  Let $({\mathcal{X}},g)$ be a closed Riemmanian manifold of dimension $n>0$.
Let $\Delta$ be the Laplacian on ${\mathcal{X}}$, and let $(e\_k)\_k$ be an
$L^2$-orthonormal and dense family of Laplace eigenfunctions with respective
eigenvalues $(\lambda\_k)\_k$. We assume that $(\lambda\_k)\_k$ is
non-decreasing and that the $e\_k$ are real-valued. Let $(\xi\_k)\_k$ be a
sequence of iid $\mathcal{N}(0,1)$ random variables. For each $L>0$ and
$s\in{\mathbb{R}}$, possibly negative, set\[f^s\_L=\sum\_{0<\lambda\_j\leq
L}\lambda\_j^{-\frac{s}{2}}\xi\_je\_j\, .\]Then, $f\_L^s$ is almost surely
regular on its zero set. Let $N\_L$ be the number of connected components of
its zero set. If $s<\frac{n}{2}$, then we deduce that there exists
$\nu=\nu(n,s)>0$ such that $N\_L\sim \nu {Vol}\_g({\mathcal{X}})L^{n/2}$ in
$L^1$ and almost surely. In particular, ${\mathbb{E}}[N\_L]\asymp L^{n/2}$. On
the other hand, we prove that if $s=\frac{n}{2}$ then\[{\mathbb{E}}[N\_L]\asymp
\frac{L^{n/2}}{\sqrt{\ln\left(L^{1/2}\right)}}\, .\]In the latter case, we also
obtain an upper bound for the expected Euler characteristic of the zero set of
$f\_L^s$ and for its Betti numbers. In the case $s>n/2$, the pointwise variance
of $f\_L^s$ converges so it is not expected to have universal behavior as
$L\rightarrow+\infty$.
