ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let $[L(x)]$ denote the number of square-full integers not exceeding $[x]$. It is proved in [1] that$$[L(x) sim frac{{zeta (3/2)}}{{zeta (3)}}{x^{1/2}} + frac{{zeta (2/3)}}{{zeta (2)}}{x^{1/3}}asbegin{array}{*{20}{c}}{x to infty }end{array}]$$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.$$[int_1^T {frac{{{Delta ^2}(t)}}{{{t^{6/5}}}}dt} sim clog T]$$where $[c > 0]$ is a constant.