Small values of $| L^\prime/L(1,χ) |$
Youness Lamzouri, Alessandro Languasco
TL;DR
The paper addresses the minimal magnitude of the logarithmic derivative $|L'/L(1,\chi)|$ across nontrivial Dirichlet characters modulo large prime $q$, establishing the bound $m_q \ll \dfrac{\log\log q}{\sqrt{\log q}}$ via a probabilistic model $\textup{Ld}(1, \mathbb{X})$ and a tight discrepancy estimate. It proves asymptotic moment formulas for $L'/L(1,\chi)$, reduces large-power behavior to short Dirichlet polynomials, and uses zero-density results to justify approximations. The discrepancy between the actual distribution and the random model is bounded, enabling the main bound to follow from positivity of the random model’s density at the origin. Complementing the theory, extensive computations for odd primes $q \le 10^7$ yield precise numerical bounds $\dfrac{21}{200q} < m_q < \dfrac{5}{\sqrt{q}}$ and confirm $L'(1,\chi) \neq 0$ for all nontrivial $\chi$ in this range, with data publicly available to support related conjectures and further research.
Abstract
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}.