Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals

Biao Wang

REVIEW

Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals

Biao Wang

Number Theory · Submitted 7 Sep 2026

Pith's verdict: The short-interval pair-correlation formula and the variational minimization are solid and worth preserving, but Theorem 1.1 is unproven as written because the multiset Z in (3.6) fails the conjugation-invariance hypothesis of Lamzouri's inequality.

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arXiv ID2609.07918 (version 2)

Pith ID7AGCDGBS, Pith's permanent record of this paper and its review

CitationBiao Wang. "Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals." arXiv:2609.07918 (2026). https://pith.science/paper/7AGCDGBS

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Claim paper

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Editor's note

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Short version: the main theorem is not proved as written. The stress-test note is right. In Section 3, Z is defined using only ordinates γ in (T,T+H]. For a zero ρ with γ in that interval, the conjugate zero has ordinate −γ, which is not in I, so Z is not invariant under complex conjugation. Proposition 2.1 is then applied to this Z. That step is load-bearing, so (3.7)–(3.9) and Theorem 1.1 do not follow.

What is actually good: Theorem 2.2 is a genuine short-interval extension of the Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh pair-correlation theorem, with careful error terms. Lemma 3.1, removing the rational weight, is clean. Proposition 4.1 is a nice exact variational calculation; the constant C(λ) looks right, and the approximation by f=η² is handled honestly. The writing is clear and the references are used appropriately.

The soft spot is central, not cosmetic. Adding the conjugate zeros with ordinates in (−T−H,−T] does not fix the present argument: Lamzouri's inequality would then count simple real elements in the symmetric interval, not in the one-sided interval (T,T+H], and the pair sum would acquire cross terms the paper does not compute. So this is not a typo; the theorem needs a genuinely different argument or a different formulation.

A smaller note, which the paper acknowledges: the simple-zero bound is positive only for θ>0.55019, barely below Steuding's 0.552, and the distinct-zero bound overlaps the range of Karatsuba's result. That limits the novelty but is not a logical flaw.

The paper is for analytic number theorists working on zero proportions and pair correlation. A corrected version would deserve serious attention and likely citation. As it stands, I would not cite the main theorem.

Recommendation: send to a competent referee, with instructions to check the invariance hypothesis carefully. This is exactly the kind of gap refereeing exists to catch, and the surrounding technique may be salvageable. Do not accept the current version.

Referee report

Major objections (1)

[§3, Eq. (3.6) and the application of Proposition 2.1] Lamzouri's inequality (Proposition 2.1) requires the finite multiset Z to be invariant under complex conjugation. The multiset defined in (3.6) as Z = { i(ρ−1/2)L/(2π) : γ ∈ I } with I = (T, T+H] contains only zeros with positive ordinates. For ρ = 1/2+δ+iγ with γ>0, the complex conjugate of the corresponding element is not in Z; indeed, the element arising from the conjugate zero ρ̅ = 1/2+δ−iγ would have imaginary part −γL/(2π), which lies outside the interval. Hence Z is not conjugation-invariant. The paper provides no justification or modification to satisfy this hypothesis. Consequently, the key inequalities (3.7) and the subsequent derivations of (3.8)–(3.9) lack a foundation in Proposition 2.1, and Theorem 1.1 is unproven.

Minor objections (4)

[§2, Eq. (2.13)] The bound 1/|1+((t−γ)+iδ)^2| ≪ 1/(1+(t−γ)^2) is stated; a short justification would be helpful, though it is standard.
[§4, Proposition 4.1] The minimization is carried out among functions with ∫ f = 1, but the class from (3.1) requires f = η^2 with ∫ η^2 = 1, which is a stricter normalization; the proof that the infimum coincides requires the approximation argument at the end, which is fine.
[Throughout] The notation O_g(·) and O_f(·) is used but not formally defined. Clarify that the implied constants depend on the test function.
[References] Reference [11] (Lamzouri) is cited as arXiv:2609.02882v1, which appeared after the present preprint. The paper may benefit from noting any updates.

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The paper

The paper claims

For every fixed short interval of power length, the paper gives explicit lower bounds on the proportion of simple critical zeros and of distinct zeros of the Riemann zeta-function, expressed as functions of the interval exponent θ.

The reading

This paper extends recent global results on the zeros of the Riemann zeta-function to short intervals of length T^θ. The author proves that, as T grows, the proportion of simple zeros on the critical line in such an interval is at least 2 − θ/2 − (1/√2)cot(θ/√2), and the proportion of distinct zeros is at least 3/2 − θ/4 − 1/(2√2)cot(θ/√2). These lower bounds are positive for θ > 0.550... (simple critical) and θ > 0.346... (distinct). The proof combines a short-interval version of Montgomery's pair-correlation formula (developed using the method of Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh) with Lamzouri's inequality for complex-number multisets invariant under conjugation, and then minimizes a certain quadratic functional to obtain the optimal constants.

