Let $ S $ be a finite subset of the primes and let $ \mu_{S} $ the multiplicative arithmetic function defined by $ \mu_{S}(p)=-1_{p\not\in S} $ for all prime $ p $. Has any such mock Möbius function been used in sieve methods ? My idea is that it can be useful if $ S $ is adapted to fit the considered problem.
One can also consider the case $ \mu_{S,\chi} $ for some Dirichlet character $ \chi $ such that $ \mu_{S,\chi}(p)=\mu_{S}(p)-\chi(p).1_{p\in S} $.