# What do we know about the number $\prod_p \left( 1 - \frac{1}{p}\right)^4\left( 1 + \frac{4}{p} + \frac{1}{p^2} \right)$?

I am looking at a result of Peyre, and he says for a certain variety, the number of rational points of height less than $B$ is:

$$N(B) \sim \frac{1}{3} \color{#3DB08E}{\prod_p \left( 1 - \frac{1}{p}\right)^4\left( 1 + \frac{4}{p} + \frac{1}{p^2} \right) }B (\log B)^3$$

The manifold is some sort of Veronese del Pezzo type variety surface:

$$V(\mathbb{Q}) = \big\{ \big( (x_0:y_0), (x_1:y_1),(x_2:y_2) \big) : x_0 x_1 x_2 = y_0 y_1 y_2 \big\} \subset \text{P}^1_\mathbb{Q}\times \text{P}^1_\mathbb{Q} \times \text{P}^1_\mathbb{Q}$$

The height is some sort of exotic height -- probably a Weil height of some kind.

$$H( ...) = \sup \big(|x_0|, |y_0|\big)\sup \big(|x_1|, |y_1|\big) \sup \big(|x_2|, |y_2|\big)$$

This is the height associated to the anti-canonical divisor $\omega_V^{-1} = \mathcal{O}_V(1,1,1)$. He has removed some exceptional divisors which come from variables being zero, $x = 0$ or $y = 0$. So that $U = V \,\backslash \bigcup\{ x_i = y_j = 0 \}$.

What do we know about this constant? Is it a period? It looks like the Euler product of some type of L-function of some scheme. An algebraic geometer, might try to read the product over primes as the "Adelic" part and the $B \log^3 B$ part as places over some ring of polynomials.

Do the Cohen-Lenstra heuristics predict anything here? Those are more about estimating ranks of class groups, or Galois groups, but they momentarily look like a probability.

The variety $V$ is a del Pezzo surface of degree $6$ (this has nothing to do with Veronese varieties). One has
$$\#V(\mathbb{F}_p) = 1 + 4p + p^2,$$ which is how the Euler factors naturally arise (the factor $(1 - 1/p)^4$ is just a convergence factor). The constant has an interpretation as an adelic volume; all this is explained in the cited paper of Peyre.