Let $f(n)$ be an arithmetic function such that $\frac{\log n}{\log 2}\leq f(n)\leq 2n$, for all positive integer $n$. By using other properties of $f(n)$ I can obtain $$ \sum_{n\leq x}f(n) = O\left(\dfrac{x^2}{(\log x)^{1/3}}\right). $$ Now, I would like to find some upper bound for $|\sum_{n\leq x}(-1)^{n+1}f(n)|$.
In fact, my prediction is that $|\sum_{n\leq x}(-1)^{n+1}f(n)|=O(x^{3/2})$. However, I don't know too much tools to attack this kind of problem. Any one can give me some hint?
P.S. $f(n)$ is the smallest positive integer $k$ such that $n$ divides the $k$th Fibonacci number $F_k$.