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On completely dense sequences. (English) Zbl 1024.11052
The sequence \((x_n)\) of real numbers is completely dense in an interval \(I\) if for every \(s =1, 2, \dots \) the \(s\)-dimensional sequence \(((x_{n+1}, x_{n+2},\dots , x_{n+s}))\) is dense in \(I^s\). The authors give the following sufficient condition for the complete density of a multiplicative arithmetic function: Let \(p_n\) denote the \(n\)-th prime number, and let \(f (n)\) be a positive multiplicative function satisfying the following conditions: (i) \(\lim _{n\rightarrow \infty } f(p_n)^n = 1\), (ii) \( \prod _{p_n, f (p_n) > 1} f (p_n) = + \infty \), \(\prod _{p_n , f(p_n) < 1} f (p_n) = 0\). Then the sequence \((f (n))\) is completely dense in the set of positive real numbers. In the second part of the paper it is shown that in the space of all real sequences endowed with the Fréchet metric, the family of all completely dense sequences is residual.
MSC:
11K36 Well-distributed sequences and other variations
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References:
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