Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910675744325632 |
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| author | Arguin, Louis-Pierre Bailey, Emma |
| author_facet | Arguin, Louis-Pierre Bailey, Emma |
| contents | Let $δ>0$ and $σ=\frac{1}{2}+\tfracδ{\log T}$. We prove that, for any $α>0$ and $V\sim α\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|ζ(σ+\rm{i} τ)|>V\big\}\geq C_α(δ)\int_V^\infty \frac{e^{-y^2/\log\log T}}{\sqrt{π\log\log T}} \rm{d} y,$ where $δ$ is large enough depending on $α$. The result is unconditional on the Riemann hypothesis. As a consequence, we recover the sharp lower bound for the moments on the critical line proved by Heap & Soundararajan and Radziwiłł & Soundararajan. The constant $C_α(δ)$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_19803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem Arguin, Louis-Pierre Bailey, Emma Number Theory Probability Let $δ>0$ and $σ=\frac{1}{2}+\tfracδ{\log T}$. We prove that, for any $α>0$ and $V\sim α\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|ζ(σ+\rm{i} τ)|>V\big\}\geq C_α(δ)\int_V^\infty \frac{e^{-y^2/\log\log T}}{\sqrt{π\log\log T}} \rm{d} y,$ where $δ$ is large enough depending on $α$. The result is unconditional on the Riemann hypothesis. As a consequence, we recover the sharp lower bound for the moments on the critical line proved by Heap & Soundararajan and Radziwiłł & Soundararajan. The constant $C_α(δ)$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments. |
| title | Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2403.19803 |