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A functional limit theorem related to natural divisors. (English) Zbl 0937.11040

Let \({\mathbf D}[0,1]\) be the space of all real-valued functions on \([0,1]\) which are right continuous and have left-hand limit, endowed with the Skorokhod topology and the Borel \(\sigma\)-algebra \(\mathcal D\). Given a non-negative multiplicative function \(f\) let \(F(m,v)=\sum f(d)\), the sum being over all \(d\mid m\) such that \(d\leq v\). Then for \(t\in[0,1]\) set \(X_n(m,t)=F(m,m)^{-1}F(m,n^t)\in{\mathbf D}[0,1]\). For \(B\in\mathcal D\) the frequency \(\nu_n(X_n\in B)\) defines a probability measure on \(\mathcal D\) which is denoted by \(\nu_n\cdot X_n^{-1}\). The main result is that if \(f(p)=\varkappa>0\) and \(f(p^k)\geq 0\) for all primes \(p\) and integers \(k\geq 2\), then \(\nu_n\cdot X_n^{-1}\) converges weakly to a limiting measure defined on \(\mathcal D\). This extends results of J.-M. Deshouillers, F. Dress and G. Tenenbaum [Acta Arith. 34, 273-285 (1979; Zbl 0408.10035)] concerning the divisor function.

MSC:

11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions in probabilistic number theory
11N60 Distribution functions associated with additive and positive multiplicative functions

Citations:

Zbl 0408.10035
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