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A mean value theorem for the square of class number times regulator of quadratic extensions. (English) Zbl 1231.11111
Summary: Let $$k$$ be a number field. In this paper, we give a formula for the mean value of the square of class number times regulator for certain families of quadratic extensions of $$k$$ characterized by finitely many local conditions. We approach this by using the theory of the zeta function associated with the space of pairs of quaternion algebras. We also prove an asymptotic formula of the correlation coefficient for class number times regulator of certain families of quadratic extensions.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11S90 Prehomogeneous vector spaces 11S40 Zeta functions and $$L$$-functions
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##### References:
 [1] Datskovsky, B., A mean value theorem for class numbers of quadratic extensions, Contemporary Mathematics, 143, 179-242, (1993) · Zbl 0791.11058 [2] Datskovsky, B.; Wright, D. J., The adelic zeta function associated with the space of binary cubic forms II: local theory, J. Reine Angew. Math., 367, 27-75, (1986) · Zbl 0575.10016 [3] Datskovsky, B.; Wright, D. J., Density of discriminants of cubic extensions, J. Reine Angew. Math., 386, 116-138, (1988) · Zbl 0632.12007 [4] Granville, A.; Soundararajan, K., The distributions of values of $${L}(1,χ _d),$$ Geom. Funct. Anal., 13, 992-1028, (2003) · Zbl 1044.11080 [5] Kable, A. C.; Wright, D. J., Uniform distribution of the Steinitz invariants of quadratic and cubic extensions, Compos. Math., 142, 84-100, (2006) · Zbl 1113.11065 [6] Kable, A. C.; Yukie, A., The mean value of the product of class numbers of paired quadratic fields, I, Tohoku Math. J., 54, 513-565, (2002) · Zbl 1020.11079 [7] Kable, A. C.; Yukie, A., The mean value of the product of class numbers of paired quadratic fields, II, J. Math. Soc. Japan, 55, 739-764, (2003) · Zbl 1039.11087 [8] Kable, A. C.; Yukie, A., The mean value of the product of class numbers of paired quadratic fields, III, J. Number Theory, 99, 185-218, (2003) · Zbl 1039.11086 [9] Mumford, D.; Fogarty, J., Geometric invariant theory, Springer-Verlag, (1982) · Zbl 0504.14008 [10] Peter, M., Momente der klassenzahlen binärer quadratischer formen mit ganzalgebraischen koeffizienten, Acta Arithm., 70, 43-77, (1995) · Zbl 0817.11025 [11] Software, Waterloo Maple, Maple V, Waterloo Maple Inc., Waterloo, Ontario, (1994) [12] Taniguchi, T., Distributions of discriminants of cubic algebras [13] Taniguchi, T., Distributions of discriminants of cubic algebras II [14] Taniguchi, T., On propotional constants of the mean value of class numbers of quadratic extensions, Trans. Amer. Math. Soc., 359, 5517-5524, (2007) · Zbl 1134.11041 [15] Taniguchi, T., On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras, Ann. Inst. Fourier, 57, 1331-1358, (2007) · Zbl 1173.11050 [16] Vignéras, M. F., Lecture Notes in Mathematics, 800, Arithmétique des algèbres de quaternions, (1980), Springer-Verlag, Berlin, Heidelberg, New York · Zbl 0422.12008 [17] Weil, A., Basic number theory, Springer-Verlag, (1974) · Zbl 0823.11001 [18] Wright, D. J.; Yukie, A., Prehomogeneous vector spaces and field extensions, Invent. Math., 110, 283-314, (1992) · Zbl 0803.12004
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