  We investigate the magnetic susceptibility $\chi(T)$ of quantum spin chains
of $N=1280$ spins with power-law long-range antiferromagnetic coupling as a
function of their spatial decay exponent $\alpha$ and cutoff length $\xi$. The
calculations are based on the strong disorder renormalization method which is
used to obtain the temperature dependence of $\chi(T)$ and distribution
functions of couplings at each renormalization step. For the case with only
algebraic decay ($ \xi = \infty$) we find a crossover at $\alpha^*=1.066$
between a phase with a divergent low-temperature susceptibility
$\chi(T\rightarrow 0) $ for $\alpha > \alpha^*$ to a phase with a vanishing
$\chi(T\rightarrow 0) $ for $\alpha < \alpha^*$. For finite cutoff lengths
$\xi$, this crossover occurs at a smaller $\alpha^*(\xi)$. Additionally we
study the localization of spin excitations for $ \xi = \infty$ by evaluating
the distribution function of excitation energies and we find a delocalization
transition that coincides with the opening of the pseudo-gap at
$\alpha_c=\alpha^*$.
