A Hilbert space of Dirichlet series and systems of dilated functions in L2(0,1)

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