A Note on Twisted Moments of Dirichlet $L$-functions

In this paper, we establish an asymptotic formula for the twisted second moments of Dirichlet $L$-functions with one and two twists when averaged over all primitive Dirichlet characters of modulus $R$, where $R$ is a monic polynomial in ${\mathbb{F}}_q[T]$. The main result in this paper generalizes the work of Djanković [`The reciprocity law for the twisted second moment of Dirichlet $L$-functions over rational function fields’, Bull. Aust. Math. Soc. 98 (2018), no. 3, 382–388].