# $n^{1/2}-f(n)=O(n^\epsilon)$ for all $\epsilon>0$?

Define $$f(n)$$ as follows:

$$f(1)=0$$

$$f(n)=\frac{(n^{1/2}+1)\Lambda(n)+f(n-1)\psi(n-1)}{\psi(n)}$$ for $$n>1$$

where $$\Lambda(n)$$ is the Von Mangoldt function and $$\psi(n)$$ is the second Chebyshev function.

Show or disprove that $$n^{1/2}-f(n)=O(n^{\epsilon})$$ for all $$\epsilon>0$$