  An improved uniform error bound at $O\left(h^m+\varepsilon^2 \tau^2\right)$
is established in $H^{\alpha/2}$-norm for the long-time dynamics of the
nonlinear space fractional Klein-Gordon equation (NSFKGE). A second-order
exponential wave integrator (EWI) method is used to semi-discretize NSFKGE in
time and the Fourier spectral method in space is applied to derive the
full-discretization scheme. Regularity compensation oscillation (RCO) technique
is employed to prove the improved uniform error bounds at $O\left(\varepsilon^2
\tau^2\right)$ in temporal semi-discretization and $O\left(h^m+\varepsilon^2
\tau^2\right)$ in full-discretization up to the long-time $T_{\varepsilon}=T /
\varepsilon^2$ ($T>0$ fixed), respectively. Complex NSFKGE and oscillatory
complex NSFKGE with nonlinear terms of general power exponents are also
discussed. Finally, the correctness of the theoretical analysis and the
effectiveness of the method are verified by numerical examples.
