论文标题
$ |的小值l^\ prime/l（1，χ）| $
Small values of $| L^\prime/L(1,χ) |$
论文作者
论文摘要
在本文中，我们调查了数量$ m_q：= \ min_ {χ\neχ_0} | l^\ prime/l（1，χ）| $，AS $ q \ to \ to \ infty $，其中$ l（s，χ）$是dirichlet $ l $ function to nontivial dirichlet carture modulo $ q $。我们的主要结果表明，$ M_Q \ ll \ log \ log q/\ sqrt {\ log q} $。我们还计算每个奇数$ q $ $ q $ to $ 10^7 $的$ m_q $。结果，我们从数字上验证，对于每个奇数$ q $，$ 3 \ le q \ le 10^7 $，我们都有$ C_1/q <m_q <5/\ sqrt {q} $，$ c_1 = 21/200 $。特别是，这表明每个非琐碎的dirichlet字符$χ$ mod $ q $ $ l^\ prime（1，χ）\ ne 0 $其中$ 3 \ leq Q \ leq Q \ leq 10^7 $是Prime，回答了此范围内的枪支，Murty和Rath的问题。我们还提供有关$ M_Q $ - 值的一些统计和散点图，请参见第6节。此处所述的程序和计算结果可在以下网址上找到：\ url {http://wwwwwwwwww.math.unipd.it/~languanguce/smallvalues/smallvalues.html}。
In this paper, we investigate the quantity $m_q:=\min_{χ\ne χ_0} | L^\prime/L(1,χ)|$, as $q\to \infty$ over the primes, where $L(s,χ)$ is the Dirichlet $L$-function attached to a non trivial Dirichlet character modulo $q$. Our main result shows that $m_q \ll \log\log q/\sqrt{\log q}$. We also compute $m_q$ for every odd prime $q$ up to $10^7$. As a consequence we numerically verified that for every odd prime $q$, $3 \le q \le 10^7$, we have $c_1/q< m_q<5/\sqrt{q}$, with $c_1=21/200$. In particular, this shows that $L^\prime(1,χ) \ne 0$ for every non trivial Dirichlet character $χ$ mod $q$ where $3\leq q\leq 10^7$ is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the $m_q$-values, see Section 6. The programs used and the computational results described here are available at the following web address: \url{http://www.math.unipd.it/~languasc/smallvalues.html}.
