Siegel's mass formula and averages of Dirichlet $L$-functions over function fields

Abstract

Let $\mathfrak{G}$ be a genus of definite ternary lattices over ${\mathbb{F}}_q[t]$ of square-free determinant. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representation numbers over the classes of $\mathfrak{G}$ and for the mass of $\mathfrak{G}$. Our proof of the mass formula shows an interesting and seemingly new relation with certain averages of Dirichlet $L$-functions.