Abstract
Let L be a number field. For a given prime p, we define integers
α
p
L
$ \alpha_{p}^{L} $
and
β
p
L
$ \beta_{p}^{L} $
with some interesting arithmetic properties. For instance,
β
p
L
$ \beta_{p}^{L} $
is equal to 1 whenever p does not ramify in L and
α
p
L
$ \alpha_{p}^{L} $
is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of
α
p
L
$ \alpha_{p}^{L} $
is not zero for all p then such residues determine the genus of the integral trace.
The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type I × f N