Selberg Conjectures

Selberg has made two conjectures concerning the Dirichlet series in the Selberg class ${\mathcal S}$:

Conjecture A. For each $F\in \mathcal S$ there exists an integer $n_F$ such that

$$\sum_{p\le X} \frac{\vert a_p(F)\vert^2}{p} = n_F \log\log x + O(1).$$

Conjecture A follows from

Conjecture B. 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 [ MR 98h:11106], and that the factorization of elements into primitives is unique [ MR 95f:11064].

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 prime number theorem for $F(s)$.

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