Estimates of the levy constant for\(\sqrt p \) and class number one criterion for ℚ(\(\sqrt p \))
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.
Unable to display preview. Download preview PDF.
- 1.H. Jager and P. Liardet, “Distributions arithmétiques des dénominateurs des convergents de fraction continues,”Indag. Math.,50, 181–197 (1988).Google Scholar
- 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.E. P. Golubeva, “On the length of the period of quadratic irrational,”Mat. Sb.,123, 120–129 (1984).Google Scholar
- 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.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.C. Faivre, “Distributions of Lévy constants for quadratic numbers,”Acta Arithm.,61, 13–34 (1992).Google Scholar
- 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.H. Iwaniec, “Fourier coefficients of modular forms of half-integral weight,”Invent. Math.,82, 385–401 (1987).Google Scholar
- 9.W. Duke, “Hyperbolic distribution problems and half-integral Maass forms,”Invent. Math.,92, 73–90 (1988).Google Scholar
- 10.E. P. Golubeva, “Representation of large numbers by ternary quadratic forms,”Mat. Sb.,129, 40–54 (1986).Google Scholar
- 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.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.P. J. Stephens, “Optimizing the size ofL(1,χ),”Proc. London Math. Soc. (3),24, 1–14 (1972).Google Scholar