Selberg has made two conjectures concerning the Dirichlet series in the Selberg class :
there exists an
integer such that
Conjecture A follows from
is primitive, then
, and if
are distinct and primitive,
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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