The Distribution of Logarithmic Derivatives of Quadratic L-functions in Positive Characteristic

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.