On the error function of the square-full integers (Q757470)

A number n is squarefull if \(p^ 2| n\) for every prime divisor p of n. \textit{P. T. Bateman} and \textit{E. Grosswald} [Ill. J. Math. 2, 88-98 (1958; Zbl 0079.071)] proved that the sum function for squarefull numbers has an asymptotic formula with an error term satisfying \(\Delta (x)=o(x^{1/6})\) as \(x\to \infty\). Assuming the Riemann Hypothesis, \textit{D. Suryanarayana} and \textit{R. Sita Rama Chandra Rao} [Ark. Mat. 11, 195- 201 (1973; Zbl 0266.10037)] proved that \(\Delta (x)=O(x^{13/81+\epsilon})\) for every \(\epsilon >0\), and \textit{D. Suryanarayana} [Period. Math. Hung. 10, 261-271 (1979; Zbl 0422.10036) and 14, 69-75 (1983; Zbl 0513.10044)] also proved that the average order of \(\Delta\) (x) and \(| \Delta (x)|\) are \(O(x^{1/12+\epsilon})\) and \(O(x^{1/10+\epsilon})\) respectively. Here the author proves, also under the assumption of the Riemann Hypothesis, that there is a positive constant c such that \[ \int^{x}_{1}(\Delta^ 2(t)/t^{6/5})dt\quad \sim \quad c \log x, \] which implies that the average order of \(| \Delta (x)|\) is \(O(x^{1/10} \log^{1/2}x)\), and also that \(\Delta (x)=\Omega (x^{1/10})\).