The mean value of the product of class numbers of paired quadratic fields III

This is the final part in a series of three papers. In this part, we evaluate the previously unevaluated local densities at dyadic places that appear in the density theorem stated in the first part. For this purpose we introduce an invariant, the level, attached to a pair of ramified quadratic extensions of a dyadic local field. This invariant measures how close the fields are in their arithmetic properties and its evaluation may be of interest independent of its application here.