On the exponential sum over squarefree integers
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fej\'er majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.