Zeta zeros on the critical line

The nontrivial zeros of the Riemann zeta function $\zeta(s)$ are its zeros $\rho$ with $0<\operatorname{Re}\rho<1$. The critical line is $\operatorname{Re}s=\tfrac12$; the Riemann hypothesis says that every nontrivial zero lies on it. A zero of multiplicity $m$ counts $m$ times, and a simple zero is a zero of multiplicity $1$. Let $N(T)$ be the number of nontrivial zeros with $0<\operatorname{Im}\rho\le T$, and let $N_0^{s}(T)$ be the number of those that are simple and lie on the critical line. The quantity is $\kappa^{*}=\liminf_{T\to\infty}N_0^{s}(T)/N(T)$, the share of the zeros that are simple and on the critical line, in the worst case as $T$ grows. A result is a proof that $\kappa^{*}\ge c$, and it must not assume an unproved hypothesis. A higher $c$ is better. Several results below use the Montgomery–Taylor constant $H=\tfrac32-\tfrac{1}{\sqrt2}\cot\tfrac{1}{\sqrt2}\approx0.6725007$. Under the Riemann hypothesis, larger values are known (for example $19/27\approx0.7037$ by Bui and Heath-Brown). Such conditional bounds do not count here. For zeros on the critical line with no condition on multiplicity, the human record before 2026 was $0.417293$ (more than five-twelfths), by the same four authors as the baseline. Anthropic's post gives that record as 41.6%.

Classic baseline: $0.407511$, published by Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler in 2020.

These results are Claimed: their authors published them, and we have not checked them.

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