Radial Density in Apollonian Packings

Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0 = \partial B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\epsilon} = \partial B(z_0, r_0 + \epsilon)$ is red, and what is the behavior of this quantity as $\epsilon \rightarrow 0$? Using equidistribution of closed horocycles on the modular surface $\mathbb{H}^2/SL(2, \mathbb{Z})$, we show that the answer is $\frac{3}{\pi} = 0.9549\dots$ We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann Hypothesis. For the analogous problem for Soddy Sphere packings, we find that the limiting radial density is $\frac{\sqrt{3}}{2V_T}=0.853\dots$, where $V_T$ denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles $\pi/3$.Comment: New section based on an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann Hypothesi

Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses

For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ of the cyclotomic polynomial $\Phi_{pqr}(x)$, where $r$ and $q$ are free primes satisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that $M(p)\le(p+1)/2$. In 2009 Gallot and Moree showed that $M(p)\ge 2p(1-\epsilon)/3$ for every $p$ sufficiently large. In this article Kloosterman sums (`cloister man sums') and other tools from the distribution of modular inverses are applied to quantify the abundancy of counter-examples to Sister Beiter's conjecture and sharpen the above lower bound for $M(p)$.Comment: 2 figures; 15 page