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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Zeta regularized products

Authors: J. R. Quine, S. H. Heydari and R. Y. Song
Journal: Trans. Amer. Math. Soc. 338 (1993), 213-231
MSC: Primary 58G26; Secondary 11F72, 11M41, 30D10, 33B15, 81T30
MathSciNet review: 1100699
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Abstract: If $ {\lambda _k}$ is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, $ \prod\nolimits_k {{\lambda _k}} $, to be $ \exp ( - Z\prime (0))$ where $ Z(s) = \sum\nolimits_{k = 0}^\infty {\lambda _k^{ - s}} $. We assume that $ Z(s)$ has analytic continuation to a neighborhood of the origin. If $ {\lambda _k}$ is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as $ \det \prime \Delta $, the determinant of the Laplacian, and $ \prod\nolimits_k {({\lambda _k} - \lambda )} $ is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as $ \lambda \to - \infty $ and its relationship to the Weierstrass product. We give some applications to the theory of Barnes' multiple gamma functions and elliptic functions. A new proof is given for Kronecker's limit formula and the product expansion for Barnes' double Stirling modular constant.

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Keywords: Determinant of the Laplacian, zeta function, Barnes' double gamma function, multiple gamma function, zeta regularization, Weierstrass product, Kronecker limit formula
Article copyright: © Copyright 1993 American Mathematical Society

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