More generally, if X is the spectrum of the ring of integers of an algebraic number field, then ζX (s) is the Dedekind zeta function.
The zeta function of affine and projective spaces over a scheme X are given by The latter equation can be deduced from the former using that, for any X that is the disjoint union of a closed and open subscheme U and V, respectively, Even more generally, a similar formula holds for infinite disjoint unions.
In particular, this shows that the zeta function of X is the product of the ones of the reduction of X modulo the primes p: Such an expression ranging over each prime number is sometimes called Euler product and each factor is called Euler factor.
There are a number of conjectures concerning the behavior of the zeta function of a regular irreducible equidimensional scheme X (of finite type over the integers).
Many (but not all) of these conjectures generalize the one-dimensional case of well known theorems about the Euler-Riemann-Dedekind zeta function.
Hasse and Weil conjectured that ζX (s) has a meromorphic continuation to the complex plane and satisfies a functional equation with respect to s → n − s where n is the absolute dimension of X.
It is a consequence of the Weil conjectures (more precisely, the Riemann hypothesis part thereof) that the zeta function has a meromorphic continuation up to
Subject to the analytic continuation, the order of the zero or pole and the residue of ζX (s) at integer points inside the critical strip is conjectured to be expressible by important arithmetic invariants of X.
An argument due to Serre based on the above elementary properties and Noether normalization shows that the zeta function of X has a pole at s = n whose order equals the number of irreducible components of X with maximal dimension.
More generally, Soulé conjectured[4] The right hand side denotes the Adams eigenspaces of algebraic K-theory of X.
However, there are still very few proven results about the L-factors of arithmetic schemes in characteristic zero and dimensions 2 and higher.
Ivan Fesenko initiated[5] a theory which studies the arithmetic zeta functions directly, without working with their L-factors.
Two other applications of Fesenko's theory are to the poles of the zeta function of proper models of elliptic curves over global fields and to the special value at the central point.