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: $\zeta (s)=\chi (s)\zeta (1-s)$.