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Numerical verification of Littlewood's bounds for |L(1,χ)|
Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-01-28 , DOI: 10.1016/j.jnt.2020.12.017
Alessandro Languasco

Let L(s,χ) be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined modq, where q is an odd prime. In this paper we introduce a fast method to compute |L(1,χ)| using the values of Euler's Γ function. We also introduce an alternative way of computing logΓ(x) and ψ(x)=Γ/Γ(x), x(0,1). Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on |L(1,χ)|, where χ runs over the non-principal primitive Dirichlet characters modq, for every odd prime q up to 107. The programs used and the results here described are collected at the following address http://www.math.unipd.it/~languasc/Littlewood_ineq.html.



中文翻译:

Littlewood边界的数值验证| L(1,χ)|

大号sχ是与定义的非主要本原Dirichlet字符χ相关的Dirichlet L-函数q,其中q是奇数素数。在本文中,我们介绍了一种快速的计算方法|大号1个χ|使用EulerΓ函数的值。我们还介绍了另一种计算方式日志ΓXψX=Γ/ΓXX01个。使用这样的算法,我们在数值上验证了经典的Littlewood边界和最近的Lamzouri-Li-Soundararajan估计|大号1个χ|,其中χ遍历非主要原始Dirichlet字符q,对于每个奇数质数q最多10 7。在以下地址http://www.math.unipd.it/~languasc/Littlewood_ineq.html上收集了所使用的程序和此处描述的结果。

更新日期:2021-02-03
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