Infinite sets of primes of density 0

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.

There is a general strategy, which unfortunately is a failure because the information needed for success is scarcely ever available.

Let $A$ denote a set of primes, and $B$ the set of all products of powers of the primes in $A$. Then

$$ \prod_{p \in A}(1 - p^{-s})^{-1} = \sum_{n \in B}n^{-s} $$

by the Euler product formula. The Dirichlet series on the right hand side has nonnegative coefficients, so by a theorem of Landau, the point where the line of convergence crosses the real axis is a singularity of the sum function. If $A$ is finite, this singularity is at the origin. Thus if it can be established that the line of convergence is to the right of the imaginary axis, $A$ must be infinite (The only reason to invoke the theorem of Landau is to guarantee that there always is a singularity on the line of convergence). Of course, existence of a singularity of the sum function somewhere in the open right hand half plane would also work, even if we do not know the line of convergence.

This tends to fail because we can't get a good grip on $B$. Suppose $A$ is the set of twin primes (primes $p$ such that $p+2$ is also prime). Nothing really useful is known about the set $B$ of integers that are products of powers of twin primes. But if the accepted conjecture about the distribution of twin primes holds, there will be a singularity at $s = 1$, so $B$ cannot be really sparse.

As an example, there are infinitely many primes $p \equiv 1\pmod{4}$. Letting these primes, together with $2$, constitute $A$, we see that $B$ contains the values of the polynomial $n^2 + 1$, since the latter is never divisible by any prime $q \equiv 3\pmod{4}$. Then

$$ \sum_{n \in B}n^{-\sigma} \geq \sum_{m = 1}^{\infty}(m^2 + 1)^{-\sigma} $$

and thus the Dirichlet series over $B$ has a singularity in the half plane $\sigma \geq 1/2$. So $A$ has to be infinite.

Admittedly, this example is not that interesting.

This is essentially a comment on Kevin Buzzard's answer, but because of its length and (I feel) importance, I am leaving it separately.

It is a consequence of Serre's open image theorem that the set of primes of supersingular reduction for a non-CM elliptic curve $E_{/ \mathbb{Q}}$ has density zero. (Conversely, for a CM elliptic curve, the set of such primes has density $\frac{1}{2}$.) Therefore, when Elkies showed -- by a method of proof strongly reminiscent of Euclid's argument, as you say! -- that there are infinitely many supersingular primes for every $E$, he was in particular giving a very interesting and natural example of an infinite set of primes of density zero.

It is interesting to try to generalize this example to analogous cases. For instance, Barry Mazur (my thesis advisor) asked me once if I could prove infinitude of primes of supersingular reduction for abelian surfaces $A_{/\mathbb{Q}}$ with quaternionic multiplication. (I had previously proven the analogue of Serre's open image theorem in this case, or rather reproven it, since Ohta had done it many years before.) I had no idea how to do this. My roommate, David Jao, ended up writing a thesis on generalizations of Elkies' proof to elliptic curves corresponding to $\mathbb{Q}$-rational points on Atkin-Lehner quotients of genus $0$ modular curves $X_0(N)$, in particular giving new examples of infinitude of supersingular primes for certain elliptic curves over quadratic imaginary fields. He tried the QM case (in which $X_0(N)$ is replaced by a Shimura curve) but couldn't make the argument work for a funny reason: although a QM abelian surface is much like a non-CM elliptic curve in its reduction behavior at most primes, it is like a CM elliptic curve in that it has some primes of guaranteed supersingular reduction, namely those primes dividing the discriminant of the quaternion algebra. He was able to set up a Euclid-Elkies style argument where you use the supersingular primes you already have to get another one, but he couldn't show that this "other one" wasn't one of the finitely many primes dividing the quaternionic discriminant! More recently, Baba and Granath established this result for QM surfaces of quaternionic discriminant $6$ and $10$.

(There is an analogous problem for Drinfeld modules in the function field case, but Bjorn Poonen famously gave an example of a Drinfeld module with no primes of supersingular reduction.)

I seem to vaguely remember that Serre's 1981 paper on applications of the Cebotarev density theorem and/or his 1997 JAMS paper on equidistribution contains further examples of this general type, but I wouldn't swear to it.