THE RAMANUJAN JOURNAL, Vol. 10, No. 3 (2005), 291-303.

Generalized Zeta Regularizations, Quantum Class Number, and Appel's $0$-Functions

NOBUSHIGE KUROKAWA
Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo, 152-0033, Japan
Email: kurokawa@math.titech.ac.jp

MASATO WAKAYAMA
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
Email: wakayama@math.kyushu-u.ac.jp

Received July 30, 2002; Accepted October 8, 2002

Abstract. We introduce a generalized notion of zeta regularization. As applications we show the quantum analogue of the Lerch formula and of the Dirichlet class number formulas. We also give further examples related to Appel's $0$-functions.

Keywords: zeta regularization, Hurwitz zeta function, quantum class number, Appell's $O$-function, $q$-gamma function

2000 Mathematics Classification: Primary 11M36