Atkinson's formula for Hardy's function

An analogue of Atkinson's formula is proved for the integral function of Hardy's function Z(t). As an application of this formula, we analyze the behavior of the function F(T) showing that it can be approximated by a simple step-function. It follows that F(T)=O(T1/4) and F(T)=Ω±(T1/4); these results were recently obtained by M.A. Korolev using an alternative method.