A neat way to show that a set of primes has zero density (within the primes) is to use the following form of the Green-Tao theorem:

Any set of positive density within the primes has arbitrarily long arithmetic progressions.

In particular, any set of primes which does not contain three (say) elements in arithmetic progression must have zero density.

If a set $P$ of primes has the property that $p_1,p_2\in P$ implies that $(p_1+p_2)/2\not\in P$, then the set $P$ has zero density in the primes.

As an immediate corollary, we get the (unconditional) result that the Mersenne primes (ones of the form $2^p-1$) have zero density.

This trick seems like it could be applied to many other natural sets of primes.

A neat way to show that a set of primes has zero density (within the primes) is to use the following form of the Green-Tao theorem:

Any set of positive density within the primes has arbitrarily long arithmetic progressions.

In particular, any set of primes which does not contain three (say) elements in arithmetic progression must have zero density.

If a set $P$ of primes has the property that $p_1,p_2\in P$ implies that $(p_1+p_2)/2\not\in P$, then the set $P$ has zero density in the primes.

As an immediate corollary, we get the (unconditional) result that the Mersenne primes (ones of the form $2^p-1$) have zero density.

This trick seems like it could be applied to many other natural sets of primes.

EDIT: Zero density for Mersenne primes is easy to get anyway, as Ben Weiss points out, and so is zero density for primes of form $n^2+1$, which would also follow from this method.