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Multiple sine functions and Selberg zeta functions. (English) Zbl 0738.11041
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.

MSC:
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11M41 Other Dirichlet series and zeta functions
58J52 Determinants and determinant bundles, analytic torsion
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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