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The Theory of Hardy’s Z-Function by Aleksandar Ivi

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7

The moments of Hardy’s function

7.1 The asymptotic formula for the moments

In this chapter we shall deal with the moments

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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

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then on integrating by parts we find that

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so that the properties of Fk(x) are reflected on . Conversely, the Mellin inversion formula gives ...

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