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

Abstract

Let G be a genus of definite ternary lattices over Fq[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 representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows an interesting and seemingly new relation with certain averages of Dirichlet L-functions.