This talk is based on joint papers with J.-P. Kahane and D. Mauldin.
Our starting point is the following problem of J.A. Haight and H.V. Weizsacker:
Given a non-negative measurable function f defined for positive realnumbers is it true that Sumn f(nx) converges almosteverywhere or diverges almost everywhere?
This problem was unsolved for almost thirty years and was answered inthe negative in 1998. We discuss the history of this problem, recent generalizations,and results.