More than two thirds of the zeta zeros are simple and on the critical line
Abstract
We prove unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function, counted with multiplicity, are simple and lie on the critical line, and that at least five sixths are distinct; the previous unconditional records are 5/12 and $0.6603$. With the Montgomery--Taylor window the constants become $0.6725$ and $0.8362$. The argument makes Montgomery's 1973 deduction unconditional: the Riemann hypothesis, classically needed to read the zero side as a positive sum over real ordinates, is replaced by a rank-trace inequality applied to a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs. The analytic inputs are those of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. The results extend to primitive Dirichlet L-functions and are formally verified in Lean 4.
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BibTeX
@misc{alpoge2026more,
title = {{More than two thirds of the zeta zeros are simple and on the critical line}},
author = {Levent Alpöge and Ralph Furman},
howpublished = {arXiv (Cornell University)},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.13637},
url = {https://arxiv.org/abs/2608.13637},
note = {CC BY},
}
RIS
TY - UNPB
TI - More than two thirds of the zeta zeros are simple and on the critical line
AU - Levent Alpöge
AU - Ralph Furman
PY - 2026
DA - 2026/08/13
PB - arXiv (Cornell University)
DO - 10.48550/arxiv.2608.13637
UR - https://arxiv.org/abs/2608.13637
AB - We prove unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function, counted with multiplicity, are simple and lie on the critical line, and that at least five sixths are distinct; the previous unconditional records are 5/12 and $0.6603$. With the Montgomery--Taylor window the constants become $0.6725$ and $0.8362$. The argument makes Montgomery's 1973 deduction unconditional: the Riemann hypothesis, classically needed to read the zero side as a positive sum over real ordinates, is replaced by a rank-trace inequality applied to a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs. The analytic inputs are those of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. The results extend to primitive Dirichlet L-functions and are formally verified in Lean 4.
SN - 2331-8422
ER -