  Given a transitive Anosov diffeomorphism or flow on a closed connected
Riemannian manifold $M$, the Livshits theorem states that a H\"{o}lder function
$\varphi : M \to \mathbb{R}$ is a coboundary if all of its periods vanish. We
explain how a finer statistical understanding of the distribution of these
periods can be used to obtain a stronger version of the classical Livshits
theorem where one only has to check that the periods of $\varphi$ vanish on a
set of positive asymptotic upper density. We also include a strengthening of
the nonpositive Livshits theorem.
