Numerical estimates on the Landau-Siegel zero and other related quantities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910815466029056 |
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| author | Languasco, Alessandro |
| author_facet | Languasco, Alessandro |
| contents | Let $q$ be a prime, $χ$ be a non-principal Dirichlet character $\bmod\ q$ and $L(s,χ)$ be the associated Dirichlet $L$-function. For every odd prime $q\le 10^7$, we show that $L(1,χ_\square) > c_{1} \log q$ and $β< 1- \frac{c_{2}}{\log q}$, where $c_1=0.0124862668\dotsc$, $c_2=0.0091904477\dotsc$, $χ_{\square}$ is the quadratic Dirichlet character $\bmod\ q$ and $β\in (0,1)$ is the Landau-Siegel zero, if it exists, of such a set of Dirichlet $L$-functions. As a by-product of the computations here performed, we also obtained some information about the Littlewood and Joshi bounds on $L(1,χ_\square)$ and on the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-q})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_10722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical estimates on the Landau-Siegel zero and other related quantities Languasco, Alessandro Number Theory 11M20, 11-04, 11Y60 Let $q$ be a prime, $χ$ be a non-principal Dirichlet character $\bmod\ q$ and $L(s,χ)$ be the associated Dirichlet $L$-function. For every odd prime $q\le 10^7$, we show that $L(1,χ_\square) > c_{1} \log q$ and $β< 1- \frac{c_{2}}{\log q}$, where $c_1=0.0124862668\dotsc$, $c_2=0.0091904477\dotsc$, $χ_{\square}$ is the quadratic Dirichlet character $\bmod\ q$ and $β\in (0,1)$ is the Landau-Siegel zero, if it exists, of such a set of Dirichlet $L$-functions. As a by-product of the computations here performed, we also obtained some information about the Littlewood and Joshi bounds on $L(1,χ_\square)$ and on the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-q})$. |
| title | Numerical estimates on the Landau-Siegel zero and other related quantities |
| topic | Number Theory 11M20, 11-04, 11Y60 |
| url | https://arxiv.org/abs/2301.10722 |