We determine, up to a constant factor, the L1 mean of the exponential sum formed with the r-free integers. This improves earlier results of Brdern, Granville, Perelli, Vaughan and Wooley. As an application, we improve
the known bound for the L1 norm of the exponential sum defined with the Mbius function.
begun by Indlekofer and Kátai. Here |Yn|,|Xp| ≦ 1 and α is irrational. We prove one conjecture of Kátai, disprove another by both authors, and give what may be a close to best possible
result valid for all irrational α.