We show that the proportion of permutations of length $n$ with principal Möbius function equal to zero, $Z(n)$, is asymptotically bounded below by 0.3995. If a permutation $π$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then we show that the value of the principal Möbius function $μ[1, π]$ is zero, and we use this result to find the lower bound for $Z(n)$. We also show that if a permutation $φ$ has certain properties, then any permutation $π$ which contains an interval order-isomorphic to $φ$ has $μ[1, π] = 0$.