New large value estimates for Dirichlet polynomials

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3\mathrm{/}4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(\sigma ,T)\le T^{30(1-\sigma )\mathrm{/}13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17\mathrm{/}30+o(1)}$.