Skip to main content
Log in

Arithmetic-analytic properties of binary positive-definite quadratic forms

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

One considers the arithmetic and analytic properties of positive-definite binary quadratic forms of discriminant −Dn2. The arithmetic structure of the set of these forms is described by forms of discriminant −D and by the Hecke operators T(n). One gives some arithmetic and analytic consequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature cited

  1. V. A. Bykovskii, “On a certain summation formula in the spectral theory of automorphic functions and its applications to analytic number theory,” Dokl. Akad. Nauk SSSR,264, No. 2, 275–277 (1982).

    Google Scholar 

  2. V. A. Bykovskii, “The spectral expansions of certain automorphic functions and their number-theoretic applications,” J. Sov. Math.,36, No. 1 (1987).

  3. A. B. Venkov, “The spectral theory of automorphic functions,” Trudy Mat. Inst. Akad. Nauk SSSR,153, 1–171 (1981).

    Google Scholar 

  4. B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen (1970).

    Google Scholar 

  5. K. F. Gauss, Works in Number Theory [Russian translation], Moscow (1959).

  6. E. P. Golubeva, “The asymptotic distribution of lattice points, belonging to given residue classes, on hyperboloids of a special form,” Mat. Sb.,123, No. 4, 510–533 (1984).

    Google Scholar 

  7. P. G. Lejeune Dirichlet. Vorlesungen uber Zahlentheorie, F. Vieweg und Sohn, Braunschweig (1894).

    Google Scholar 

  8. N. V. Kuznetsov, “The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. I. Sums of Kloosterman sums,” Mat. Sb.,111, No. 3, 334–383 (1980).

    Google Scholar 

  9. N. V. Kuznetsov, “Convolution of the Fourier coefficients of the Eisenstein-Maass series,” J. Sov. Math.,29, No. 2 (1985).

  10. Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Springer, New York (1968).

    Google Scholar 

  11. A. V. Malyshev, “Representation of integers by positive quadratic forms,” Tr. Mat. Inst. Akad. Nauk SSSR,65 (1962).

  12. L. D. Faddeev, “Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group in the Lobachevskii plane,” Trudy Mosk. Mat. Obshch.,17, 323–350 (1967).

    Google Scholar 

  13. A. Selberg, “Uber die Fourierkoeffizienten elliptischen Modulformen negative Dimension,” in: Neuviéme Congrés des Mathématiciens Scandinaves, Helsingfors, 1938, pp. 320–322.

  14. A. Selberg, “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” Indian J. Math. Soc.,20, 47–87.

  15. F. Hirzebruch and D. Zagier, “Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus,” Invent. Math.,36, 57–113 (1976).

    Google Scholar 

Download references

Authors

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 5–20, 1985.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bykovskii, V.A. Arithmetic-analytic properties of binary positive-definite quadratic forms. J Math Sci 38, 2029–2040 (1987). https://doi.org/10.1007/BF01474435

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01474435

Keywords

Navigation