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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
Number theory
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