More than 83.69% of the zeros of the Riemann zeta function are distinct
Abstract
The lower asymptotic proportion of distinct nontrivial zeros of the Riemann zeta function, relative to the total number counted with multiplicity, is at least 0.8369928814…. Earlier work proves 0.83625…, and a report we have not verified claims 0.83672…. As in the proof of the bound $0.83625$, an unconditional version of Montgomery's pair-correlation theorem gives an asymptotic energy estimate. The new ingredient is a short matrix inequality with a free clipping parameter. It strengthens the lower bound for this energy in terms of the number of distinct zeros. The gain is a nonnegative correction from overlaps between different nearby zeros on the critical line, which is retained even when some of these zeros are double. The matrix inequality, the threshold lemma, the block dichotomy, the counting assembly and the exact arithmetic are proved formally in Lean 4. The constant relies on a computer-assisted local inequality from recent work that has not yet been refereed. That computation was re-run independently, and every imported input is listed. This paper is primarily an experiment in AI-assisted mathematical research (Section 4).
Related
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BibTeX
@misc{knausgard2026more,
title = {{More than 83.69% of the zeros of the Riemann zeta function are distinct}},
author = {Kristian Muri Knausgård},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33043},
eprint = {2609.33043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33043},
}
RIS
TY - UNPB
TI - More than 83.69% of the zeros of the Riemann zeta function are distinct
AU - Kristian Muri Knausgård
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.33043
UR - https://arxiv.org/abs/2609.33043
AB - The lower asymptotic proportion of distinct nontrivial zeros of the Riemann zeta function, relative to the total number counted with multiplicity, is at least 0.8369928814…. Earlier work proves 0.83625…, and a report we have not verified claims 0.83672…. As in the proof of the bound $0.83625$, an unconditional version of Montgomery's pair-correlation theorem gives an asymptotic energy estimate. The new ingredient is a short matrix inequality with a free clipping parameter. It strengthens the lower bound for this energy in terms of the number of distinct zeros. The gain is a nonnegative correction from overlaps between different nearby zeros on the critical line, which is retained even when some of these zeros are double. The matrix inequality, the threshold lemma, the block dichotomy, the counting assembly and the exact arithmetic are proved formally in Lean 4. The constant relies on a computer-assisted local inequality from recent work that has not yet been refereed. That computation was re-run independently, and every imported input is listed. This paper is primarily an experiment in AI-assisted mathematical research (Section 4).
ER -