The moments of Hardy’s function
7.1 The asymptotic formula for the moments
In this chapter we shall deal with the moments
so that, in particular, F(T) ≡ F1(T) is a primitive of Hardy’s function Z(t). The study of a primitive of a highly oscillating function, such as Z(t), certainly is of interest. If we set
then on integrating by parts we find that
so that the properties of Fk(x) are reflected on . Conversely, the Mellin inversion formula gives ...