Average value of the divisor class numbers of real cubic function fields (Q6611510)

Let \(k=\mathbb{F}_q(T)\) be the rational function field with \(q \equiv 1 (\mathrm{mod}\: 3)\). Let \(m \in \mathbb{F}_q[T]\) be a cube-free polynomial where the degree of \(m\) is divisible by \(3\).\N\NThe main goal of this paper is to establish the mean value of \(|L(1,\chi)|^2\) when \(\chi\) average runs through the primitive cubic even Dirichlet characters of \(\mathbb{F}_q[T]\), where \(L(s, \chi)\) is the associated Dirichlet \(L\)-function. As consequence, the authors derive an asymptotic formula for the mean value of the divisor class numbers of real cubic function fields \(K_m=k(\sqrt[3]{m})\).