Arithmetic zeta function

In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory.

The arithmetic zeta function ζ_X (s) is defined by an Euler product analogous to the Riemann zeta function:

where the product is taken over all closed points x of the scheme X. Equivalently, the product is over all points whose residue field is finite. The cardinality of this field is denoted N(x).

If X is the spectrum of a finite field with q elements, then

For a variety X over a finite field, it is known by Grothendieck's trace formula that

where ${\displaystyle Z(X,t)}$ is a rational function (i.e., a quotient of polynomials).

Given two varieties X and Y over a finite field, the zeta function of ${\displaystyle X\times Y}$ is given by

...
Wikipedia