On series of dilated functions

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number theoretic factors, but precise results exist only in a few special cases. In this paper we use connections with orthogonal function theory and GCD sums to prove several new results, improve old ones and also to simplify and unify the theory.