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

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4

The approximate functional equations for ζ k (s)

4.1 A simple AFE for ζ(s)

We recall the AFE (2.3), which we write again as

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The purpose of this section is to provide a more thorough discussion of the AFEs for ζ k (s), k ∈ ℕ and fixed k. Most of the results can be generalized to more general L-functions, notably to those belonging to the Selberg class S. However, the accent here is to acquaint the reader with the methods used in obtaining AFEs, and not to provide the results in any form of generality. If one wants to obtain AFEs for Zk(t), one only has to use the AFEs for (ζ( + it))k and the asymptotic formula for ( + it) (see the defining ...

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