Half-sized gaps between zeros of the Riemann zeta function

Assuming the Riemann hypothesis, we prove that the lower limit of the normalised gaps between consecutive zeros of the Riemann zeta function is at most $1/2$. Our methods utilise a more efficient method for detecting gaps with short Dirichlet polynomials. Specifically, we apply resonance to a quotient of shifted zeta functions, rather than to the number of zeros in a short interval, using a Möbius resonator adapted to the symmetry between the shifts.