L-функциялар класының аксиомалық анықтамасы

Axiomatic definition of a class of L functions
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 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.

Conjectures 1 and 2 imply that if F has a pole of order m at s = 1, then F(s)/ζ(s)m is entire. In particular, they imply Dedekind's conjecture. M. Ram Murty showed in that conjectures 1 and 2 imply the Artin conjecture. 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. The functions in S also satisfy an analogue of the prime number theorem: F(s) has no zeroes on the line Re(s) = 1. As mentioned above, conjectures 1 and 2 imply the unique factorization of functions in S into primitive functions. Another consequence is that the primitivity of F is equivalent to nF = 1.