Cheolmin Kim

math.NT — Number Theory

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

Contributed by Cheolmin Kim

We study isolation in the full sequence of nontrivial zeros of the Riemann zeta function with positive ordinate, counted with multiplicity. For every fixed $0<c\le10^{-5}$, we prove unconditionally that the lower limiting proportion of zeros whose two neighboring gaps both exceed $2\pi c/\log X$ is at least $0.672468$. Here $X$ is the common height cutoff and the neighbors are taken from the infinite sequence. This refines Lamzouri's small-fixed-gap bound $0.6552\ldots$. 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 $6\pi^2c^2$ 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.