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Additive arithmetical functions and statistical independence. (English) Zbl 0022.00903
Important results are obtained concerning additive functions, i.e. functions \(f(n)\) which satisfy \(f(n_1 n_2) = f(n_1)+f(n_2)\), whenever \((n_1,n_2) = 1\); so that \(f(n)\) is determined by the values of \(f(p^k)\), for all primes \(p\) and all \(k\). It is shown that such a function has an asymptotic distribution function \(\sigma\) if and only if \(\sum p^{-1} g(p)\) and \(\sum' p^{-1} g(p)^2\) are convergent, when \(g(p) = f(p)\) or \(g(p) = 1\) according as \(|f(p)| < 1\) or \(|f(p)| \geq 1\). Furthermore, if \(\sigma_p\) is the asymptotic distribution function of the function \(f_p(n)\), which is defined by \(f_p(n)=f(p^k)\) if \(p^k|n\) and \(p^{k+1} \nmid n\), then \(\sigma\) is the infinite convolution of the \(\sigma_p\) and the above condition for the existence of \(\sigma\) is identical with the condition that this infinite convolution be convergent. The complete proof of which large parts are given in earlier publications [cf. P. Erdős, J. Lond. Math. Soc. 13, 119–127 (1938; Zbl 0018.29301)] is long and involves delicate operations with prime numbers related to Brunn’s method.

MSC:
11N60 Distribution functions associated with additive and positive multiplicative functions
11K65 Arithmetic functions in probabilistic number theory
Keywords:
Number theory
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