Some mean value results related to Hardy's function

Abstract

Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for $\int_0^T Z^3(t)\chi^{\alpha}(1/2+it)dt$ for $-1/2<\alpha<1/2$, where $\chi(s)$ is the function which appears in the functional equation of the Riemann zeta function: $\zeta(s)=\chi(s)\zeta(1-s)$.