Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the mean value of the zeta and $L$-functions. XIII

Roelof Wichert Bruggeman and Yoichi Motohashi

Full-text: Open access

Abstract

Extending the discussion in the previous note [6] of this series, the group $\mathrm{PSL}_2(\mathbf{C})$ will be dealt with in place of $\mathrm{PSL}_2(\mathbf{R})$. We shall indicate that the functional structure that supports the spectral theory of Kloosterman sums in the complex case is essentially the same as in the real case, though it is more involved as can be expected.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 6 (2002), 87-91.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392681

Digital Object Identifier
doi:10.3792/pjaa.78.87

Mathematical Reviews number (MathSciNet)
MR1913937

Zentralblatt MATH identifier
1116.11066

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Keywords
Mean values of zeta-functions local functional equations of Jacquet-Langlands Gamma-functions of representations Bessel functions of representations

Citation

Bruggeman, Roelof Wichert; Motohashi, Yoichi. A note on the mean value of the zeta and $L$-functions. XIII. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 6, 87--91. doi:10.3792/pjaa.78.87. https://projecteuclid.org/euclid.pja/1148392681


Export citation

References

  • Bruggeman, R.W., and Motohashi, Y.: Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field (submitted).
  • Bruggeman, R.W., and Motohashi, Y.: A note on the mean value of the zeta and $L$-functions. X. Proc. Japan Acad., 77A, 111–114 (2001).
  • Goodman, R., and Wallach, N.R.: Whittaker vectors and conical vectors. J. Funct. Anal., 39, 199–279 (1980).
  • Jacquet, H., and Langlands, R.P.: Automorphic Forms on $\RM{GL}(2)$. Springer Verlag, Berlin, pp. 1–548 (1970).
  • Motohashi, Y.: New analytic problems over imaginary quadratic number fields. Number Theory, in Memory of Kustaa Inkeri (eds. Jutila, M., and Metsänkylä,T.). de Gruyter, Berlin-New York, pp. 255–279 (2001).
  • Motohashi, Y.: A note on the mean value of the zeta and $L$-functions. XII. Proc. Japan Acad., 78A, 36–41 (2002).
  • Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge, pp. 1–804 (1944).