Small values of $|L^{\prime }/L(1,\chi )|$

Abstract

In this paper, we investigate the quantity $m_q:={\min }_{\chi \ne {\chi }_0}|L^{\prime }/L(1,\chi )|$, as $q\to \infty$ over the primes, where $L(s,\chi )$ 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 $L^{\prime }(1,\chi )\ne 0$ for every non trivial Dirichlet character $\chi$ mod $q$ where $3\le q\le 10^7$ 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}.

arXiv comment20 pages, 3 figures, 1 table