Edit. Apparently (this is suggested by some remarks he made elsewhere), Scholz called the expression$$ \prod_p \Phi_K(p)/\phi(p) $$the ideal density of a number field $K$, where $\phi$ and $\Phi_K$ denote Euler's phi function in the rationals and in $K$, respectively. This expression occurs in the product formula for the zeta function. I still don't know where to go from here.
As for Robin's remark on the density of fields ordered by discriminants, Scholz claimed, in a letter to Hasse dated Sept. 27, 1938, the following: The Dirichlet series$$ G(s) = \sum_{Gal(K)=G} D_K^{-s}, $$where the sum is over all quartic fields whose normal closure has Galois group $G$, have abscissa of convergence $\alpha(D)=1$, $\alpha(Z) = \alpha(V) = \frac{1}{2}$ andprobably $\alpha(S)=1$, $\alpha(A)=\frac{1}{2}$, where $D$, $Z$, $V$, $A$, $S$ denotethe dihedral, cyclic, four, alternating and symmetric group. Moreover,$$ \lim_{s \to 1/2} \frac{Z(s)}{V(s)} = 0, $$where $Z(s)$ and $V(s)$ are the Dirichlet series defined above for $G=Z$ and $G=V$. This is all correct, as we know now, but how could Scholz have discovered (and, for $G = D$, $Z$, $V$, proved) these results in the 1930s?
