# Arithmetic zeta functions of products and fibrations

Suppose $$X$$ and $$Y$$ are schemes of finite type over $$\mathbb{Z}$$.

How is the arithmetic zeta function of their product, $$\zeta_{X \times Y} (s)$$, related to their individual zeta functions, $$\zeta_X(s)$$ and $$\zeta_Y(s)$$? More generally if $$Z$$ is a fiber bundle over $$X$$ with fiber $$Y$$, does a similar relation hold?

(I have been wondering if $$\ln \zeta(s)$$ is a ring homomorphism to an appropriate target ring from the Grothendieck ring of schemes of finite type over $$\mathbb{Z}$$.)

We have the relation $$\zeta_{X\times Y}(s) = \zeta_X(s)* \zeta_Y(s)$$ where $$*$$ is the Witt product in the Witt ring of $$\mathbb Z[[t]]$$. For any commutative ring A, the (big) Witt ring $$W(A)$$ is defined by:
• Under addition, $$W(A), +$$ is isomorphic to the group $$(1 + tA[[t]],\times)$$
• The multiplication $$*$$ is uniquely determined by $$(1-at)^{-1}(1-bt)^{-1} = (1-abt)^{-1}.$$
That this is indeed true can be verified on each Euler product: the $$k$$-points of $$X\times Y$$ are in bijection with pairs of $$k$$-points on $$X,Y$$. In fact, the same relation holds if we only assume that $$Z$$ is a fiber bundle over $$X$$ with fiber $$Y$$ by the same fact about points: to give a point $$P$$ on $$Z$$ is to give a point on $$X$$ (the image of $$P$$ under projection) and a point on the fiber.
[The reference only talks about this formula over finite fields but I think it holds for the zeta function of any scheme $$X$$ finite type over $$\mathbb Z$$ because such a zeta function is the product of a zeta function $$X/\mathbb F_p$$ over every finite field. As an example, the Riemann zeta function is the product of the zeta function of a point over every $$\mathbb F_p$$. Since the formula is proven Euler product by euler product, this is not an issue.