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An Erd6s-Wintner type criterion is given for the convergence of the distributions D (z) = [x]-#{1 < n < x: f(n + 1) f(n) < z}, where f is a real-valued additive function. A corollary of this result is that an additive function f, for which f(n + 1) f(n) tends to zero on a set of density one, must be of the form f = A log for some constant A. This had been conjectured by Erd6s.

An Erdős-Wintner theorem for differences of additive functions