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

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3

The Selberg class of L-functions

3.1 The axioms of Selberg’s class

The Riemann zeta-function images is the simplest and most important example of a Dirichlet series (or L-function) of the form

images

with an arithmetic significance. One feels that the essential properties of ζ(s), namely the Euler product (1.1) and the functional equation (1.5) should, in some form, hold true for a general class of L-functions given by (3.1). The best known such generalization is the Selberg class , defined axiomatically by A. Selberg [Sel] as follows.

(a)  The series in ...

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