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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).

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 if $\varphi: [1,\infty) \to \mathbb{R}$ is any compactly supported or fast enough decaying function, it follows that the mean value of $\varphi(\frac{\pi}{3}y_Q)$ (over all reduced forms $Q$ if the given discriminant $-D$) is equal to $\int_1^k \varphi(t) \frac{dt}{t^2} + o_{\varphi}(1)$, as $D \to \infty$. The asymptotic $\mathrm{Avg}_Q \varphi(\frac{\pi}{3}y_Q) \sim \int_1^k \varphi(t) \frac{dt}{t^2}$ turns out to be also satisfied for $\varphi(t) = t$ (assuming $L(s,\chi_D)$ satisfies the Riemann hypothesis), for a very special reason: the Kronecker-Chowla-Selberg limit formula expresses $$ \mathrm{Avg}_Q \big(\frac{\pi}{3} y_Q - \log{y_Q} \big) = \log{k} + \frac{L'}{L}(1,\chi_D) + O(1), $$ with an absolutely bounded $O(1)$ term.

On the other hand it is plain that $\mathrm{Avg}_Q \varphi(\frac{\pi}{3}y_Q) \sim \int_1^k \varphi(t) \frac{dt}{t^2}$ fails to hold with $\varphi(t) = t^{1+\epsilon}$, for any fixed $\epsilon > 0$: already the principal form (with $a = 1$) contributes an $\sim k^{\epsilon} \frac{k}{h}$ to this average ($h$ denoting the class number), a contribution that can be as big as $k^{\epsilon} \log{\log{k}}$.

I am interested in what happens for $\varphi(t) = t\log{t}$.

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 $D \to \infty$?

Evidently this mean value lies somewhere between $\log{k}$ and $(\log{k})^2$. Should the limit $\lim_{D \to \infty} \mathrm{Avg}_Q(y_{Q}\log{y_Q}) \big/ (\log{k})^2$ even exist?

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

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 $\int_1^k t \frac{dt}{t^2} = \log{k}$, equidistribution suggests (!) an expectation of $\log{\sqrt{|D|}} = \log{k} + O(1)$ for the mean value of $\frac{\pi}{3} y_Q$.

  This heuristic cannot be turned into a proof since the equidistribution result only applies to a compactly supported test function (and our mean value is much larger than $\log{|D|}$ under a putative 'Siegel zero'); indeed, it is probably not a priori clear if such an expectation is reasonable. Nonetheless, assuming the Riemann hypothesis for $L(s,\chi_D)$, this can be proved by virtue of the limit formula of Kronecker, Chowla and Selberg, which gives $\mathrm{Avg}_Q (\frac{\pi}{3} y_Q - \log{y_Q} )= \log{\sqrt{|D|}} + \frac{L'}{L}(1,\chi_D) + O(1)$ (and the mean value of $\log{y_Q}$ can be proved to be bounded under ERH). Our expected $\sim \log{k}$ asymptotic thus follows under ERH, and we do know that the heuristic is right in this particular case.

This direct (Kronecker) relation of the mean value of $\varphi(\frac{\pi}{3}y_Q)$ to the zeros of $L(s,\chi_D)$ is unique to $\varphi(\frac{\pi}{3}y) = -4\log{|\eta(q)|}$, a constant multiple of the logarithm of the Dedekind eta function, which is approximately a linear function in $y$ (asymptotically this function is $\frac{\pi}{3} y - \log{y} +O(e^{-y})$ as $y \to \infty$). This is what allows to treat the linear case $\varphi(y) = y$, in addition to the compactly supported functions $\varphi$. But (presumably?) the same heuristic would suggest $\int_1^k \varphi(t) \frac{dt}{t^2}$ for the expected value, for certain other reasonable unbounded functions $\varphi$.

Do we have $\mathrm{Avg}_Q \frac{\pi}{3} y_Q \log{y_Q} \sim \frac{1}{2} (\log{k})^2$ and $\mathrm{Avg}_Q (\frac{\pi}{3} y_Q)^2 \sim k$, assuming GRH and possibly other mainstream analytic hypotheses? Are there results out there that will allow to compute a GRH-conditional asymptotic for these averages?

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

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).