Average value of the divisor class numbers of real cubic function fields

We compute an asymptotic formula for the divisor class numbers of real cubic function fields K m = k ( m 3 ) {K}_{m}=k\left(\sqrt[3]{m}) , where F q {{\mathbb{F}}}_{q} is a finite field with q q elements, q ≡ 1 ( mod 3 ) q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) , k ≔ F q ( T ) k:= {{\mathbb{F}}}_{q}\left(T) is the rational function field, and m ∈ F q [ T ] m\in {{\mathbb{F}}}_{q}\left[T] is a cube-free polynomial; in this case, the degree of m m is divisible by 3. For computation of our asymptotic formula, we find the average value of ∣ L ( s , χ ) ∣ 2 {| L\left(s,\chi )| }^{2} evaluated at s = 1 s=1 when χ \chi goes through the primitive cubic even Dirichlet characters of F q [ T ] {{\mathbb{F}}}_{q}\left[T] , where L ( s , χ ) L\left(s,\chi ) is the associated Dirichlet L L -function.