Selberg class

The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions.

Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis.

[1] The formal definition of the class S is the set of all Dirichlet series absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them): where Q is real and positive, Γ the gamma function, the ωi real and positive, and the μi complex with non-negative real part, as well as a so-called root number such that the function satisfies with and, for some θ < 1/2, The condition that the real part of μi be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μi is negative.

Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Peterssen conjecture holds, and have a functional equation, but do not satisfy the Riemann hypothesis.

It is a consequence of 4. that the an are multiplicative and that The prototypical example of an element in S is the Riemann zeta function.

[2] Dirichlet L-functions associated with primitive characters modulo

[3] All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.

[4] The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2.

[5] As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s).

If F and G are in the Selberg class, then so is their product and A function F ≠ 1 in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2.

Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.

Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters.

In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.

[11] Combined with the Generalized Riemann hypothesis, different versions of Conjectures 1 and 2 imply certain growth rates for the function and its logarithmic derivative.

[12][13][14] The functions in S also satisfy an analogue of the prime number theorem: F(s) has no zeroes on the line Re(s) = 1.