Let (a(j)) be an unbounded convex sequence of natural numbers. In 1983, Carleson [7] proved that a necessary and sufficient condition for the (C, 1) means of \((S_{a(j)}f)\) (partial Fourier sums) to converge uniformly to f in the supremum norm is \(\sup _j j^{-1/2}\log a(j) <+\infty \) . In this paper, we prove that if we consider Riesz logarithmic means instead of (C, 1) means, then almost everywhere convergence holds for any integrable function and for any convex (a(j)). That is (see Corollary 1.3): Let (a(j)) be any unbounded convex sequence of natural numbers and \(f\in L^1\) . Then the following convergence holds almost everywhere: \( \frac{1}{\log n}\sum _{j=1}^{n}\frac{S_{a(j)}f}{j} \rightarrow f. \) In fact, we verify a more general statement (see Theorem 1.1).
