Lithuanian Mathematical Journal

, Volume 32, Issue 1, pp 129–134 | Cite as

Discrete fundamental region and its asymptotics

  • A. I. Vinogradov
Article
  • 15 Downloads

Conclusion

The calculus presented here may be called multiplicative in view of an essential dependence of it on the multiplicative structure of the function ζ D (s) from the equality (3.1).

But at the same time, the function ζ D (s) possesses also an additive structure of the form of the sum over the roots ω j from Lemma 1.1. Therefore, it is possible to calculate the square mean (3.2) using the sums of Kloosterman-Salié sums in the average with respect toD, and to obtain its asymptotics by a spectral approach using Kuznetsov's formulas (see [6]) for summing on Weil's metaplectic group [7]. The possibility of connecting the Riemann hypothesis with the square mean estimates of the discrete spectrum of the metaplectic group appears in this way.

Keywords

Discrete Spectrum Additive Structure Riemann Hypothesis Fundamental Region Spectral Approach 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. I. Vinogradov and L. A. Tahtadjan, Analogs of the Vinogradov-Gauss formula in the critical strip,Trudy Mat. Inst. Steklova,158, 45–68 (1981).Google Scholar
  2. 2.
    A. I. Vinogradov and L. A. Tahtadjan, On the Linnik-Skubenko asymptotics,Sov. Math. Dokl.,22, 136–140 (1980).Google Scholar
  3. 3.
    T. Kubota,Elementary Theory of Eisenstein Series, Kodansha Ltd., Tokyo/ John Wiley and Sons, New York (1973).Google Scholar
  4. 4.
    V. A. Bykovskij and A. I. Vinogradov, The mean value of series of the Hecke cusp forms convoluted with a quadratic character on the critical line,Trudy Mat. Inst. Steklova,200, 57–74 (1991).Google Scholar
  5. 5.
    F. A. Lavrik, The functional equation for DirichletL-functions and the problem of divisors in arithmetic progressions,Izv. Akad. Nauk SSSR, Ser. Mat.,30, 433–448 (1968).Google Scholar
  6. 6.
    N. V. Kuznetsov, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums,Math. USSR Sb.,39, 299–342 (1981).Google Scholar
  7. 7.
    A. Weil, Sur certains groupes d'opérateurs unitaires,Acta Math.,111, 143–211 (1964).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • A. I. Vinogradov

There are no affiliations available

Personalised recommendations