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New large value estimates for Dirichlet polynomials
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DOI：10.4007/annals.2026.203.2.6.png)
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Abstract
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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 N3/4, 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) <= T30(1-sigma )/13+o(1) and asymptotics for primes in short intervals of length x17/30+o(1).
Keywords:
Zero density estimates
Dirichlet polynomials
Large values estimates
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