Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

We study Ruelle's type zeta and L-functions for a torsion free abelian group Γ of rank v < 2 defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when v = 2, 4 and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus ℝ^v/Γ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.