  We study O(N) models with power-law interactions by using functional
renormalization group methods: we show that both in Local Potential
Approximation (LPA) and in LPA' their critical exponents can be computed from
the ones of the corresponding short-range O(N) models at an effective
fractional dimension. In LPA such effective dimension is given by
$D_{eff}=2d/\sigma$, where d is the spatial dimension and $d+\sigma$ is the
exponent of the power-law decay of the interactions. In LPA' the prediction by
Sak [Phys. Rev. B 8, 1 (1973)] for the critical exponent $\eta$ is retrieved
and an effective fractional dimension $D_{eff}'$ is obtained. Using these
results we determine the existence of multicritical universality classes of
long-range O(N) models and we present analytical predictions for the critical
exponent $\nu$ as a function of $\sigma$ and N: explicit results in 2 and 3
dimensions are given. Finally, we propose an improved LPA" approximation to
describe the full theory space of the models where both short-range and
long-range interactions are present and competing: a long-range fixed point is
found to branch from the short-range fixed point at the critical value
$\sigma_* = 2-\eta_{SR}$ (where $\eta_{SR}$ is the anomalous dimension of the
short-range model), and to subsequently control the critical behavior of the
system for $\sigma < \sigma_*$.
