Selberg has made two conjectures concerning the Dirichlet series in the Selberg class :

**Conjecture A.**
*For each
there exists an
integer such that
*

Conjecture A follows from

**Conjecture B.**
*If
is primitive, then
, and if
are distinct and primitive,
then
*

The above sums are over 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 -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 for and real, then Conjecture B with is equivalent to a prime number theorem for .

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