Atkinson's formula for Hardy's function

Abstract

An analogue of Atkinson's formula is proved for the integral function F ( T ) = ∫ 0 T Z ( t ) d t 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 ( T 1 / 4 ) and F ( T ) = Ω ± ( T 1 / 4 ) ; these results were recently obtained by M.A. Korolev using an alternative method.