m at h . FA ] 1 D ec 1 99 5 A HILBERT SPACE OF DIRICHLET SERIES AND SYSTEMS OF DILATED FUNCTIONS IN L 2 ( 0 , 1 )

For a function φ in L(0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates φ(nx), n = 1, 2, 3, . . . , constitutes a Riesz basis or a complete sequence in L(0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f(s) = ∑ n ann, where the coefficients an are square summable. It proves useful to model H as the H space of the infinite-dimensional polydisk, or, which is the same, theH space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given f in H and characters χ, fχ(s) = ∑ n anχ(n)n is a vertical limit function of f . We study certain probabilistic properties of these vertical limit functions.