It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product1Let2What are the sequences {zn} for which I(r) is a bounded function of r?