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arXiv · 2301.10722

Numerical estimates on the Landau-Siegel zero and other related quantities

Alessandro Languasco
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Abstract

Let $q$ be a prime, $\chi$ be a non-principal Dirichlet character $\bmod\ q$ and $L(s,\chi)$ be the associated Dirichlet $L$-function. For every odd prime $q\le 10^7$, we show that $L(1,\chi_\square) > c_{1} \log q$ and $\beta < 1- \frac{c_{2}}{\log q}$, where $c_1=0.0124862668\dotsc$, $c_2=0.0091904477\dotsc$, $\chi_{\square}$ is the quadratic Dirichlet character $\bmod\ q$ and $\beta\in (0,1)$ 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 $L(1,\chi_\square)$ and on the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-q})$.

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Alessandro Languasco. 2023-01-25. Numerical estimates on the Landau-Siegel zero and other related quantities. https://doi.org/10.1016/j.jnt.2023.04.008

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