  In this paper we prove the main conjectures of Berkovich and Jagy about
weighted averages of representation numbers over an S-genus of ternary lattices
(defined below) for any odd squarefree S \in N. We do this by reformulating
them in terms of local quantities using the Siegel-Weil and Conway-Sloane
formulas, and then proving the necessary local identities. We conclude by
conjecturing generalized formulas valid over certain totally real number fields
as a direction for future work.
