  Let ${\bf X}=(X, \Sigma, m, \tau)$ be a dynamical system. We prove that the
bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(\tau^nx)g(\tau^{-n}x)}{n}$
converges almost everywhere for each $f,g\in L^{\infty}(X).$ We also give a
proof along the same lines of Bourgain's analog result for averages.
