We consider closed contact manifolds $(M,\xi)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which equivariant symplectic homology is a well defined invariant. Under weak index assumptions on a non-degenerate contact form $\alpha$ on $M$, we establish a sharp lower bound $r_M$ for the number of closed Reeb orbits of $\alpha$, and show that equality holds if and only if $\alpha$ is lacunary. The bound $r_M$ is essentially determined by the deviation of a suitable finite sum of the contact Betti numbers of $M$ from its asymptotic average. We also compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim \H_*(M/S^1;\Q)$. This is joint work with Miguel Abreu.