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 ∫ 0 T Z (t) ζ (1 / 2 + i t) d t and ∫ 0 T Z 2 (t) ζ (1 / 2 + i t) d t . Furthermore we derive an upper bound for ∫ 0 T Z 3 (t) χ α (1 / 2 + i t) d t 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) . [ABSTRACT FROM AUTHOR]