Half-Isolated Zeros and Zero-Density Estimates

We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.