  The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied
on a periodic lattice from the analytic and dynamic perspective by varying the
mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that
the discrete system converges to the fractional NLS as $h \rightarrow 0$ below
the energy space by directly estimating the difference between the discrete and
continuum solutions in $L^2(\mathbb{T})$ using the periodic Strichartz
estimates. The sharp convergence rate via the finite-difference method is shown
to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. On the other hand
for a fixed $h > 0$, the linear stability analysis on a family of continuous
wave (CW) solutions reveals a rich dynamical structure of CW waves due to the
interplay between nonlinearity, nonlocal dispersion, and discreteness. The gain
spectrum is derived to understand the role of $h$ and $\alpha$ in triggering
higher mode excitations. The transition from the quadratic dependence of
maximum gain on the amplitude of CW solutions to the linear dependence, due to
the lattice structure, is shown analytically and numerically.
