Some mean value results related to Hardy's function

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

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