Milnor-Selberg zeta functions and zeta regularizations

Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」By a similar idea for constructing Milnor\u27s gamma functions, we study \u22higher depth determinants\u22 of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. Moreover, it is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product