  The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an
asymptotic sum formula $$N(\lambda) := \sum_{\lambda_j \leq \lambda} \left|
\int_H e_j \, dV_H \right|^2 = C_{H,M} \lambda^{\operatorname{codim} H} +
O(\lambda^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert
basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with
$\Delta_g e_j = -\lambda_j^2 e_j$, and $H$ is an embedded submanifold. We show
for some suitable definition of `$\sim$', $$ N(\lambda) \sim C_{H,M}
\lambda^{\operatorname{codim} H} + Q(\lambda) \lambda^{\operatorname{codim} H -
1} + o(\lambda^{\operatorname{codim} H - 1}) $$ where $Q$ is a bounded
oscillating term and is expressed in terms of the geodesics which depart and
arrive $H$ in the normal directions.
  In work by Canzani, Galkowski, and Toth, they establish (as a corollary to a
stronger result involving defect measures) that if the set of recurrent
directions of geodesics normal to $H$ has measure zero, then we obtain improved
bounds on the individual terms in the sum -- the period integrals. We are able
to give a dynamical condition such that $Q$ is uniformly continuous and
`$\sim$' can be replaced with `$=$'. This implies improved bounds on period
integrals, and this condition is weaker than the recurrent directions having
measure zero. Moreover, our result implies improved bounds for period integrals
if there is no $L^1$ measure on $SN^*H$ that is invariant under the first
return map. This generalizes a theorem of Sogge and Zelditch and of Galkowski.
