Asymptotic behaviors of class number sums associated with Pell-type equations

  • Yasufumi HashimotoEmail author


It is well-known that the Pell equation \(t^2-Du^2=4\) has infinitely many integer solutions (tu) for a given \(D>0\) with \(D\equiv 0,1\bmod {4}\) and \(D\ne l^2\) for any integer l. However, for a square free integer \(N\ne 1\), the equation \(t^2-Du^2=4N\) does not always have integer solutions, and verifying its solubility/insolubility is not easy. In the present paper, we propose the asymptotic formulas of the sum of the class numbers h(D) of the primitive indefinite binary quadratic forms over the discriminants \(D>0\) for which the Pell type equation \(t^2-Du^2=4N\) has an integer solution, to study the distributions of such D.


Class numbers Pell-type equation Asymptotic behavior 

Mathematics Subject Classification

11R29 11E41 



The author would like to thank the anonymous referee for reading the previous draft of this paper carefully and giving helpful comments. He was supported by JST CREST no. JPMJCR14D6 and JSPS Grant-in-Aid for Scientific Research (C) no. 17K05181.


  1. 1.
    Ankeny, N.C., Chowla, S., Hasse, H.: On the class number of the real subfield of a cyclotomic field. J. Reine Angew. Math. 217, 217–220 (1965)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barban, M.B.: The “large sieve” method and its application to number theory. Russ. Math. Surv. 21, 49–103 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Conrey, J.B., Iwaniec, H.: The cubic moment of central values of automorphic L-functions. Ann. Math. (2) 151, 1175–1216 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Davenport, H.: The Higher Arithmetic—an Introduction to the Theory of Numbers, 8th edn. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  5. 5.
    Datskovsky, B.A.: A mean-value theorem for class numbers of quadratic extensions. Number Theory Relat. Anal. Contemp. Math. 143, 179–242 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Deitmar, A., Hoffmann, W.: Asymptotics of class numbers. Invent. Math. 160, 647–675 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gauss, C.F.: Disquisitiones Arithmeticae. Fleischer, Leipzig (1801)zbMATHGoogle Scholar
  8. 8.
    Goldfeld, D., Hoffstein, J.: Eisenstein series of \(\frac{1}{2}\)-integral weight and the mean value of real Dirichlet \(L\)-series. Invent. Math. 80, 185–208 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 5th edn. Oxford University Press, Oxford (1979)zbMATHGoogle Scholar
  10. 10.
    Halter-Koch, F.: On a class of insoluble binary quadratic Diophantine equations. Nagoya Math. J. 123, 141–151 (1991)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hashimoto, Y.: Asymptotic formulas for class number sums of indefinite binary quadratic forms on arithmetic progressions. Int. J. Number Theory 9, 27–51 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hua, L.K.: On the least solution of Pell’s equation. Bull. Am. Math. Soc. 48, 731–735 (1942)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications 53. American Mathematical Society, Providence (2004)Google Scholar
  14. 14.
    Jacobson Jr., M.J., Williams, H.C.: Solving the Pell Equation. CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC. Springer, New York (2009)Google Scholar
  15. 15.
    Lang, S.D.: Note on the class number of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 290, 70–72 (1977)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mollin, R.A.: On the insolubility of a class of Diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type. Nagoya Math. J. 105, 39–47 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Peter, M.: Momente der Klassenzahlen binarer quadratischer Formen mit ganzalgebraischen Koeffizienten. Acta Arith. 70, 43–77 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Raulf, N.: Asymptotics of class numbers for progressions and for fundamental discriminants. Forum Math. 21, 221–257 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sarnak, P.: Class numbers of indefinite binary quadratic forms. J. Number Theory 15, 229–247 (1982)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sarnak, P.: The arithmetic and geometry of some hyperbolic three-manifolds. Acta Math. 151, 253–295 (1983)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sarnak, P.: Class numbers of indefinite binary quadratic forms II. J. Number Theory 21, 333–346 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shintani, T.: On zeta functions associated with the vector space of quadratic forms. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22, 25–65 (1975)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Siegel, C.L.: The average measure of quadratic forms with given determinant and signature. Ann. Math. II(45), 667–685 (1944)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Soundararajan, K., Young, M.P.: The prime geodesic theorem. J. Reine Angew. Math. 676, 105–120 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Whiteman, A.L.: A note on Kloosterman sums. Bull. Am. Math. Soc. 51, 373–377 (1945)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yokoi, H.: On the Diophantine equation \(x^2-py^2=\pm 4q\) and the class number of real subfields of a cyclotomic field. Nagoya Math. J. 91, 151–161 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yokoi, H.: Solvability of the Diophantine equation \(x^2-py^2=\pm 2\) and new invariants for real quadratic fields. Nagoya Math. J. 134, 137–149 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zagier, D.B.: Zetafunktionen und quadratische Korper. Eine Einfuhrung in die hohere Zahlentheorie. Springer, Berlin (1981)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of RyukyusNishihara-choJapan

Personalised recommendations