The prime-curvature counting conjecture for Apollonian circle packings

Let $P$ be an Apollonian circle packing. For $x>0$, let $N_P(x)$ be the number of circles in $P$ with curvature less than $x$, and let $\psi_P(x)$ be the weighted count of circles of prime curvature, defined by

$$\psi_P(x)=\sum_{\substack{a(C)\leq x\a(C)\text{ prime}}}\log\bigl(a(C)\bigr),$$

where $a(C)$ is the curvature of $C$. Let $\chi_4$ be the character with $\chi_4(p)=1$ for $p\equiv1\pmod 4$ and $\chi_4(p)=-1$ for $p\equiv3\pmod 4$.

Prime-curvature counting conjecture. As $x\to\infty$,

$$\psi_P(x)\sim L(2,\chi_4)\,N_P(x),$$

where $L(2,\chi_4)=0.9159\dots$.

This heuristic is asserted to be independent of the packing and is supported experimentally for the packings examined in the paper. Sarnak's result that every packing contains infinitely many circles of prime curvature is known, but this precise asymptotic remains open in the source.