  We investigate the full counting statistics (FCS) of a harmonically confined
1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite
temperature. The particles interact with each other through a repulsive
power-law interaction with an exponent $k>1$ which includes the Calogero-Moser
model for $k=2$. We examine the probability distribution of the number of
particles in a finite domain $[-W, W]$ called number distribution, denoted by
$\mathcal{N}(W, N)$. We analyze the probability distribution of $\mathcal{N}(W,
N)$ and show that it exhibits a large deviation form for large $N$
characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation
function of the fraction $c = \mathcal{N}(W, N)/N$ of the particles inside the
domain and $W$. We show that the density profiles that create the large
deviations display interesting shape transitions as one varies $c$ and $W$.
This is manifested by a third-order phase transition exhibited by the large
deviation function that has discontinuous third derivatives. Monte-Carlo (MC)
simulations show good agreement with our analytical expressions for the
corresponding density profiles. We find that the typical fluctuations of
$\mathcal{N}(W, N)$, obtained from our field theoretic calculations are
Gaussian distributed with a variance that scales as $N^{\nu_k}$, with $\nu_k =
(2-k)/(2+k)$. We also present some numerical findings on the mean and the
variance. Furthermore, we adapt our formalism to study the index distribution
(where the domain is semi-infinite $(-\infty, W])$, linear statistics (the
variance), thermodynamic pressure and bulk modulus.
