  In \cite{Cipriani2016}, the authors proved that, with the appropriate
rescaling, the odometer of the (nearest neighbours) divisible sandpile on the
unit torus converges to a bi-Laplacian field. Here, we study
$\alpha$-long-range divisible sandpiles, similar to those introduced in
\cite{Frometa2018}. We show that, for $\alpha \in (0,2)$, the limiting field is
a fractional Gaussian field on the torus with parameter $\alpha/2$. However,
for $\alpha \in [2,\infty)$, we recover the bi-Laplacian field. This provides
an alternative construction of fractional Gaussian fields such as the Gaussian
Free Field or membrane model using a diffusion based on the generator of L\'evy
walks. The central tool for obtaining our results is a careful study of the
spectrum of the fractional Laplacian on the discrete torus. More specifically,
we need the rate of divergence of the eigenvalues as we let the side length of
the discrete torus go to infinity. As a side result, we obtain precise
asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore,
we determine the order of the expected maximum of the discrete fractional
Gaussian field with parameter $\gamma=\min \{\alpha,2\}$ and $\alpha \in
\mathbb{R}_+\backslash\{2\}$ on a finite grid.
