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A note on the distribution function of additive arithmetical functions in short intervals. (English) Zbl 0638.10049

Let \(f\) be an additive arithmetic function having a distribution \(F\). For any sequence \(1\leq b(n)\leq n\), \(b(n)\to \infty\), let \[ Q_n(b,f)(x)=card\{n\leq m\leq n+b(n): f(m)\leq x\}/b(n). \] In this note, we determine the slowest growing function \(b\) so that \(Q_n(b,f)\) tends weakly to \(F\), for various \(f\).
Reviewer: G.J.Babu

MSC:

11K65 Arithmetic functions in probabilistic number theory
60F05 Central limit and other weak theorems
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