprint is by a reputable author, and the rest of the paper is self-contained. If Lamzouri’s lemma turns out to have a flaw, the present results would need adjustment, but the structural ideas would still stand.

Overall, this is a serious piece of work that answers a natural question with clean methods. The reliance on a single unreviewed lemma is a concern but not a disqualifier—it is common in fast-moving areas to cite preprints, and the authors do not hide it. A careful referee should check Lemma 1’s derivation, but the present paper’s own contributions are sound.

I would bring this to a reading group, cite it, and recommend that a serious editor send it to peer review. The main recommendation for the authors (in confidence) would be to either include a short self-contained proof of Lemma 1 or to note that it will appear in Lamzouri’s forthcoming paper; in the meantime the result is credible.

Referee report

Major objections (2)

[Lemma 1 (Section 2.2, also Section 8)] The entire numerical lower bound of Theorem 1 depends on the constant C_MT and the error term in Lemma 1. Lemma 1 is stated as a theorem from Lamzouri [7, Lemma 3.2], but [7] is an unreviewed arXiv preprint. The present paper does not independently derive Lemma 1 from the unconditional pair-correlation estimate of Baluyot et al. [2]; it only sketches the construction of η_δ and asserts that 'the argument in the proof of [7, Lemma 3.2]' yields (12). The convergence C_η_δ → C_MT is not verified, and the error estimate O_δ(1/√{log T}) is not justified here. Since this step is load-bearing, the paper should either (a) provide a self-contained proof of Lemma 1 using [2], or (b) give a detailed verification of the crucial estimates (the convergence and the error term) within the present framework. As it stands, the result is contingent on the correctness of an unpublished manuscript.
[Lemma 11 (Section 6, proof near Equation after line 'It follows that')] The derivation of the inequality tr Ψ(H) - 3 ≥ 2||H - I||_F - 1 is not fully justified. The step '(μ-1)^2 ≤ (μ-1)^2 + (λ1-1)^2 + (λ2-1)^2 ≤ ||H - I||_F^2' is true, but then the inequality tr Ψ(H) - 3 ≥ 2||H - I||_F - 1 needs a more explicit justification, especially the transition from ||H - I||_F^2 - (||H - I||_F - 1)^2 = 2||H - I||_F - 1. The paper uses this to deduce 4K_δ(c) - 1 ≥ 3K_δ(c)^2, which relies on the assumption that ||H - I||_F ≥ 2K_δ(c). It would be helpful to explain the chain of inequalities more completely.

Minor objections (5)

[Equation (7) and throughout] The constant C_MT is written with inconsistent spacing: 'C_MT' (no space) and 'C_MT' (space) appear interchangeably. Please unify the notation to 'C_MT'.
[Section 2.2, definition of η_δ] In the line 'η_δ = eη_δ(x) √{A(x)}/√{A_δ}', the function name 'eη_δ' uses a tilde; it might be clearer to denote the smooth cutoff by a different letter (e.g., χ_δ) to avoid confusion with the final η_δ.
[Page 5, line after (11)] The text says 'All these functions are extended by zero outside I.' This is already implied by the definition of A and the cutoff; it can be stated more briefly.
[Section 7, Proposition 1 proof] The bound that each cluster in Z_m^T invalidates at most ℓ-1 strings is plausible but should be justified with a brief reasoning (e.g., because the cluster occupies at most ℓ consecutive positions in the ordering when interlaced with R_1 elements). Adding a short argument would improve clarity.
[Section 8, final display] The term ℓ/N(T) in the inequality before taking limits should be written as ℓ/N(T) to match the parentheses; as written it is ambiguous but the intended meaning is clear.

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The paper

The paper claims

Unconditional positive proportion lower bounds for consecutive moderate gaps between Riemann zeta zeros

The reading

This paper proves that, for every fixed moderate-gap size up to a quarter of the mean spacing, runs of two, three, and four consecutive gaps of at least that size occur a positive proportion of the time among all pairs, triples, and quadruples of consecutive gaps between zeros of the Riemann zeta function. The existence of such runs was previously known only under the Riemann hypothesis and the pair correlation conjecture. The proof combines a Hilbert-space formulation of the Montgomery–Taylor second-moment method with spectral estimates for close zeros and a cluster-counting argument that handles multiple and off-critical-line zeros. The numerical lower bounds on the asymptotic densities are 0.3450014… for two gaps, 0.1812517… for three gaps, and 0.0175021… for four gaps, derived from the constant C_MT ≈ 1.327499.

What carries the argument

The key machinery is a finite-dimensional Hilbert-space setup derived from Lamzouri's framework: to each zero one associates a function f_z in $L^{2}$, and the Gram matrix G of these functions is decomposed as P + Q, where P corresponds to simple critical zeros. The Frobenius norm ║G║$_F^{2}$ is expressed as the pair correlation sum with a carefully constructed kernel K_δ. Spectral estimates (Lemmas 10–12) bound trΨ(P) from below in terms of K_δ(c)^2 and the number of close simple zeros. A cluster-counting argument (Proposition 1) then translates these spectral bounds into a lower bound on w_ℓ(T,c), the count of consecutive moderate gaps.

