On Möbius functions from automorphic forms and a generalized Sarnak's conjecture

In this paper, we consider Möbius functions associated with two types of $L$-functions: Rankin-Selberg $L$-functions of symmetric powers of distinct holomorphic cusp forms and $L$-functions of Maass cusp forms. We show that these Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.