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On the non-homogeneous quadratic Bessel zeta function

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 February 2010

M. Spreafico
M. Spreafico
Affiliation:
Dipartimento Matematica ed Applicazioni, Università Milano Bicocca, Milano, Italy. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, CEP 13560-970 São Carlos, SP, Brazil.
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Abstract

This article studies the non-homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

MSC classification

Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 123 - 130
Copyright
Copyright © University College London 2004

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References

1.Actor, A. and Bender, I.. The zeta function constructed from the zeros of the Bessel function. J. Phys. A, 29 (1996), 6555–6580.CrossRefGoogle Scholar
2.Atiyah, M., Bott, R. and Patodi, V. K.. On the heat equation and the index theorem. Invent. Math., 19 (1973), 279–330.CrossRefGoogle Scholar
3.Burghelea, D., Friedlander, L. and Kappeler, T.. On the determinant of elliptic differential and finite difference operators in vector bundles over S1. Comm. Math. Phys., 138 (1991), 1–18.CrossRefGoogle Scholar
4.Burghelea, D., Friedlander, L. and Kappeler, T.. On the determinant of elliptic boundary value problems on a line segment. Proc. Amer. Math. Soc., 123 (1995), 3027–3038.CrossRefGoogle Scholar
5.Bransom, T. P. and Gilkey, P. B.. The functional determinant of a four-dimensional boundary value problem. Trans. Amer. Math. Soc., 344 (1994), 479–531.CrossRefGoogle Scholar
6.Bransom, T. P. and Orsted, B.. Conformal geometry and local invariants. Diff. Geom. Appl., 1 (1991), 279–308.CrossRefGoogle Scholar
7.Bruning, J. and Seeley, R.. Regular singular asymptotics. Advances Math., 58 (1985), 133–148.CrossRefGoogle Scholar
8.Bruning, J. and Seeley, R.. The resolvent expansion for second order regular singular operators. J. Fund. Anal., 73 (1988), 369–415.CrossRefGoogle Scholar
9.Callias, C.. The heat equation with singular coefficients. Comm. Math. Phys., 88 (1983), 357–385.CrossRefGoogle Scholar
10.Cheeger, J.. On the spectral geometry of spaces with conical singularities. Proc. Nat. Acad. Sci., 76 (1979), 2103–2106.CrossRefGoogle Scholar
11.Cheeger, J.. Spectral geometry of singular riemannian spaces. J. Diff. Geom., 18 (1983), 575–657.Google Scholar
12.Choi, J. and Quine, J. R.. Zeta regularized products and functional determinants on spheres. Rocky Mountain J. Math., 26 (1996), 719–729.Google Scholar
13.Gilkey, P. B.. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (second edition). (CRC press, 1995).Google Scholar
14.Hawking, S. W.. Zeta function regularization of path integrals in curved space time. CMP 55 (1977), 133–148.Google Scholar
15.Lawson, H. B. and Michelsohn, M. L.. Spin Geometry (Princeton Math. Series 38 1989).Google Scholar
16.Lesh, M.. Determinants of regular singular Sturm-Liouville operators. Math. Nachr., 194 (1998), 139–170.CrossRefGoogle Scholar
17.Ray, D. B. and Singer, I. M.. R-torsion and the Laplacian on riemannian manifolds. Advances Math., 7 (1974), 145–210.CrossRefGoogle Scholar
18.Rosenberg, S. The Laplacian on a riemannian manifold. LMSST, 31.Google Scholar
19.Spreafico, M.. Zeta function and regularized determinant on projective spaces. Rocky Mountain J. Math., 33 (2003), 1499–1512.CrossRefGoogle Scholar
20.Spreafico, M. Zeta invariants on a disc and on a cone. J. Geo. Phys. (to appear).Google Scholar
21.Stolarsky, K. B.. Singularities of Bessel-zeta functions and Hawkins' polynomials. Mailwinatika, 32 (1985), 96–103.Google Scholar
22.Watson, G. N.. A Treatise on the Theory of Bessel Functions. (Cambridge University Press 1922).Google Scholar