CN-TST 2014 Problem 7

Prove that for any positive integers $k$ and $N$ , $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},$$ where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$ ). Note: For integer $n>1$ , $\omega (n)$ denotes number of distinct prime factors of $n$ , and $\omega (1)=0$ .