Sunyaev-Zel'dovich Effect (SZE) cluster surveys are anticipated to yield
tight constraints on cosmological parameters such as the equation of state of
dark energy. In this paper, we study the impact of relativistic corrections of
the thermal SZE on the cluster number counts expected from a cosmological model
and thus, assuming that other cosmological parameters are known to high
accuracies, on the determination of the $w$ parameter and $\sigma_8$ from a SZE
cluster survey, where $w=p/\rho$ with $p$ the pressure and $\rho$ the density
of dark energy, and $\sigma_8$ is the rms of the extrapolated linear density
fluctuation smoothed over $8\hbox{Mpc}h^{-1}$. For the purpose of illustrating
the effects of relativistic corrections, our analyses mainly focus on $\nu=353
\hbox{GHz}$ and $S_{lim}=30\hbox{mJy}$, where $\nu$ and $S_{lim}$ are the
observing frequency and the flux limit of a survey, respectively. These
observing parameters are relevant to the {\it Planck} survey. It is found that
from two measurable quantities, the total number of SZE clusters and the number
of clusters with redshift $z\ge 0.5$, $\sigma_8$ and $w$ can be determined to a
level of $\pm 1%$ and $\pm 8%$, respectively, with $1\sigma$ uncertainties from
a survey of $10000\hbox{deg}^2$. Relativistic effects are important in
determining the central values of $\sigma_8$ and $w$. If we choose the two
quantities calculated relativistically from the flat cosmological model with
$\sigma_8=0.8284$ and $w=-0.75$ as input, the derived $\sigma_8$ and $w$ would
be 0.819 and -0.81, respectively, if relativistic effects are wrongly
neglected. The location of the resulting $\sigma_8$ and $w$ in the $\sigma_8-w$
plane is outside the $3\sigma$ region around the real central $\sigma_8$ and
$w$.Comment: ApJ in pres