On some lower bounds of some symmetry integrals.

We study the 'symmetry integral', say $$I_f$$, of some arithmetic functions $$f:\mathbb{N }\rightarrow \mathbb{R }$$; we obtain from lower bounds of $$I_f$$ (for a large class of arithmetic functions $$f$$) lower bounds for the 'Selberg integral' 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 $$f=d_k$$, the divisor function (having Dirichlet series $$\zeta ^k$$, with $$\zeta $$ the Riemann zeta function), where $$k\ge 3$$ is integer, we give lower bounds for the Selberg integrals, say $$J_k=J_{d_k}$$, of the $$d_k$$. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give $$I_f$$ lower bounds. [ABSTRACT FROM AUTHOR]