An explicit form of Ingham's zero density estimate
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909816321998848 |
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| author | Chourasiya, Shashi Simonič, Aleksander |
| author_facet | Chourasiya, Shashi Simonič, Aleksander |
| contents | Ingham (1940) proved that $N(σ,T)\ll T^{3(1-σ)/(2-σ)}\log^{5}{T}$, where $N(σ,T)$ counts the number of the non-trivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}\geqσ\geq 1/2$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this result with the exponent $(7-5σ)/(2-σ)$ of the logarithmic factor. In addition, we also provide an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An explicit form of Ingham's zero density estimate Chourasiya, Shashi Simonič, Aleksander Number Theory 11M06, 11M26 Ingham (1940) proved that $N(σ,T)\ll T^{3(1-σ)/(2-σ)}\log^{5}{T}$, where $N(σ,T)$ counts the number of the non-trivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}\geqσ\geq 1/2$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this result with the exponent $(7-5σ)/(2-σ)$ of the logarithmic factor. In addition, we also provide an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line. |
| title | An explicit form of Ingham's zero density estimate |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2507.15184 |