Number of prime factors with a given multiplicity

Abstract 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 .