THE SPECTRAL ZETA FUNCTION OF THE UNIT $ n $-SPHERE AND AN INTEGRAL TREATED BY RAMANUJAN

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概要

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• We introduce a certain Dirichlet series which is closely related to the spectral zeta function of the unit $ n $-sphere. We prove basic properties of this Dirichlet series including a meromorphic continuation, a functional equation, and closed evaluation of values at some integers. These results are consequences of an integral representation of the Dirichlet series as an integral of a product of a polynomial and the Hurwitz zeta function. This integral has already appeared in Ramanujan's work independent of the Dirichlet series. The integral representation also gives another approach for evaluating the determinant of the Laplacian on the unit $ n $-sphere.

• Faculty of Mathematics, Kyushu Universityの論文

著者

• Mizuno Yoshinori

  Department Of Mathematics Keio University

• Mizuno Yoshinori

  Department Of Applied Analysis And Complex Dynamical Systems Graduate School Of Informatics Kyoto Un

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• THE SPECTRAL ZETA FUNCTION OF THE UNIT $ n $-SPHERE AND AN INTEGRAL TREATED BY RAMANUJAN

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