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I would like to use the Erdős–Wintner theorem, but not quite sure if I can. I have a function $f$ that I believe is additive and in the process of proving/disproving, but $f(p) \nleq 1$ for all $p$. Does this mean that if I just show the first series (leftmost in link) converges I have a limiting distribution? Or does it mean that I can't use the theorem? I no longer have access to the original paper anymore since taking off from school, but when I did I was still not sure.

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