We study the links between additive and multiplicative arithmetical functions, say , and their square-free supported counterparts, i.e. (here is the square-free numbers characteristic function), regarding the (upper bound) estimate of their symmetry around in almost all short intervals .
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, and their square-free supported counterparts, i.e. 
(here 
is the square-free numbers characteristic function), regarding the (upper bound) estimate of their symmetry around 
in almost all short intervals ![$ [x-h,x+h]$](images/059_04_JIPAM/img5.gif){kind=small}
.
