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<h1 align=center> The Selberg class
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Selberg [
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<a href="http://www.ams.org/msnmain?fn=130&s1=94f:11085&pg1=MR">MR 94f:11085</a>
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which presumably describes exactly the set of arithmetic $L$-functions.
He also made \begin{rawhtml}<a href="../82a">two deep conjectures</a>\end{rawhtml} 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 \begin{rawhtml}<a href="../26a">prime number theorem</a>\end{rawhtml} for members of the 
   Selberg class. 

See Perelli and Kaczorowski [
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<a href="http://www.ams.org/msnmain?fn=130&s1=2001g:11141&pg1=MR">MR 2001g:11141</a>
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<a href="http://www.ams.org/msnmain?fn=130&s1=95f:11064&pg1=MR">MR 95f:11064</a>
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and Chapter 7 of Murty and Murty [
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<a href="http://www.ams.org/msnmain?fn=130&s1=98h:11106&pg1=MR">MR 98h:11106</a>
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and some additional consequences of Selberg's conjectures.

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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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