Prime-Detecting Identities from Dirichlet Inversion

For each integer $k\geq1$, let $σ_k(n)=\sum_{d\mid n}d^k$, let $σ_k^{-1}$ denote its Dirichlet inverse, and let $J_k$ denote the $k$th Jordan totient function. For $n\in\mathbb{N}$, define $F_k(n)=σ_k^{-1}(n)+μ(n)+J_k(n)+3$. We determine the zero set of $F_k$ completely. We prove that $F_1(n)=0$ if