A MEAN VALUE THEOREM FOR THE SQUARE OF CLASS NUMBER TIMES REGULATOR OF QUADRATIC EXTENSIONS (Algebraic number theory and related topics)

Let $k$ be a number field, and $\Delta_{k}$ , $h_{k}$ and $R_{k}$ the absolute discriminant, the class number and the regulator, respectively.In this article we will give a survey of [9] in which we found the asymptotic behavior of the mean values of $h_{F}^{2}R_{F}^{2}$ with respect to $|\Delta_{F}|$ for certain families of quadratic extensions $F$ of a fixed number field $k$ .The global zeta function of prehomogeneous vector space for the space of pairs of quaternions are used to prove the theorem.Also we give some examples of interpretations of set of rational orbits in some inner form representations.