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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An Erdős-Wintner theorem for differences of additive functions


Author: Adolf Hildebrand
Journal: Trans. Amer. Math. Soc. 310 (1988), 257-276
MSC: Primary 11K65; Secondary 11N60
MathSciNet review: 965752
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Abstract: An Erdös-Wintner type criterion is given for the convergence of the distributions $ {D_x}(z) = {[x]^{ - 1}}\char93 \{ 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965752-X
PII: S 0002-9947(1988)0965752-X
Article copyright: © Copyright 1988 American Mathematical Society