  We present a Green's-function theory of magnetic short-range order in the
$S=1/2$ easy-plane XXZ chain based on the projection method for the dynamic
spin susceptibility and a decoupling of three-spin operator products
introducing vertex parameters. The longitudinal and transverse static
susceptibilities and two-point correlation functions of arbitrary range are
calculated self-consistently for all wavenumbers, temperatures, and anisotropy
parameters $ -1\leq \Delta \leq 1$. In the easy-plane ferromagnetic region
$(\Delta < 0)$, the longitudinal correlators of spins at distance $n$ change
sign at a finite temperature $T_0(n,\Delta)$, in reasonable agreement with
recent data obtained by finite-chain diagonalizations. The temperature
dependence of the uniform static susceptibilities exhibits a maximum which is
explained as an effect of magnetic short-range order which decreases with
increasing temperature.
