Large deviations of the argument of the Riemann zeta function
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910382196523008 |
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| author | Dobner, Alexander |
| author_facet | Dobner, Alexander |
| contents | Let $S(t) = \frac{1}π\Im \logζ\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/3}$. For $V \leq (\log T)^{1/3-\varepsilon}$ our bound has a Gaussian shape with variance proportional to $\log\log T$. At the endpoint, $V \asymp \left(\frac{\log T}{\log \log T}\right)^{1/3}$, our result implies the best known $Ω$-theorem for $S(t)$ which is due to Tsang. We also explain how the method breaks down for $V \gg \left(\frac{\log T}{\log \log T}\right)^{1/3}$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis we extend our results to the range $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_01747 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Large deviations of the argument of the Riemann zeta function Dobner, Alexander Number Theory Let $S(t) = \frac{1}π\Im \logζ\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/3}$. For $V \leq (\log T)^{1/3-\varepsilon}$ our bound has a Gaussian shape with variance proportional to $\log\log T$. At the endpoint, $V \asymp \left(\frac{\log T}{\log \log T}\right)^{1/3}$, our result implies the best known $Ω$-theorem for $S(t)$ which is due to Tsang. We also explain how the method breaks down for $V \gg \left(\frac{\log T}{\log \log T}\right)^{1/3}$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis we extend our results to the range $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$. |
| title | Large deviations of the argument of the Riemann zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2101.01747 |