Density of Primes in different sets

Question

I am examining the density of Primes in other sets than the naturals. E.g. we want to have the density of Mersenne Primes. From the prime number theorem I know that in the naturals we have $$ \frac{\pi(n)}{\frac{n}{\log(n)}} \rightarrow 1. $$

But what if I have a set $M^{\leq n} := \left\{m\in\mathbb N: m\leq n, m\text{ is a Mersenne number}\right\}$, which are all Mersenne numbers less or equal to $n$. And the function $ \pi_M(n) = \mid\left\{p\leq n, p\text{ is a Mersenne prime}\right\}\mid, $ which are the Mersenne primes less or equal than $n$.

What would be the correct way to examine their destiny? Would I look for the limit of $$ \frac{\pi_M(n)}{\frac{\mid M^{\leq n}\mid}{\log(n)}} $$ or rather $$ \frac{\pi_M(n)}{\frac{\mid M^{\leq n}\mid}{\log(\mid M^{\leq n}\mid)}} $$

Answer 1

The general heuristic is that "a random integer $n$ has a $1/\log n$ probability of being prime". Therefore the first guess as to an appropriate density of primes within a set $S$ is usually $$ \#\{p\le x,\, p\in S,\, p\text{ prime}\} \approx \sum_{\substack{n\le x \\ n\in S}} \frac1{\log n}. $$ This heuristic should be modified to include any divisibility tendencies of $S$; for some sets this barely matters at all, but for others (such as the Mersenne numbers) there are notably different probabilities of being divisible by small primes than for random integers.

For the specific case of the Mersenne numbers, there are probably multiple expositions to be found that go through the heuristic in detail (in particular to justify the conjecture that there are infinitely many Mersenne primes).