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.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 numberwith 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): 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. The condition that θ < 1/2 is important, as the θ = 1/2 case includes the Dirichlet eta-function, which violates 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. Another example, is the L-function of the modular discriminant Δ where a n = τ ( n ) / n 11 / 2 {displaystyle a_{n}= au (n)/n^{11/2}} and τ(n) is the Ramanujan tau function. All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.