  Here we use Vinogradov's prime distribution theorem and a multi-dimensional
generalization due to Harman to strengthen some recent results concerning the
periodic points of Hamiltonian diffeomorphisms. In particular we establish
resonance relations for the mean indices of the fixed points of Hamiltonian
diffeomorphisms which do not have periodic points with arbitrarily large
periods in $\mathbb{P}^2$, the set of natural numbers greater than one which
have at most two prime factors when counted with multiplicity. As an
application of these results we partially recover, using only symplectic tools,
a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere
by Franks and Handel.
