Discoveries / number-theory

The Proportion of Zeta Zeros on the Critical Line

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.