Selberg class

In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. 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. The class was defined by Atle Selberg in (Selberg 1992), who preferred not to use the word "axiom" that later authors have employed.

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 ω_i real and positive, and the μ_i 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 μ_i be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μ_i 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.

The condition that θ < 1/2 is important, as the θ = 1/2 case includes the Dirichlet eta-function, which violates the Riemann hypothesis.

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