Canadian Mathematical Bulletin ( IF 0.655 ) Pub Date : 2021-05-03 , DOI: 10.4153/s0008439521000266
Ertan Elma, Yu-Ru Liu

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$. Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

$k\geqslant 1$是一个自然数，而$\omega _k(n)$表示具有多重性k的自然数n的不同质因数的数量。我们用$k\geqslant 1$估计函数$\omega _k$ 的一和二阶矩。此外，我们证明了函数$\omega _1(n)$具有正态序$\log \log n$和函数$(\omega _1(n)-\log \log n)/\sqrt {\log \ log n}$服从正态分布。最后，我们证明了函数$\omega _k(n)$$k\geqslant 2$没有正态阶$F(n)$对于任何非递减的非负函数F

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