  We study the Case sum rules, especially $C_0$, for general Jacobi matrices.
We establish situations where the sum rule is valid. Applications include an
extension of Shohat's theorem to cases with an infinite point spectrum and a
proof that if $\lim n (a_n -1)=\alpha$ and $\lim nb_n =\beta$ exist and
$2\alpha <\abs{\beta}$, then the Szeg\H{o} condition fails.
