An Erdős-Wintner theorem for differences of additive functions

Hildebrand, Adolf

doi:10.1090/S0002-9947-1988-0965752-X

An Erdős-Wintner theorem for differences of additive functions
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Trans. Amer. Math. Soc. 310 (1988), 257-276 Request permission

Abstract:

An Erdös-Wintner type criterion is given for the convergence of the distributions ${D_x}(z) = {[x]^{ - 1}}\# \{ 1 \leqslant n \leqslant x: f(n + 1) - f(n) \leqslant z\}$, 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 $f = \lambda \log$ for some constant $\lambda$. This had been conjectured by Erdős.

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