Zhan Tao.ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS[J].Chinese Annals of Mathematics B,1989,10(2):227~235
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Authors：
Zhan Tao;
Abstract：
Let $\[L(x)\]$ denote the number of square-full integers not exceeding $\[x\]$. It is proved in [1] that
$$\[L(x) \sim \frac{{\zeta (3/2)}}{{\zeta (3)}}{x^{1/2}} + \frac{{\zeta (2/3)}}{{\zeta (2)}}{x^{1/3}}as\begin{array}{*{20}{c}}
{x \to \infty }
\end{array}\]$$
where $\[\zeta (s)\]$ denotes the Riemann zeta function. Let $\[\Delta (x)\]$ denote the error function in the asymptotic formula for $\[L(x)\]$. It was shown by D. $\[{\rm{Suryanaryan}}{{\rm{a}}^{[2]}}\]$ on the Riemann hypothesis (RH) that
$$\[\frac{1}{x}\int_1^x {\left| {\Delta (t)} \right|} dt = O({x^{1/10 + s}})\]$$
for every $\[s > 0\]$. In this paper the author proves the following asymptotic formula for the mean-value of $\[\Delta (x)\]$ under the assumption of R. H.
$$\[\int_1^T {\frac{{{\Delta ^2}(t)}}{{{t^{6/5}}}}dt} \sim c\log T\]$$
where $\[c > 0\]$ is a constant.
