New large value estimates for Dirichlet polynomials
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910109613948928 |
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| author | Guth, Larry Maynard, James |
| author_facet | Guth, Larry Maynard, James |
| contents | 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 $N(σ,T)\le T^{30(1-σ)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20552 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New large value estimates for Dirichlet polynomials Guth, Larry Maynard, James Number Theory 11M26, 11N05 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 $N(σ,T)\le T^{30(1-σ)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$. |
| title | New large value estimates for Dirichlet polynomials |
| topic | Number Theory 11M26, 11N05 |
| url | https://arxiv.org/abs/2405.20552 |