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5

The derivatives of Z(t)

5.1 The θ and Γ functions

In applications it is sometimes useful not to have only an expression like (2.3), but to have similar expressions for the derivatives Z(k)(t) as well. Results of this type have been obtained by A. A. Karatsuba and S. M. Voronin [KaVo] and A. A. Lavrik [Lav1]. We shall follow the former work and derive an AFE for Z(k)(t) that is valid for all non-negative integers k. Before we formulate the result, we need the following lemma, which incidentally may serve as the basis for the proof of Stirling’s formula for Γ(s).

Lemma 5.1 Let θ(t) be defined by (1.19). Then for t ≥ 2 we have

with

where ψ

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