Half-Isolated Zeros and Zero-Density Estimates

We introduce a new zero-detecting method which is sensitive to the vertical distribution of zeros of the zeta function. This allows us to show that there are few ‘half-isolated’ zeros. If we assume that the zeros of the zeta function are restricted to finitely many vertical lines, then we can improve the classical zero-density result of Ingham-Huxley to N (σ, T ) T ^{24(1−σ)/11+o(1)} (and so give new results about primes in short intervals under this assumption). This is joint work with James Maynard.