  The weighted Selberg integral is a discrete mean-square, that is a
generalization of the classical Selberg integral of primes to an arithmetic
function $f$, whose values in a short interval are suitably attached to a
weight function. We give conditions on $f$ and select a particular class of
weights, in order to investigate non-trivial bounds of weighted Selberg
integrals of both $f$ and $f\ast\mu$. In particular, we discuss the cases of
the symmetry integral and the modified Selberg integral, the latter involving
the Cesaro weight. We also prove some side results when $f$ is a divisor
function.
