$\vert L(1,χ)\vert$的Littlewood界的数值验证
Numerical verification of Littlewood's bounds for $\vert L(1,χ)\vert$
摘要：
令$L(s,χ)$是与$\bmod\ q$定义的非平凡本原狄利克雷特征标$χ$相关联的狄利克雷$L$ -函数，其中$q$是奇素数.本文介绍了一种利用欧拉$Γ$函数值计算$\vert L(1,χ) \vert$的快速方法。我们还介绍了计算$\log Γ(x)$和$ψ(x)= Γ^\prime/Γ(x)$、$x\in(0,1)$的另一种方法。使用这样的算法，我们数值验证了经典的Littlewood界和最近的Lamzouri-Li-Soundarajan估计$\vert L(1,χ) \vert$，其中$χ$运行在非平凡的原始Dirichlet字符$\bmod\ q$，对于每个奇素数$q$到$10^7$。所使用的程序和此处描述的结果在以下地址\url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}收集。
英文摘要：
Let $L(s,χ)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $χ$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,χ) \vert$ using the values of Euler's $Γ$ function. We also introduce an alternative way of computing $\log Γ(x)$ and $ψ(x)= Γ^\prime/Γ(x)$,$x\in(0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $\vert L(1,χ) \vert$, where $χ$ runs over the non trivial primitive Dirichlet characters $\bmod\ q$, for every odd prime $q$ up to $10^7$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.
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