Log in

Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series

  • Published:
Inventiones mathematicae Aims and scope

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

References

  1. Abromovich, M.: Handbook of Mathematical Functions. New York: Dover 1965

    Google Scholar 

  2. Barban, M.B.: Linnik's “large sieve” and a limit theorem for the class number of ideals of an imaginary quadratic field. Izv. Akad. Nauk SSSR Ser. Mat.26, 573–580 (1962)

    Google Scholar 

  3. Burgess, D.A.: On character sums andL-series, II. Proc. Lond. Math. Soc.13, 524–536 (1963)

    Google Scholar 

  4. Davenport, H.: Multiplicative Number Theory. Second Edition, pp. 105–106. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  5. Gauss, C.F.: Disquisitiones Arithmeticae. New Haven-London: Yale University Press 1966

    Google Scholar 

  6. Gauss, C.F.: Werke I, Zweiter Abdruck. Göttingen (1870), 369 and 466

  7. Goldfeld, D., Hoffstein, J., Patterson, S.J.: On automorphic functions of half-integral weight with applications to elliptic curves. Number Theory Related to Fermat's Last Theorem, pp. 153–193. Boston: Birkhäuser 1982

    Google Scholar 

  8. Goldfeld, D., Viola, C.: Mean values ofL-functions associated to elliptic, Fermat and other curves at the center of the critical strip. J. Number Theory11, 305–320 (1979)

    Google Scholar 

  9. Goldfeld, D., Viola, C.: Some conjectures on elliptic curves over cyclotomic fields. Trans. Am. Math. Soc.276, 511–516 (1983)

    Google Scholar 

  10. Hardy, G., Riesz, M.: Dirichlet Series, p. 15, Cambridge: Cambridge Univ. Press (1952)

    Google Scholar 

  11. Hecke, E.: Lectures on the Theory of Algebraic Numbers. Berlin-Heidelberg-New York: Springer 1981

    Google Scholar 

  12. Jutila, M.: On the mean value ofL(1/3, χ) for real characters. Analysis1, 149–161 (1981)

    Google Scholar 

  13. Landau, E.: Elementary Number Theory. New York: Chelsea 1958

    Google Scholar 

  14. Lipschitz, R.: Sitzungsber. Akad. Berlin, pp. 174–185 (1865)

  15. Mertens, F.: Über einige asymptotische Gesetze der Zahlentheorie. J. Math.77, 312–319 (1874)

    Google Scholar 

  16. Montgomery, H., Vaughn, R.C.: The distribution of squarefree numbers. Recent Prog. Analytic Number Theory1, 247–256 (Durham 1979). London-New York: Academic Press (1981)

    Google Scholar 

  17. Rankin, R.A.: Contributions to the theory of Ramanujan's function τ(n) and similar arithmetic functions. Proc. Cambridge Philos. Soc.35, 357–372 (1939)

    Google Scholar 

  18. Selberg, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturrid43, 47–50 (1940)

    Google Scholar 

  19. Shimura, G.: On modular forms of half-integral weight. Ann. Math.97, 440–481 (1973)

    Google Scholar 

  20. Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc.31, 79–96 (1975)

    Google Scholar 

  21. Shintani, T.: Zeta functions associated with the vector space of quadratic forms. J. Fac. Sci., Univ. Tokyo, Sect. IA22, 25–65 (1975)

    Google Scholar 

  22. Siegel, C.L.: The average measure of quadratic forms with given determinant and signature. Gesammelte Abhandlungen,II, pp. 473–491. Berlin-Heidelberg-New York: Springer (1966)

    Google Scholar 

  23. Stepanov, B.V.: On the mean value of thek th power of the number of classes for an imaginary quadratic field. Dokl. Akad. Nauk SSSR124, 984–986 (1959)

    Google Scholar 

  24. Takhtadzjan, L.A., Vinogradov, A.I.: On analogues of the Vinogradov-Gauss formula. Sov. Math., Dokl.22, 555–559 (1980)

    Google Scholar 

  25. Vinogradov, I.M.: Reprinted in his Selected Works. Izv. Rossiiskoi Akad. Nauk11, 1347 (1917). Izdat. Akad. Nauk SSSR, Moscow, p. 3, 1952

    Google Scholar 

  26. Vinogradov, I.M.: Reprinted in his Selected Works. Soobsc. Har'kov. Mat. Obsc.16, (1918); Izdat. Akad. Nauk SSSR, Moscow, p. 29, 1952

  27. Vinogradov, I.M.: Reprinted in his Selected Works. Izv. Akad. Nauk SSSR, Ser. Mat.13, 97 (1949). Izdat. Akad. Nauk SSSR, Moscow, p. 366, 1952

    Google Scholar 

  28. Vinogradov, I.M.: Izv. Akad. Nauk SSSR, Ser. Mat.27, 957 (1963)

    Google Scholar 

  29. Walfisz, A., Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1963

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by NSF grant # DMS-8303621 and a grant from the Vaughn Foundation

Partially supported by NSF grant # MCS-8305527

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldfeld, D., Hoffstein, J. Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series. Invent Math 80, 185–208 (1985). https://doi.org/10.1007/BF01388603

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01388603

Keywords

Navigation