Principal forms X 2+nY 2 representing many integers

  • David Brink
  • Pieter Moree
  • Robert OsburnEmail author


In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X 2+nY 2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.


Binary quadratic forms Bernays’ constant Special values of L-series 

Mathematics Subject Classification (2000)

11E16 11M20 


  1. 1.
    Bateman, P., Chowla, S., Erdös, P.: Remarks on the size of L(1,χ). Publ. Math. (Debr.) 1, 165–182 (1950) zbMATHGoogle Scholar
  2. 2.
    Bernays, P.: Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante. Dissertation, Göttingen, 1912, available at
  3. 3.
    Chowla, S.: On the class-number of the corpus \(P(\sqrt{-k})\). Proc. Natl. Acad. Sci., India 13, 197–200 (1947) MathSciNetGoogle Scholar
  4. 4.
    Cook, R.: A note on character sums. J. Number Theory 11, 505–515 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cox, D.: Primes of the Form x 2+ny 2. Wiley, New York (1989) Google Scholar
  6. 6.
    Finch, S., Martin, G., Sebah, P.: Roots of unity and nullity modulo n. Proc. Am. Math. Soc. 138, 2729–2743 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fomenko, O.: Distribution of values of Fourier coefficients of modular forms of weight 1. J. Math. Sci. (N.Y.) 89, 1050–1071 (1998) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Granville, A., Soundararajan, K.: The distribution of values of L(1,χ d). Geom. Funct. Anal. 13, 992–1028 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hardy, G.: Ramanujan. Chelsea, New York (1959) zbMATHGoogle Scholar
  10. 10.
    Iwaniec, H.: The half dimensional sieve. Acta Arith. 29(1), 69–95 (1976) MathSciNetzbMATHGoogle Scholar
  11. 11.
    James, R.: The distribution of integers represented by quadratic forms. Am. J. Math. 60, 737–744 (1938) CrossRefGoogle Scholar
  12. 12.
    Joshi, P.: The size of L(1, χ) for real nonprincipal residue characters χ with prime modulus. J. Number Theory 2, 58–73 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kaplan, P., Williams, K.: The genera representing a positive integer. Acta Arith. 102, 353–361 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Landau, E.: Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys. 13(3), 305–312 (1908) Google Scholar
  15. 15.
    Littlewood, J.: On the class-number of the corpus \(P(\sqrt{-k})\). Proc. Lond. Math. Soc. 27(2), 358–372 (1928) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Moree, P., Osburn, R.: Two-dimensional lattices with few distances. Enseign. Math. 52, 361–380 (2006) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Shanks, D.: The second order term in the asymptotic expansion of B(x). Math. Comput. 18, 75–86 (1964) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Shanks, D., Schmid, L.: Variations on a theorem of Landau. I. Math. Comput. 20, 551–569 (1966) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sun, Z., Williams, K.: On the number of representations of n by ax 2+bxy+cy 2. Acta Arith. 122, 101–171 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Williams, K.: Note on integers representable by binary quadratic forms. Can. Math. Bull. 18, 123–125 (1975) zbMATHCrossRefGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity College DublinDublin 4Ireland
  2. 2.Max-Planck-Institut für MathematikBonnGermany

Personalised recommendations