Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Zeta determinant for double sequences of spectral type


Author: M. Spreafico
Journal: Proc. Amer. Math. Soc. 140 (2012), 1881-1896
MSC (2010): Primary 11M41; Secondary 33E20
DOI: https://doi.org/10.1090/S0002-9939-2011-11061-X
Published electronically: October 12, 2011
MathSciNet review: 2888176
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence.


References [Enhancements On Off] (What's this?)

  • 1. J. Brüning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1988) 369-415. MR 899656 (88g:35151)
  • 2. J. Choi and J.R. Quine, Zeta regularized products and functional determinants on spheres, Rocky Mount. Jour. Math. 26 (1996) 719-729. MR 1406503 (97k:58176)
  • 3. P.B. Gilkey, Invariance theorems, the heat equation, and the Atiyah-Singer index theorem (Studies in Adv. Math., CRC Press, 1995). MR 1396308 (98b:58156)
  • 4. G.H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge University Press, 1915.
  • 5. J. Jorgenson and S. Lang, Basic analysis of regularized series and products, Lect. Notes in Math. 1564, Springer-Verlag, 1993. MR 1284924 (95e:11094)
  • 6. J. Jorgenson and S. Lang, Explicit formulas for regularized products and series, Lect. Notes in Math. 1593, Springer-Verlag, 1994. MR 1329730 (96f:11110)
  • 7. J. Jorgenson and S. Lang, Artin formalism and heat kernels, J. Reine Angew. Math. 447 (1994) 165-200. MR 1263173 (95c:11106)
  • 8. J. Jorgenson and S. Lang, On Cramér's theorem for general Euler products with functional equations, Math. Ann. 297 (1993) 383-416. MR 1245398 (94k:11101)
  • 9. J.D. Murray, Asymptotic analysis, Springer-Verlag, 1984. MR 740864 (85m:34085)
  • 10. J.R. Quine, S.H. Heydari and R.Y. Song, Zeta regularized products, Trans. Amer. Math. Soc. 338 (1993) 213-230. MR 1100699 (93j:58139)
  • 11. D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1974) 145-210. MR 0295381 (45:4447)
  • 12. M. Spreafico, A generalization of the Euler gamma function, Funct. Anal. Appl. 39 (2005) 87-91. MR 2161522 (2006j:33005)
  • 13. M. Spreafico, Zeta function and regularized determinant on a disc and on a cone, J. Geom. Phys. 54 (2005) 355-371. MR 2139088 (2005k:11184)
  • 14. M. Spreafico, Zeta invariants for sequences of spectral type, special functions and the Lerch formula, Proc. Royal Soc. Edinburgh. 281 (2006) 865-889. MR 2250451 (2007e:11108)
  • 15. M. Spreafico, Zeta invariants for Dirichlet series, Pacific J. Math. 274 (2006) 185-199. MR 2231657 (2007b:11129)
  • 16. M. Spreafico and S. Zerbini, Spectral analysis and zeta determinants on the deformed spheres, Comm. Math. Phys. 273 (2007) 677-704. MR 2318862 (2008h:58063)
  • 17. A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987) 439-465. MR 891947 (89b:58173)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M41, 33E20

Retrieve articles in all journals with MSC (2010): 11M41, 33E20


Additional Information

M. Spreafico
Affiliation: ICMC, Universidade São Paulo, São Carlos, 13556-560 Brazil
Email: mauros@icmc.usp.br

DOI: https://doi.org/10.1090/S0002-9939-2011-11061-X
Received by editor(s): July 1, 2010
Received by editor(s) in revised form: February 2, 2011
Published electronically: October 12, 2011
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society