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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 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.
Giovanni Coppola (Mon,) studied this question.