Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:34:18.075Z Has data issue: false hasContentIssue false
  • English
  • Français

A zeta function connected with the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular group

Published online by Cambridge University Press:  22 January 2016

Akio Fujii
Akio Fujii*
Affiliation:
Department of Mathematics, Rikkyo Univ, Tokyo, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ; … run over the eigenvalues of the discrete spectrum of the Laplace-Beltrami operator on L2(H/yΓ), where H is the upper half of the complex plane and we take Γ = PSL(2, Z). It is well known that Let a be a positive number. Here we are concerned with the zeta function defined by

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 96 , December 1984 , pp. 167 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Delsarte, L., Formules de Poisson avec reste, J. Analyse Math., 17 (1966), 419–431.CrossRefGoogle Scholar
[ 2 ] Fujii, A., On the uniformity of the distribution of the zeros of the Riemann zeta function (II), Comment. Math. Univ. St. Paul, 31 (1982), 99–113.Google Scholar
[ 3 ] Fujii, A., The zeros of the Riemann zeta function and Gibbs’s phenomenon, Comment. Math. Univ. St. Paul, 32 (1983), 229–248.Google Scholar
[ 4 ] Guinand, A. P., A summation formula in the theory of prime numbers, Proc. London Math. Soc. Ser. 2, 50 (1945), 107–119.Google Scholar
[ 5 ] Hejhal, D., The Selberg trace formula for PSL(2, R), Lecture notes in Math. Vol. 543, Springer, 1976.Google Scholar
[ 6 ] Hejhal, D., The Selberg trace formula and the Riemann zeta function Duke Math. J., 43 (1976), 441–482.Google Scholar
[ 7 ] Kubota, T., Elementary theory of Eisenstein series, Halsted Press, New York, 1973.Google Scholar
[ 8 ] Minakshisundaram, S. and Pleijel, A., Some properties of the eigenfunction of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242–256.CrossRefGoogle Scholar
[ 9 ] Selberg, A., Harmonic analysis and discontinuous group in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc, 20 (1956), 177–189.Google Scholar
[10] Venkov, A. B., Selberg’s trace formula for the Hecke operator generated by an involution and the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular group PSL(2, Z), Math. USSR-Izv., 12 (1978), 448–462.CrossRefGoogle Scholar
[11] Venkov, A. B., Remainder term in the Weyl-Selberg asymptotic formula, J. Soviet Math., 17 (1981), 2083–2097.CrossRefGoogle Scholar