New large value estimates for Dirichlet polynomials

• 成果类型: Article
• 署名作者: Guth, Larry; Maynard, James
• 署名单位: Massachusetts Institute of Technology (MIT); University of Oxford; University of Oxford
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.203.2.6
• 发表日期: 2026-03
• 页码: 623-675
• 关键词: Zero density estimates; Dirichlet polynomials; Large values estimates; number

摘要

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).