estimation of sum over primes

Question

The problem is this: I have read that if i have a function $f: \mathbb{N}\to\mathbb{C}$ and we are interested in estimation of the sum $V(x)=\sum_{p<x} f(p)$ where $p$ runs on primes numbers then is equivalent to estimate $S(x)=\sum_{n<x} f(n)\Lambda(n)$ (where $\Lambda(n)$ is the von Mangoldt function). I don't see why this holds. Thank you in advance.

Answer 1

This is not correct as stated, if I understand correctly. The von Mangoldt function equals $\log(p)$ for primes and prime powers, and is zero otherwise. The logarithm may not disappear. The theorems of Mertens are a good example of what is true: $$ \sum_{p\le x}\frac{\log(p)}{p}=\log(x)+O(1)=\sum_{n\le x}\frac{\Lambda(n)}{n}. $$ This is for example used (and refined) for the proof of Dirichlet's theorem on primes in arithmetic progressions.