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An Explicit Lower Bound for the Proportion of Isolated Zeros of the Riemann Zeta Function

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Cheolmin Kim ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We study isolation in the full sequence of nontrivial zeros of the Riemann zeta function with positive ordinate, counted with multiplicity. For every fixed 0<c≤10−50<c\le10^{-5}, we prove unconditionally that the lower limiting proportion of zeros whose two neighboring gaps both exceed 2πc/log⁡X2\pi c/\log X is at least 0.6724680.672468. Here XX is the common height cutoff and the neighbors are taken from the infinite sequence. This refines Lamzouri's small-fixed-gap bound 0.6552…0.6552\ldots. We retain his isolation argument and the Montgomery–Taylor kernel, while replacing the linear cost of horizontal displacement by a certified majorant with a positive threshold. Young's shifted mollified second moment bounds the corresponding truncated displacement moment. We supply the boundary and global-height arguments and obtain a gap loss 6π2c26\pi^2c^2 uniform under the required smoothing. A fixed interval-arithmetic certificate completes the numerical estimate. The result is asymptotic and gives no effective first height.

Author comments: This submission corresponds to version v0.4 of the preprint available on Zenodo: https://doi.org/10.5281/zenodo.24126375 The author understands the research question, the historical context, and the meaning of the claimed result. However, the detailed analytic arguments and computer-assisted numerical certification have not been independently verified or fully understood by the author. The mathematical development, internal proof checking, and manuscript preparation relied substantially on AI assistance, primarily GPT and Claude, with supplementary input from Gemini. The Zenodo record includes the full manuscript, LaTeX source, numerical certificate, and reproducibility materials. The manuscript has not undergone independent human peer review.

Provenance statement

This research was developed with substantial assistance from AI systems, primarily GPT and Claude, with supplementary input from Gemini. AI assistance included mathematical exploration, derivation and organization of arguments, internal proof checking, numerical verification, and manuscript writing. The author understands the research question, historical context, and significance of the claimed result, but has not independently verified or fully understood the detailed analytic proofs or numerical certification. The AI-based checks do not constitute independent human mathematical review. The manuscript, LaTeX source, and numerical reproducibility materials are publicly available at https://doi.org/10.5281/zenodo.24126375.

Tools used

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References

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Show all 13 referencesShow fewer references
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Version history

  1. v1 · 21 pagesSubmitted by Cheolmin KimThis submission corresponds to version v0.4 of the preprint available on Zenodo: https://doi.org/10.5281/zenodo.24126375 The author understands the research question, the historical context, and the meaning of the claimed result. However, the detailed analytic arguments and computer-assisted numerical certification have not been independently verified or fully understood by the author. The mathematical development, internal proof checking, and manuscript preparation relied substantially on AI assistance, primarily GPT and Claude, with supplementary input from Gemini. The Zenodo record includes the full manuscript, LaTeX source, numerical certificate, and reproducibility materials. The manuscript has not undergone independent human peer review.CurrentNov 11, 2026