On the exponential sum over \(r\)-free integers (Q5933729)

The authors determine up to a constant factor the \(L^1\) mean of the exponential sum formed with \(r\)-free integers. Let \(a_n\) be the characteristic function of the \(r\)-free integers, then they prove the estimates \(N^{\frac {1}{r+1}}\ll\int^1_0\left|\sum^N_{n=1}a_n e^{2\pi inx}\right| dx\ll N^{\frac 1{r+1}}\). From this they conclude that \(N^{\frac 1 6}\ll \int^1_0\left|\sum^N_{n=1}\mu(n)e^{2\pi inx}\right|dx\) holds, where \(\mu(n)\) denotes the Möbius function.