Let $p$ be a fixed prime number. Roughly speaking, I am interested in the following ratio

$ \frac{ \#\{all\ CM\ number\ fields\ of\ degree\ 2g\}} {\#\{CM\ fields\ of\ degree\ 2g,\, such\ that\ p\ splits\ completely in K\}} $

A possible definition could the following: let $d_{K}$ be the discriminant of K, then we can define this ratio as

$lim_{d \to \infty}( \frac {\#{all CM fields of degree 2g and d_{K} \le d\}} {\#\{CM fields of degree 2g, such that p splits completely and d_{K} \le d \}})$

Was it studied by anyone? I would appreciate any reference.