Lower Bounds for the Disk Packing Constant

An osculatory packing of a disk, $U$, is an infinite sequence of disjoint disks, $\{ {U_n}\}$, contained in $U$, chosen so that, for $n \geqq 2$, ${U_n}$ has the largest possible radius, ${r_n}$, of all disks fitting in $U\backslash ({U_1} \cup \cdots \cup {U_{n - 1}})$. The exponent of the packing, $S$, is the least upper bound of numbers, $t$, such that $\sum {r_n^t} = \infty$. Here, we present a number of methods for obtaining lower bounds for $S$, based on obtaining weighted averages of the curvatures of the ${U_n}$. We are able to prove that $S > 1.28467$. We use a number of well-known results from the analytic theory of matrices.