## Complex Moments and the distribution of Values of $L(1,\chi_D)$ over Function Fields with Applications to Class Numbers

We investigate the moments and the distribution of $L(1,\chi_D)$, where $\chi_D$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_q$, as $n\to\infty$. Specializing $n=2g+1$ and making use of one case of Artin's class number formula, we obtain similar results for the class number $h_D$ associated to $\mathbb{F}_q(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin's class number formula and deduce analogous results for $h_DR_D$ where $R_D$ is the regulator of $\mathbb{F}_q(T)[\sqrt{D}]$.

(Mathematika, 65(2), 236-271. doi:10.1112/S0025579318000396)