Zeta function of algebraic variety over $\mathbb{F}_q$, formula for product of Witt vectors. Let $X$ be an algebraic variety over $\mathbb{F}_q$. We have the definition of the zeta function of $X$ as follows:$$Z(X, t) = \prod_{x \, = \,\text{Fr}_q\text{-orbit in }X(\overline{\mathbb{F}}_q)}(1 - t^{\deg x})^{-1},$$where we write $\deg x = \text{number of elements in }x$.
Question. What is $Z(X_1 \times X_2, t)$ in terms of $Z(X_1, t)$ and $Z(X_2, t)$? The result supposedly be a formula for the product of Witt vectors...
 A: The zeta function can be considered as a map 
$$
Var_k \to 1 + t \mathbb{Z}[[t]] =: W
$$
from isomorphism classes of varieties over $k = \mathbb{F}_q$ to the multiplicative group of power series with coefficient $1$. More precisely we should be working with the Grothendieck group of varieties $K_0(Var_k)$. 
$W$ is an abelian group under multiplication of power series which I will denote by $\times$. However $W$ also has a product $*$ that makes $(W,\times, *)$ a ring with addition given by $\times$ and product given by $*$. The product $*$ is characterized by the identity
$$
\frac{1}{1 - at} * \frac{1}{1 - bt} = \frac{1}{1 - abt}
$$
for $a,b \in \mathbb{Z}$. This is the (big) Witt vector ring. 
Then the following is true:
$$
Z(X \times Y, t) = Z(X,t) * Z(Y,t)
$$
This result is Theorem 2.1(i) in the following paper: 
Ramachandran, N. (2015), "Zeta functions, Grothendieck groups, and the Witt ring." Bull. Sci. Math. 139 (2015), no. 6, 599–627. 
This paper is also relevant: 
Ramachandran, N., Tabuada, G., "Exponentiable motivic measures." J. Ramanujan Math. Soc. 30 (2015), no. 4, 349–360. 
PDFs of both are available on Ramachandran's website. 
