SCHWEITZER 2014 Problem 4

For a positive integer $n$ , define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$ . Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$ , i.e $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2$ Prove that [list] (a) $$\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha)$$ (b) There is no real number $\beta<\alpha$ such that $$\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta)$$ [/list]