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More than 8/9 zeros of the Riemann zeta function are simple or on the critical line

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Cristian Cobeli, Alexandru Zaharescu

Number Theory · Submitted 16 Oct 2026 · 25 references

Pith's verdict: Solid incremental advance past 8/9, but the computational verification of Lemma 5.2 has tight margins and no shipped code—a real vulnerability, not a dealbreaker.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 19, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv ID2610.23467 (version 1)
Pith IDFL66S22R, Pith's permanent record of this paper and its review
CitationAlexandru Zaharescu, Cristian Cobeli. "More than 8/9 zeros of the Riemann zeta function are simple or on the critical line." arXiv:2610.23467 (2026). https://pith.science/paper/FL66S22R
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Referee report

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The paper

The paper claims

This paper unconditionally proves that more than 8/9 of nontrivial zeros of the Riemann zeta function are simple or lie on the critical line.

The reading

The paper establishes a new unconditional lower bound for the proportion of Riemann zeta zeros that are either simple or on the critical line: the proportion exceeds 8/9 (approximately 0.88893). This improves the previous best bound of about 0.88762 by Lamzouri. The proof combines a refined Hilbert-space inequality with a carefully chosen kernel and a computer-assisted local correlation inequality. The result does not rely on any unproven conjectures such as the Riemann Hypothesis.

What carries the argument

Three components: (A) a spectral inequality that extracts a remainder $R_\tau(H)$ from the Gram matrix of real simple zeros, improving the Bessel-type bound of Lamzouri; (B) a seven-point inequality (Lemma 5.2) bounding a weighted sum of squared correlations plus a penalty term, verified by a binary subdivision algorithm with interval arithmetic; (C) a three-cosine perturbation of the Montgomery–Taylor density kernel that yields a geometric gain sufficient to push the bound past $8/9$.

Extended reading notes

Core claim

The authors prove that $\liminf_{T\to\infty} N^{s\cup 0}(T)/N(T) > 8/9$, where $N^{s\cup 0}(T)$ counts zeros of the Riemann zeta function that are simple or on the critical line (with multiplicity). The constant in the lower bound is approximately $0.88893$.

Load-bearing premise

The proof depends on a computer-assisted verification (Lemma 5.2) that a certain inequality involving seven ordered real points holds in a six-dimensional box; if that verification contains an error, the main result could fail.

Editorial extensions

If this is right

  • Any zero not on the critical line and of multiplicity at least 2 can occupy at most about 11.1% of the total zero density.
  • The methods provide a new template for improving unconditional bounds on zero statistics without assuming the Riemann Hypothesis.
  • The computer-assisted verification demonstrates that local correlation constraints can be rigorously optimized using interval analysis.

Reading between the lines

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

  • The computer-algebra verification of Lemma 5.2, while described in detail, would benefit from an independent implementation to confirm the absence of software bugs.
  • The same spectral inequality approach might be applied to other $L$-functions or to higher-order correlations, potentially pushing the proportion further.
  • The constant $0.88893$ appears close to the limit of the current method; further improvement may require new ideas beyond the three-term cosine perturbation.
Original abstract

We prove unconditionally that the lower limiting proportion of nontrivial zeros of the Riemann zeta function, which are simple or lie on the critical line, is greater than $8/9$, with zeros counted with multiplicity.

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, 2026-10-20: 1 request, 1,675 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · 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 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We prove unconditionally that the lower limiting proportion of nontrivial zeros of the Riemann zeta function, which are simple or lie on the critical line, is greater than $8/9$, with zeros counted with multiplicity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove unconditionally that the lower limiting proportion of nontrivial zeros of the Riemann zeta function, which are simple or lie on the critical line, is greater than $8/9$, with zeros counted with multiplicity. result 0.97 0.96 0.97 asserted 1.00

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

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

Figures
Figure 1. The density kernel f∗(u). 5 0 5 0.2 0.0 0.2 0.4 0.6 0.8 1.0 K¤ (x)

Figures reproduced from arXiv:2610.23467 by the authors.

References

Discussion

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