On the exponential sum over squarefree integers

Let $μ$ be the Möbius function and $e(t)=e^{2πit}$. We prove that if $N\ge2$, $α\in\mathbb{R}$, $(a,q)=1$, and $|α-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}μ^2(n)e(αn)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the