Extended reading notes

Core claim

The paper's main result (Theorem 1) states that for every fixed 0 < c ≤ 1/4 and ℓ = 2,3,4, the lower asymptotic density of ℓ consecutive c-moderate gaps (each at least 2πc/log T) is at least (ℓ+4 − (ℓ+2)C_MT)/2, where C_MT = 1/2 + (1/√2) cot(1/√2) ≈ 1.3274992963205. Concretely, the lower bounds are 0.3450014… (ℓ=2), 0.1812517… (ℓ=3), and 0.0175021… (ℓ=4). These hold unconditionally, without assuming the Riemann hypothesis or the pair correlation conjecture, answering a question raised by Gonek and Sahay.

Load-bearing premise

The entire derivation depends on the correctness of the unconditional pair-correlation estimate proved in Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [2], which provides the fundamental analytic input for the kernel estimate.

Editorial extensions

If this is right

For ℓ=2,3,4, the proportion of ℓ-tuples of consecutive gaps with length at least 2πc/log T is bounded away from zero, providing the first unconditional positive-proportion result for runs of moderate gaps.
The explicit numerical densities (0.345, 0.181, 0.0175) serve as benchmarks that subsequent work on consecutive gaps can attempt to improve.
The connection between the Montgomery–Taylor constant C_MT and consecutive gaps reveals that the extremal window used in pair correlation also dictates the possibility of long runs of moderate gaps.
The method of decomposing the Gram matrix into contributions from simple, multiple, and off-line zeros, together with the cluster-counting argument, may be applicable to other L-functions for which an unconditional pair-correlation estimate exists.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Any future improvement to the unconditional pair-correlation estimate (currently limited by C_MT) would directly increase the numerical lower bounds for consecutive moderate gaps.
The condition that K_δ(c)^2 > ℓ/(ℓ+2) restricts the theorem to ℓ ≤ 4; obtaining positive bounds for ℓ ≥ 5 would require a kernel with a larger lower bound or a refined spectral estimate.
Because the proof handles zeros off the critical line via the tr(Q) term, the results are compatible with the possibility that all zeros lie on the line, but do not require it.
The cluster-counting argument shows that the defect from multiple and off-line zeros can be absorbed by the spectral bound, indicating that even if such zeros exist, the density of consecutive moderate gaps remains positive.

Original abstract

We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function. More precisely, for every fixed $0<c\leq 1/4$, the lower asymptotic densities, normalized by $N(T)$, of the occurrences of two, three, and four consecutive gaps, each of length at least $2\pi c/\log T$, are at least $0.3450014\ldots$, $0.1812517\ldots$, and $0.0175021\ldots$, respectively. Here, zeros are ordered by ordinate and counted with multiplicity, and $N(T)$ denotes the number of nontrivial zeros with ordinates in $(0,T]$. These bounds address the corresponding cases of a question of Gonek and Sahay. The proof combines the methods of Alp\"oge--Furman and Lamzouri with local spectral estimates and cluster counting.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 5,266 input tokens.

What the abstract itself states, as probabilities: claims first 0.24 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.92 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function. More precisely, for every fixed $0<c\leq 1/4$, the lower asymptotic densities, normalized by $N(T)$, of occurrences of two, three, and four consecutive gaps, each of length at least $2\pi c/\log T$, are at least $0.3450014\ldots$, $0.1812517\ldots$, and $0.0175021\ldots$, respectively. Here zeros are ordered by ordinate and counted with multiplicity, and $N(T)$ denotes the number of nontrivial zeros with ordinates in $(0,T]$. This answers unconditionally and quantitatively a question studied by Gonek and Sahay under the Riemann hypothesis and the pair correlation conjecture. In particular, at the time of their work, it was not known unconditionally that even two consecutive moderate gaps occur infinitely often for any fixed $c>0$. The proof combines Lamzouri's Hilbert-space formulation of the Montgomery--Taylor second-moment method with local spectral estimates for simple critical-line zeros and a cluster-counting argument that accounts for multiple zeros and zeros off the critical line.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function.	result 0.55	0.97	0.92	asserted 1.00
s1	More precisely, for every fixed $0<c\leq 1/4$, the lower asymptotic densities, normalized by $N(T)$, of occurrences of two, three, and four consecutive gaps, each of length at least $2\pi c/\log T$, are at least $0.3450014\ldots$, $0.1812517\ldots$, and $0.0175021\ldots$, respectively.	result 1.00	0.95	0.76	asserted 1.00
s2	Here zeros are ordered by ordinate and counted with multiplicity, and $N(T)$ denotes the number of nontrivial zeros with ordinates in $(0,T]$.	background 0.71	0.32	0.03	asserted 0.90
s3	This answers unconditionally and quantitatively a question studied by Gonek and Sahay under the Riemann hypothesis and the pair correlation conjecture.	interpretation 0.70	0.74	0.14	asserted 1.00
s4	In particular, at the time of their work, it was not known unconditionally that even two consecutive moderate gaps occur infinitely often for any fixed $c>0$.	gap 0.99	0.06	0.05	asserted 0.99
s5	The proof combines Lamzouri's Hilbert-space formulation of the Montgomery--Taylor second-moment method with local spectral estimates for simple critical-line zeros and a cluster-counting argument that accounts for multiple zeros and zeros off the critical line.	method 1.00	0.83	0.03	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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