Denote $\pi(x)$ be the number of primes $\leq x,$ $p(n)$ be the $n$-th prime number. We have $\pi(p(n))=n.$
It's well known that $$\pi(x)\sim \frac{x}{\log x} \\p(n)\sim n\log n.$$
Is it always true that if $f(x)$ is a function and
$f(x)>0, \forall x>0$
$f(x)$ is a monotonic function
$f(x)=O(x^r)$ for some $r>0$
$\sum_{n=1}^{\infty}f(n)=\infty$
then $$\sum_{p\leq x}f(p)\sim \sum_{t\leq \pi(x)}f(t\log t)$$?