  The structure factor and correlation energy of a quantum wire of thickness
$b\ll a_B$ are studied in random phase approximation and for the less
investigated region $r_s<1$. Using the single-loop approximation, analytical
expressions of the structure factor have been obtained. The exact expressions
for the exchange energy are also derived for a cylindrical and harmonic wire.
The correlation energy $\epsilon_c$ is found to be represented by $\epsilon_c
(b,r_s)= \frac{\alpha(r_s)}{b} + \beta(r_s)\; ln(b) + \eta(r_s)$, for small $b$
and high densities. For a pragmatic width of the wire, the correlation energy
is in agreement with the quantum Monte Carlo simulation data.
