Let μ be a Borel measure on the unit circle, and let e_n denote the L² norm of the monic orthogonal polynomial of degree n with respect to μ. By the classical Szegő theorem e_n→ 0 if and only if the logarithmic integral of μ diverges. In this talk, we discuss the rate of decay of the quantities e_n for certain classes of measures μ with divergent logarithmic integral. This talk is based on the joint work with A. Borichev and M. Sodin [1, 2].