Multiple sine functions and Selberg zeta functions (Q1180780)

Forming suitable Weierstraß products the author defines a multiple gamma function \(G_ r\) (related to the Barnes multiple gamma function) and a multiple sine function \(F_ r\) of order \(r\geq 2\). The multiple sine functions are related with the polylogarithm function \(Li_ k(x)\) and may be used to express special values of zeta functions such as \(\zeta(2m+1)\) (\(m\geq 1\)). The main result is an announcement of the calculation of the gamma factors involved in the Selberg-Gangolli- Wakayama zeta functions of rank one locally symmetric spaces. A typical example is the case of an even dimensional real hyperbolic space.