  We study in detail the Ramanujan smooth expansions, for arithmetic functions;
we start with the most general ones, for which we supply the "$P-$local
expansions", for arguments with all prime-factors $p\le P$ (namely, $P-$smooth
arguments), that are also square-free; then, we supply general results for
interesting subsets of arithmetic functions, regarding both their $P-$local and
(global) Ramanujan smooth expansions.
