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Concept
Selberg class
Summary
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 , 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 case includes whose zeros are not on the critical line.
Without the condition there would be 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 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.
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.
Official source
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