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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^3/4, 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 (, T) T^30 (1-) /13+o (1) and asymptotics for primes in short intervals of length x^17/30+o (1).
Guth et al. (Thu,) studied this question.