  We consider the $d$-dimensional fractional Anderson model $(-\Delta)^\alpha+
V_\omega$ on $\ell^2(\mathbb Z^d)$ where $0<\alpha\leq 1$. Here $-\Delta$ is
the negative discrete Laplacian and $V_\omega$ is the random Anderson potential
consisting of iid random variables. We prove that the model exhibits Lifshitz
tails at the lower edge of the spectrum with exponent $ d/ (2\alpha)$. To do
so, we show among other things that the non-diagonal matrix elements of the
negative discrete fractional Laplacian are negative and satisfy the two-sided
bound $$ \frac{c_{\alpha,d}}{|n-m|^{d+2\alpha}} \leq -(-\Delta)^\alpha(n,m)\leq
\frac{C_{\alpha,d}}{|n-m|^{d+2\alpha}} $$ for positive constants
$c_{\alpha,d}$, $C_{\alpha,d}$ and all $n\neq m\in\mathbb Z^d$.
