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.
Definition The formal definition of the class S is the set of all Dirichlet series
F ( s ) = ∑ n = 1 ∞ a n n s {\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}
absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them):
Comments on definition Without the condition a n = O ( n ε ) {\displaystyle a_{n}=O(n^{\varepsilon })} , there would be:
L ( s + 1 / 3 , χ 4 ) L ( s − 1 / 3 , χ 4 ) {\displaystyle L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})}
which violates the Riemann hypothesis. Without this functional equation we would have Dirichlet L-functions for any imprimitive character. If χ {\textstyle \chi } is Dirichlet character induced by χ ⋆ {\textstyle \chi ^{\star }} , then we have:
L ( s , χ ) = L ( s , χ ⋆ ) ∏ p | q ( 1 − χ ⋆ ( p ) p − s ) {\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-\chi ^{\star }(p)p^{-s}\right)}
Despite this function satisfies any other axiom, from that additional factors follows that it has infinitely many zeros on Re ( s ) = 0 {\displaystyle \operatorname {Re} (s)=0} . They are not symmetric with respect to Re ( s ) = 1 2 {\displaystyle \operatorname {Re} (s)={\tfrac {1}{2}}} and without functional equation proposed by Selberg it is hard to distinguish between trivial and nontrivial zeros of this function. If function F satisfies functional equation with different gamma factors, then using Stirling formula for gamma function one can show:
γ 2 ( s ) = C ⋅ γ 1 ( s ) where: C ∈ R {\displaystyle \gamma _{2}(s)=C\cdot \gamma _{1}(s)\quad {\text{where: }}C\in \mathbb {R} } . However, by the multiplication formula the same gamma factor can be expressed in many different ways, involving different number of gamma functions with different constants. Despite this, Selberg proved that the sum ∑ i = 1 k ω i {\textstyle \sum _{i=1}^{k}\omega _{i}} is independent of the choice of the gamma factor formula. 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–Petersson conjecture holds, and which have a functional equation, but do not satisfy the Riemann hypothesis. Having Euler product is essential, as notable counterexample given by Davenport and Heilbronn:
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