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<h1 align=center> Selberg Conjectures
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Selberg has made two conjectures concerning the Dirichlet
series in the Selberg class ${\mathcal S}$:

\noindent{\bf Conjecture A.}
{\it   For each $F\in \mathcal S$ there exists an
integer  $n_F$ such that
$$
\sum_{p\le X} \frac{|a_p(F)|^2}{p} = n_F \log\log x + O(1).
$$
}



Conjecture A follows from

\noindent{\bf Conjecture B.}
{\it  If $F\in \mathcal S$ is primitive, then
$n_F=1$, and if $F,F'\in \mathcal S$ are distinct and primitive,
then
$$
\sum_{p\le X} \frac{a_p(F){\overline{a_p(F'}}}{p} =  O(1).
$$
}


The above sums are over $p$ prime.

Conjecture B can be interpreted as saying that the primitive
functions form an orthonormal system.  This conjecture  is
very deep.  It implies, among other things, Artin's conjecture
on the holomorphy of non-abelian $L$-functions [
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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 that the
factorization of elements into primitives is unique [
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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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If you extend the Selberg Class to include $G(s)=F(s+iy)$
for $F\in \mathcal S$ and $y$ real, then Conjecture~B with
$F'(s)=\zeta(s-iy)$ is equivalent to a \begin{rawhtml}<a href="../26a">prime number theorem</a>\end{rawhtml}
for $F(s)$.\begin{rawhtml}
<p> <br>
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<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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