TY - JOUR
T1 - Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups
AU - Kurokawa, Nobushige
AU - Wakayama, Masato
AU - Yamasaki, Yoshinori
PY - 2008/9
Y1 - 2008/9
N2 - 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 ℝv/Γ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.
AB - 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 ℝv/Γ 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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U2 - 10.1142/S0129167X08004959
DO - 10.1142/S0129167X08004959
M3 - Article
AN - SCOPUS:51449109166
VL - 19
SP - 957
EP - 979
JO - International Journal of Mathematics
JF - International Journal of Mathematics
SN - 0129-167X
IS - 8
ER -