Geodesic
A note on the distribution of some additive functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 262-265.

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Let $f$ denote an additive arithmetical function with continuous limiting distribution $F$ on the integers. Then $f$ also has a limiting distribution $G$ on shifted primes. Under some growth conditions on the values of $f$ at primes, we provide optimal lower bounds for the modulus of continuity of $F$ and $G$, at all points from a specified infinite set. 
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Gérald Tenenbaum. A note on the distribution of some additive functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 262-265. http://geodesic.mathdoc.fr/item/TM_2012_276_a21/

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