Cyclic Vectors in the Dirichlet Space

We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if |f(z)| |g (z)| at all points and if g is cyclic, then f is cyclic. Theorems 3-5 give a sufficient condition (f is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function f to be cyclic.