The number of cubefull numbers in an interval (supplement) (Q617789)

Let \(Q_3(x)\) be the number of cubefull numbers not exceeding \(x\). The author proves that \[ Q_3(x+x^{\frac{2}{3}+\mu})-Q_3(x)=Cx^\mu(1+o(1)), \] with a positive constant \(C\) and any \(\mu>\frac{5}{42}\). Previous estimates were given by \textit{P. Shiu} [Glasg. Math. J. 33, No. 3, 287--295 (1991; Zbl 0732.11044)] who obtained \(\mu>140/1123=0.1246607..\), \textit{H. Menzer} [Exponential sums and generalized divisor problems. Jena: Univ. Jena, Math. Fak. 147 pp. (1992; Zbl 0807.11043)] who gave a slight refinement of Shiu's result, the author [Acta Arith. 67, No. 1, 1--12 (1994; Zbl 0810.11054) who obtained \(11/92=0.1195..\) and \textit{J. Wu} [Monatsh. Math. 126, 353--367 (1998; Zbl 0917.11044)] who proved \(\mu>19/159=0.11949..\)