Selberg [ MR 94f:11085] has given an elegant a set of axioms which presumably describes exactly the set of arithmetic -functions. He also made two deep conjectures about these -functions which have far reaching consequences.
The collection of Dirichlet series satisfying Selberg's axioms is called ``The Selberg Class.'' This set has many nice properties. For example, it is closed under products. The elements which cannot be written as a nontrivial product are called ``primitive,'' and every member can be factored uniquely into a product of primitive elements.
In some cases it is useful to slightly relax the axioms so that
the set is closed under the operation
Some of the important problems concerning the Selberg Class are:
1. Show that the members of the Selberg Class are arithmetic -functions.
2. Prove a prime number theorem for members of the Selberg class.
See Perelli and Kaczorowski [ MR 2001g:11141], Conrey and Ghosh [ MR 95f:11064] and Chapter 7 of Murty and Murty [ MR 98h:11106] for more details and some additional consequences of Selberg's conjectures.
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