The Proportion of Zeta Zeros on the Critical Line

The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg established a positive proportion, Levinson reached a third in 1974, Conrey two fifths in 1989, and the record stood at $\tfrac{5}{12}$ for zeros that are simple and on the line, and $0.6603$ for distinct zeros. Under the Riemann hypothesis, Montgomery deduced $\tfrac23$ simple from the pair-correlation second moment in 1973. His prime-side evaluation was already unconditional; RH entered only to read the zero side as a positive sum over real ordinates. Goldston and Suriajaya isolated that termwise positivity as the remaining obstacle and asked what would follow if it could be removed. This removes it, proving unconditionally that at least $\tfrac23$ of zeros are simple and on the line and at least $\tfrac56$ are distinct - $67.25\ldots\%$ and $0.83625$ with the Montgomery-Taylor window.

An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate. What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgomery's conditional $\tfrac23$ to a single obstruction, the termwise positivity that fails for zeros off the line, and asked what would follow without it. Theorem A replaces that positivity with a rank-trace inequality on a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs; reading the negative index of truncations as a count of off-line pairs is Bombieri's device. The paper also proves the bound sharp for this route: improving on $\tfrac23$ this way would need pair-correlation information beyond Fourier support 1.