# Selberg Conjectures

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 .

Back to the main index for The Riemann Hypothesis.