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

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

2007-02-08

arxiv.org/abs/math/0702209v2

Internat. J. Math., 19 (2008) No. 8, 957-979.

Mathematics

Number Theory

20 pages

Scientific paper

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ 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 $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

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