Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

Solution 1:

This example morally comes from class field theory. It is just easy enough that I can do it using elementary algebraic number theory but not class field theory.

Write $\theta = (1+\sqrt{-23})/2$; it obeys $\theta^2-\theta+6=0$. Let $R$ be the ring $\mathbb{Z}[\theta]$. This is the ring of integers in the field $\mathbb{Q}(\sqrt{-23})$. It turns out that the class group of $R$ is $\mathbb{Z}/3$, representatives for the three ideal classes are: $$I_0:= \langle 1 \rangle,\ I_1:=\langle 2, \theta -1 \rangle,\ I_2:=\langle 2, \theta \rangle.$$ I leave this for you to check from the standard Minkowski bounds. Note that $I_1 I_2 = \langle 4, 2(\theta-1), 2 \theta, 6 \rangle = \langle 2 \rangle$, so $I_1^{-1} = (1/2) I_2$ and vice versa. Note for future use that we are giving a $\mathbb{Z}$ basis for the latter two ideals, and that $\langle 1, \theta \rangle$ is a $\mathbb{Z}$-basis for $I_0$.

An arbitrary ideal equivalent to $I_0$ is of the form $\langle x+y\theta \rangle$, and we have $N(\langle x+y\theta \rangle) = x^2 - xy + 6 y^2$. An arbitrary ideal equivalent to $I_0$ is of the form $\langle x+y\theta \rangle$ in precisely two ways, because the unit group of $R$ is $\pm 1$. So $$\# \{ I \subseteq R : \ I \sim I_0 \ \mbox{and} \ N(I) = n \} = \frac{1}{2} \# \{(x,y) : x^2 - xy + 6y^2 =n \}. \quad (1)$$

Similarly, an ideal equivalent to $I_1$ is of the form $\alpha I_1$, for $\alpha \in I_1^{-1} = (1/2) I_2$. This $\alpha$ looks like $(1/2) (2 x + y(\theta -1))$ for some integers $x$ and $y$ and we have $N(\alpha I_1) = N(\alpha) N(I_1) = (x^2 - xy +3 y^2/2) \cdot 2 = 2x^2 - 2xy + 3 y^2$ (exercise!). Running through the same argument as before: $$\# \{ I \subseteq R : \ I \sim I_1 \ \mbox{and} \ N(I) = n \} = \frac{1}{2} \# \{(x,y) : 2 x^2 - xy + 3y^2 =n \}. \quad (2)$$ Similarly, $$\# \{ I \subseteq R : \ I \sim I_2 \ \mbox{and} \ N(I) = n \} = \frac{1}{2} \# \{(x,y) : 2 x^2 + xy + 3y^2 =n \}. \quad (3)$$

Call these numbers $a_0(n)$, $a_1(n)$ and $a_2(n)$. We want to know that $a_2(p)$ is positive for infinitely many primes.

We have unique factorization into ideals in the ring $R$. So we have the analog of the Euler product $$\sum_n \frac{a_0(n) + a_1(n) + a_2(n)}{n^s} = \prod_{p} \frac{1}{(1- p^{-s})^{a_0(p)+a_1(p)+a_2(p)}}.$$

Let $\omega$ be a primitive cube root of unity. Since the class group is $\mathbb{Z}/3$, with $[I_1]^2=[I_2]$ and $[I_1]^3=[I_0]$, we also have the identities: $$\sum_n \frac{a_0(n) + \omega a_1(n) + \omega^2 a_2(n)}{n^s} = \prod_{p} \frac{1}{(1-p^{-s})^{a_0(p)}} \frac{1}{(1-\omega p^{-s})^{a_1(p)}} \frac{1}{(1-\omega^2 p^{-s})^{a_2(p)}}.$$ $$\sum_n \frac{a_0(n) + \omega^2 a_1(n) + \omega a_2(n)}{n^s} = \prod_{p} \frac{1}{(1-p^{-s})^{a_0(p)}} \frac{1}{(1-\omega^2 p^{-s})^{ a_1(p)}} \frac{1}{(1-\omega p^{-s})^{ a_2(p)}}.$$

Define these three sums to be $L_0(s)$, $L_1(s)$ and $L_2(s)$. Using equations $(1)$, $(2)$ and $(3)$, and approximating sums by integrals, we see that $L_0(s)$ has a simple pole at $s=1$ while $L_1(s)$ and $L_2(s)$ are bounded as $s \to 1$. Using explicit numerical computation, one can check that $L_1(s)$ and $L_2(s)$ do not vanish at $s=1$.

Now the end game is exactly like the proof of Dirichlet's theorem on primes in arithmetic progressions: take logs of the above equations to deduce that $$\log \left( \frac{1}{s-1} \right) + O(1) = \sum \frac{a_0(p)+a_1(p)+a_2(p)}{p^s} + O(1)$$ $$O(1) = \sum \frac{a_0(p)+\omega a_1(p)+ \omega^2 a_2(p)}{p^s} + O(1)$$ $$O(1) = \sum \frac{a_0(p)+\omega^2 a_1(p)+ \omega a_2(p)}{p^s} + O(1)$$

We thus have $$\sum \frac{a_2(p)}{p^s} = (1/3)\log \left( \frac{1}{s-1} \right) + O(1) \ \mbox{as}\ s \to 1^{+}$$ and, in particular, $a_2(p)$ is nonzero infinitely often.

So, where is the class field theory? For $R$ the ring of integers of a number field, $H$ the class group of $R$, and $\chi: H \to \mathbb{C}^*$ a nontrivial character, define $L(\chi, s) = \sum_{I \subseteq R} \chi([I]) N(I)^{-s}$. If you wanted to generalize this proof to an arbitrary quadratic number field, you would want to know that $L(\chi, s) \neq 0$. For any particular $R$ and $\chi$, you can prove this by hand. In Dirichlet's theorem, the analogous result can be proved by looking at the zeta function of the cyclotomic field. In the general case, it can be proved by looking at the $\zeta$ function of the class field -- but only once you know this exists!