For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\pi(x)}{x/\ln x}=1
$$
with $\pi(x)$ is the prime counting function. David Speyer gives a nice heuristic [here][1] in order to explain why Mertens formulas aren't enough for pnt. However, I'm concerned with finding a series of integer numbers $a$ such that
$$
\sum_{a \le x}\frac{1}{a} = \ln\ln x + C + O(1/x)
$$  
with $C$ suitable constant, but it isn't true that
$$
\lim_{x\to\infty}\frac{f(x)}{x/\ln x}=1
$$
where $f$ is the counting function of numbers $a$. This would give a a concrete counterexample for 
$$
\text{Mertens}\rightarrow\text{pnt}.
$$

  [1]: https://mathoverflow.net/questions/95743/why-could-mertens-not-prove-the-prime-number-theorem