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The zeta function of a mathematical operator is a function defined as : for those values of ''s'' where this expression exists, and as an analytic continuation of this function for other values of ''s''. Here "tr" denotes a functional trace. The zeta function may also be expressible as a spectral zeta function〔Lapidus & van Frankenhuijsen (2006) p.23〕 in terms of the eigenvalues of the operator by :. It is used in giving a rigorous definition to the functional determinant of an operator, which is given by : The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically. ==References== * * 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Zeta function (operator)」の詳細全文を読む スポンサード リンク
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