Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = aX^2+bXY+cY^2$ of discriminant $b^2-4ac = -D$ (thus $\mathrm{gcd}(a,b,c) = 1$, $-a < b \leq a \leq c$). Those are the ordinates of the CM points $z_Q = \frac{b + \sqrt{-D}}{2a}$ in the standard fundamental domain for $\mathcal{H} / \mathrm{PSL}(2,\mathbb{Z})$. By Duke's theorem, the $z_Q$ are equidistributed in the measure $\frac{3}{\pi} \frac{dx \, dy}{y^2}$. The highest lying point has ordinate $k := \sqrt{|D|}/2$, and since $\int_1^k t \frac{dt}{t^2} = \log{k}$, a naive heuristic suggests $\log{k}$ for the expected mean value of $\frac{\pi}{3}y_Q$. This turns out to be correct assuming the Riemann hypothesis for $L(s,\chi_D)$, as can be proved departing from the limit formula of Kronecker, Chowla and Selberg.


**Question.** *Assume GRH, and possibly some other standard analytic hypotheses. What can be said about the mean value of $\frac{\pi}{3} y_Q \log{y_Q}$*, asymptotically as $D \to \infty$?

The naive heuristic would suggest here a main term of $\frac{1}{2} (\log{k})^2$. Is this any close to the truth?

Of course, the same question could be asked for the positive discriminants (indefinite binary quadratic forms and closed geodesics on the modular surface).