Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Zeta regularized products
HTML articles powered by AMS MathViewer

by J. R. Quine, S. H. Heydari and R. Y. Song PDF
Trans. Amer. Math. Soc. 338 (1993), 213-231 Request permission

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.
References
  • Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834, DOI 10.1007/978-1-4612-0999-7
    • E. W. Barnes, Genesis of the double gamma function, Proc. London Math. Soc. 31 (0000), 358-381. —, The theory of the double gamma function, Philos. Trans. Royal Soc. (A) 196 (1901), 265-388. —, The theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374-425.
  • J. R. Quine and J. Choi, Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math. 26 (1996), no. 2, 719–729. MR 1406503, DOI 10.1216/rmjm/1181072081
  • Eric D’Hoker and D. H. Phong, The geometry of string perturbation theory, Rev. Modern Phys. 60 (1988), no. 4, 917–1065. MR 969998, DOI 10.1103/RevModPhys.60.917
  • H. Dym and H. P. McKean, Fourier series and integrals, Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972. MR 0442564
  • Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
  • H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
  • Tobias Orloff, Another proof of the Kronecker limit formulae, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 273–278. MR 894327
  • B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5
    • J. R. Quine and R. Y. Song, A double Stirling asymptotic formula, preprint.
  • D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
  • Giovanni Sansone and Johan Gerretsen, Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, 1960. MR 0113988
  • Takuro Shintani, A proof of the classical Kronecker limit formula, Tokyo J. Math. 3 (1980), no. 2, 191–199. MR 605088, DOI 10.3836/tjm/1270472992
  • Takuro Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167–199. MR 460283
  • Ilan Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19 (1988), no. 2, 493–507. MR 930041, DOI 10.1137/0519035
  • Marie-France Vignéras, L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire $\textrm {PSL}(2,\,\textbf {Z})$, Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978) Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 235–249 (French). MR 556676
  • A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), no. 3, 439–465. MR 891947
  • André Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88, Springer-Verlag, Berlin-New York, 1976. MR 0562289
Similar Articles
Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 338 (1993), 213-231
  • MSC: Primary 58G26; Secondary 11F72, 11M41, 30D10, 33B15, 81T30
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1100699-1
  • MathSciNet review: 1100699