Mertens' Theorem states that $$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(\log x).$$ This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number theorem is equivalent to $$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + o(\log x).$$ However no reference or proof is given, it just simply says that "Although this equivalence is not explicitly mentioned there, it can for instance be easily derived from the material in chapter I.3 of: G. Tenenbaum. Introduction to analytic and probabilistic number theory."

So what is a proof/reference for this fact? I naturally tried to prove it myself using partial summation, and it is easy to see where the $\log \log x$ comes from, however the constant $M$ and $o(\log x)$ are a bit more mysterious.

Standard proofs in prime number theory often proceed by introducing a logarithmic weight via the Von Mangoldt function, then proving an asymptotic and going back again. I would prefer to avoid such an approach as I have a different problem in mind coming from counting rational points on varieties where I can't introduce a logarithmic weight.