Journal of Mathematical Sciences

, Volume 95, Issue 3, pp 2185–2191

Estimates of the levy constant for\(\sqrt p \) and class number one criterion for ℚ(\(\sqrt p \))

  • E. P. Golubeva


Let p ≡ = 3 (mod 4) be a prime, let ℓ(\(\sqrt p \)) be the length of the period of the expansion of\(\sqrt p \) into a continued fraction, and let h(4p) be the class number of the field ℚ(\(\sqrt p \)). Our main result is as follows. For p > 91, h(4p)=1 if and only if ℓ(\(\sqrt p \)) > 0.56\(\sqrt p \)L4p(1), where L4p(1) is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of\(\sqrt p \) into a continued fraction. Bibliography: 13 titles.


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  1. 1.
    H. Jager and P. Liardet, “Distributions arithmétiques des dénominateurs des convergents de fraction continues,”Indag. Math.,50, 181–197 (1988).Google Scholar
  2. 2.
    A. S. Pen and B. F. Skubenko, “Upper bound of the period of a quadratic irrational,”Mat. Zametki,5, 413–417 (1969).Google Scholar
  3. 3.
    E. P. Golubeva, “On the length of the period of quadratic irrational,”Mat. Sb.,123, 120–129 (1984).Google Scholar
  4. 4.
    E. P. Golubeva, “On the class number of real quadratic fields of discriminant 4p,”Zap. Nauchn. Semin. POMI,204, 11–36 (1993).Google Scholar
  5. 5.
    P. C. Sarnak, “Class number of indefinite binary quadratic forms. I, II,”J. Number Theory,15, 229–247 (1982);21, 333–346 (1985).Google Scholar
  6. 6.
    C. Faivre, “Distributions of Lévy constants for quadratic numbers,”Acta Arithm.,61, 13–34 (1992).Google Scholar
  7. 7.
    M. Pollicott, “Distribution of closed geodesics on the modular surface and quadratic irrationals,”Bull. Soc. Math. France,114, 431–446 (1986).Google Scholar
  8. 8.
    H. Iwaniec, “Fourier coefficients of modular forms of half-integral weight,”Invent. Math.,82, 385–401 (1987).Google Scholar
  9. 9.
    W. Duke, “Hyperbolic distribution problems and half-integral Maass forms,”Invent. Math.,92, 73–90 (1988).Google Scholar
  10. 10.
    E. P. Golubeva, “Representation of large numbers by ternary quadratic forms,”Mat. Sb.,129, 40–54 (1986).Google Scholar
  11. 11.
    L. A. Takhtajan, “An asymptotic formula for the sum of lengths of periods of quadratic irrationals of discriminantD,”Zap. Nauchn. Semin. LOMI,91, 134–144 (1979).Google Scholar
  12. 12.
    E. P. Golubeva, “Quadratic irrationals of fixed length of period of expansion into a continued fraction,”Zap. Nauchn. Semin. POMI,196, 5–30 (1991).Google Scholar
  13. 13.
    P. J. Stephens, “Optimizing the size ofL(1,χ),”Proc. London Math. Soc. (3),24, 1–14 (1972).Google Scholar

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© Kluwer Academic/Plenum Publishers 1999

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  • E. P. Golubeva

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