Multiple sine functions and Selberg zeta functions
Copyright policy )Multiple sine functions and Selberg zeta functions
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.
polylogarithm function, hyperbolic space, Spectral theory; trace formulas (e.g., that of Selberg), Selberg-Gangolli-Wakayama zeta functions, multiple gamma function, \(\zeta (s)\) and \(L(s, \chi)\), symmetric spaces, 11M41, Other Dirichlet series and zeta functions, multiple sine functions, 11F72, Determinants and determinant bundles, analytic torsion
polylogarithm function, hyperbolic space, Spectral theory; trace formulas (e.g., that of Selberg), Selberg-Gangolli-Wakayama zeta functions, multiple gamma function, \(\zeta (s)\) and \(L(s, \chi)\), symmetric spaces, 11M41, Other Dirichlet series and zeta functions, multiple sine functions, 11F72, Determinants and determinant bundles, analytic torsion
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