More than two thirds of the zeta zeros are simple and on the critical line
Levent Alpöge, Ralph Furman
math.NT
Aug 13, 2026 · v2
TL;DR
The main unconditional results on the proportions of simple and distinct zeta zeros, extended to Dirichlet L-functions, are stated to be formally verified in Lean 4.
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 $\frac{5}{12}$ 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.
Problem
Montgomery's 1973 argument gives that at least two thirds of the nontrivial zeros of the Riemann zeta function are simple and on the critical line, but only under the Riemann hypothesis. The best unconditional records were 5/12 for simple zeros on the line and 0.6603 for distinct zeros.
Approach
Montgomery's deduction is made unconditional. A rank-trace inequality for Hermitian matrices is applied to a finite compression of Weil's explicit-formula Hermitian form. Sylvester's law of inertia bounds the contribution of off-line zero pairs. The analytic pair-correlation inputs come from Aryan and from Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh, and the results are formally verified in Lean 4.
Results
Unconditionally, at least 2/3 of zeros are simple and on the critical line, and at least 5/6 are distinct. With the Montgomery–Taylor window these become 0.6725 and 0.8362. The results extend to primitive Dirichlet L-functions.
| Proportion | Previous record | Indicator window | Montgomery–Taylor window |
|---|
| Simple zeros on the critical line | 5/12 | 2/3 | 0.6725 |
| Distinct zeros | 0.6603 | 5/6 | 0.8362 |
Unconditional lower bounds on zero proportions (new results vs. previous records)
