  We consider the classical trapped Riesz gas, i.e., $N$ particles at positions
$x_i$ in one dimension with a repulsive power law interacting potential
$\propto 1/|x_i-x_j|^{k}$, with $k>-2$, in an external confining potential of
the form $V(x) \sim |x|^n$. We focus on the equilibrium Gibbs state of the gas,
for which the density has a finite support $[-\ell_0/2,\ell_0/2]$. We study the
fluctuations of the linear statistics ${\cal L}_N = \sum_{i=1}^N f(x_i)$ in the
large $N$ limit for smooth functions $f(x)$. We obtain analytic formulae for
the cumulants of ${\cal L}_N$ for general $k>-2$. For long range interactions,
i.e. $k<1$, which include the log-gas ($k \to 0$) and the Coulomb gas ($k =-1$)
these are obtained for monomials $f(x)= |x|^m$. For short range interactions,
i.e. $k>1$, which include the Calogero-Moser model, i.e. $k=2$, we compute the
third cumulant of ${\cal L}_N$ for general $f(x)$ and arbitrary cumulants for
monomials $f(x)= |x|^m$. We also obtain the large deviation form of the
probability distribution of ${\cal L}_N$, which exhibits an "evaporation
transition" where the fluctuation of ${\cal L}_N$ is dominated by the one of
the largest $x_i$. In addition, in the short range case, we extend our results
to a (non-smooth) indicator function $f(x)$, obtaining thereby the higher order
cumulants for the full counting statistics of the number of particles in an
interval $[-L/2,L/2]$. We show in particular that they exhibit an interesting
scaling form as $L/2$ approaches the edge of the gas $L/\ell_0 \to 1$, which we
relate to the large deviations of the emptiness probability of the
complementary interval on the real line.
