## Tohoku Mathematical Journal 2002 December SECOND SERIES VOL. 54, NO. 4

 Tohoku Math. J. 54 (2002), 513-565

Title THE MEAN VALUE OF THE PRODUCT OF CLASS NUMBERS OF PAIRED QUADRATIC FIELDS I

Author Anthony C. Kable and Akihiko Yukie

(Received September 11, 2000, revised March 19, 2001)
Abstract. Let $k$ be a number field and $\tilde k$ a fixed quadratic extension of $k$. In this paper and its companions, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\tilde k$ contained in the biquadratic extensions of $k$ containing $\tilde k$.

2000 Mathematics Subject Classification. Primary 11S90, Secondary 11R45.

Key words and phrases. Density theorem, prehomogeneous vector spaces, binary Hermitian forms, local zeta functions.