  We study the approach to jamming in hard-sphere packings, and, in particular,
the pair correlation function $g_{2}(r)$ around contact, both theoretically and
computationally. Our computational data unambiguously separates the narrowing
delta-function contribution to $g_{2}$ due to emerging interparticle contacts
from the background contribution due to near contacts. The data also shows with
unprecedented accuracy that disordered hard-sphere packings are strictly
isostatic, i.e., the number of exact contacts in the jamming limit is exactly
equal to the number of degrees of freedom, once rattlers are removed. For such
isostatic packings, we derive a theoretical connection between the probability
distribution of interparticle forces $P_{f}(f)$, which we measure
computationally, and the contact contribution to $g_{2}$. We verify this
relation for computationally-generated isostatic packings that are
representative of the maximally jammed random state. We clearly observe a
maximum in $P_{f}$ and a nonzero probability of zero force. We computationally
observe an unusual power-law divergence in the near-contact contribution to
$g_{2}$, persistent even in the jamming limit, with an exponent of -0.4. We
also present the first computational data on the contact-contribution to
$g_{2}$ for vacancy-diluted FCC crystal packings and also investigate partially
crystallized packings along the transition from maximally disordered to fully
ordered packings. Unlike previous studies, we find that ordering has a
significant impact on the shape of $P_{f}$ for small forces.
