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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