Degree 1 elements of the Selberg class

Degree 1 Elements of the Selberg

In [5] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class.” We recall that an element F of the Selberg class S satisfies the following axioms. Axiom 1. In the half-plane σ > 1 the function F (s) is given by an absolutely convergent Dirichlet series ∑∞ n=1 a(n)n −s with a(1) = 1 and a(n) ≪ n for every ǫ > 0. Axiom 2. There is a natural number m such that ...

Strong Multiplicity One for the Selberg Class

In [7] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class” which is expected to coincide with the class of all arithmetically interesting L-functions. We recall that an element F of the Selberg class S satisfies the following axioms. • In the half-plane σ > 1 the function F (s) is given by a Dirichlet series ∑∞n=1 aF (n)n with aF (1) = 1 and aF (n) ≪ǫ n ...

On the Zeros of Functions in the Selberg Class

It is proved that under some suitable conditions, the degree two functions in the Selberg class have infinitely many zeros on the critical line.

On the Selberg Class of Dirichlet Series: Small Degrees

On the Structure of the Selberg Class, Iii: Sarnak’s Rigidity Conjecture

We further recall that under Selberg orthonormality conjecture, has unique factorization into primitive functions, the only primitive function with a pole at s = 1 is the Riemann zeta function ζ(s), and Fθ(s) is a primitive function if θ ∈R and if F ∈ are primitive and entire (see [4, Section 4]). We say a primitive function F ∈ is normal if θF = 0. Assuming Selberg orthonormality conjecture, w...