  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.
