  We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some
arithmetic functions $f:\N \rightarrow \R$; we obtain from lower bounds of
$I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \lq
\lq Selberg integral\rq \rq \thinspace of $f$, say $J_f$ (both these integrals
give informations about $f$ in almost all the short intervals $[x-h,x+h]$, when
$N\le x\le 2N$). In particular, when \thinspace $f=d_k$, the divisor function
(having Dirichlet series \thinspace $\zeta^k$, with \thinspace $\zeta$
\thinspace the Riemann zeta function), where $k\ge 3$ is integer, we give lower
bounds for the Selberg integrals, say \thinspace $J_k=J_{d_k}$, of the
\thinspace $d_k$. We apply elementary methods (Cauchy inequality to get Large
Sieve type bounds) in order to give $I_f$ lower bounds.
