Small and large gaps between zeros of Dirichlet $L$-functions

For every fixed primitive Dirichlet $L$-function and every $\epsilon >0$, we prove unconditionally that infinitely many consecutive zeros have spacing less than $1\mathrm{/}2+\epsilon$ times the mean. Assuming the Riemann Hypothesis, we also prove a positive proportion of such spacings for the Riemann zeta-function, and infinitely many spacings greater than $3.63$ times the mean for every fixed primitive Dirichlet $L$-function.