Let L be a ring additively isomorphic to ℤd. The zeta function of L is defined to bewhere the sum is taken over all subalgebras H of finite index in L. This zeta function
has a natural Euler product decomposition:These functions were introduced in a paper of Grunewald, Segal and Smith [5] where
the local factors ζL[otimes ]ℤp(s) were shown to always be rational functions in p−s. The proof
depends on representing the local zeta function as a definable p-adic integral and then
appealing to a general result of Denef’s [1] about the rationality of such integrals.
The proof of Denef relies on Macintyre’s Quantifier Elimination for ℚp [8] followed
by techniques developed by Igusa [6] which employ resolution of singularities.