The Selberg class

Selberg [ MR 94f:11085] has given an elegant a set of axioms which presumably describes exactly the set of arithmetic $L$-functions. He also made two deep conjectures about these $L$-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

$$F(s)\mapsto F(s+iy)$$

for real $y$.

Some of the important problems concerning the Selberg Class are:

1. Show that the members of the Selberg Class are arithmetic $L$-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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