What carries the argument

The paper has two main components. First, it establishes a short-interval analogue of Montgomery's pair-correlation formula (Theorem 2.2) by adapting the unconditional approach of Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh. Second, it applies Lamzouri's inequality (Proposition 2.1) to the multiset Z = { i(ρ−1/2)L/(2π) : γ ∈ I }, which is invariant under complex conjugation. The pair-correlation sum reduces to a functional C(f) that is minimized over a class of smooth functions; the unique minimizer is found via calculus of variations (Proposition 4.1), yielding the explicit constants c(θ) and d(θ).

Extended reading notes

Core claim

Theorem 1.1 states that for every fixed 0 < θ < 1, we have liminf_{T→∞} N^s_0(T,T^θ)/N(T,T^θ) ≥ c(θ) = 2 − θ/2 − (1/√2)cot(θ/√2) and liminf_{T→∞} N_d(T,T^θ)/N(T,T^θ) ≥ d(θ) = 3/2 − θ/4 − 1/(2√2)cot(θ/√2), where N^s_0 counts simple zeros on the critical line and N_d counts distinct nontrivial zeros. These are the first explicit lower bounds for these proportions that depend on the interval exponent θ.

Load-bearing premise

The error term O(H+T^λ $L^{2}$) in Theorem 2.2 rests on the unconditional explicit formula and the Montgomery-Vaughan mean value theorem, whose estimates must hold uniformly for x up to T^λ. If those estimates fail for the range of x needed (because of cancellation issues or exceptional zeros), the derived constant c(θ) and d(θ) would not be guaranteed.

Editorial extensions

If this is right

For any fixed θ > θ₀ = 0.55019..., every interval (T, T+T^θ] contains a positive proportion of simple zeros on the critical line, explicitly bounded below by c(θ).
For any fixed θ > θ_d = 0.34665..., every such interval contains a positive proportion of distinct zeros, bounded below by d(θ).
The constants are sharp in the sense that they arise from the variational minimizer; improvements would require a different approach or stronger assumptions on the zeros.
The method shows that the pair-correlation formula can be localized to intervals of power length while preserving the structure needed for Lamzouri's inequality.

Reading between the lines

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

The variational argument might be adapted to other L-functions (e.g., Dirichlet L-functions, modular form L-functions) that satisfy a similar pair-correlation formula in short intervals, giving analogous proportion bounds.
One could attempt to relax the smoothness condition on the test function f to obtain the same minimizer with weaker assumptions, possibly extending the range of θ for which the bounds remain valid.
The constants c(θ) and d(θ) increase with θ; numerical values for moderate θ (e.g., θ=3/4) are given in the paper, but explicit computations for other θ could guide numerical investigations of zero statistics.
The author used ChatGPT-6 Astra for implementing ideas; this acknowledgment does not affect the mathematical content but highlights a shift in how computational tools assist in proof design.

Original abstract

Recently, on the non-trivial zeros of the Riemann zeta function, it is obtained by Alp\"oge and Furman that more than 67.25% of the zeros are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, we will use the method of Lamzouri to give lower bounds on the number of the non-trivial zeros of the Riemann zeta function in short intervals. To prove the main result, we establish Montgomery's theorem on the pair correlation of zeros of the zeta function in short intervals by following the approach of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and then use Lamzouri's inequality on any finite multiset of complex numbers which is invariant under complex conjugation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,789 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.29 · compares with prior results 0.66 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Recently, on the non-trivial zeros of the Riemann zeta function, it is discovered by Claude and verified by Alp\"oge and Furman that more than 67.25% of the zeros are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, we will use the method of Lamzouri to give lower bounds on the number of the non-trivial zeros of the Riemann zeta function in short intervals. To prove the main result, we establish Montgomery's theorem on the pair correlation of zeros of the zeta function in short intervals by following the approach of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and then use Lamzouri's inequality on any finite multiset of complex numbers which is invariant under complex conjugation.

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	Recently, on the non-trivial zeros of the Riemann zeta function, it is discovered by Claude and verified by Alp\"oge and Furman that more than 67.25% of the zeros are simple and on the critical line, and more than 83.62% are distinct.	background 1.00	0.03	0.09	asserted 1.00
s1	Later, Lamzouri gave a different and more direct proof.	background 0.99	0.03	0.02	asserted 1.00
s2	In this article, we will use the method of Lamzouri to give lower bounds on the number of the non-trivial zeros of the Riemann zeta function in short intervals.	contribution 0.93	0.71	0.08	asserted 0.92
s3	To prove the main result, we establish Montgomery's theorem on the pair correlation of zeros of the zeta function in short intervals by following the approach of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and then use Lamzouri's inequality on any finite multiset of complex numbers which is invariant under complex conjugation.	method 0.99	0.79	0.10	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Discussion

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