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. [1]
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 ωi real and positive, and the μi 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 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. [2] Another example, is the L-function of the modular discriminant Δ
where and τ(n) is the Ramanujan tau function. [3]
All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree. [4]
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. [5]
As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s). The other zeroes are referred to as the non-trivial zeroes of F. These will all be located in some strip 1 − A ≤ Re(s) ≤ A. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by NF(T), [6] Selberg showed that
Here, dF is called the degree (or dimension) of F. It is given by [7]
It can be shown that F = 1 is the only function in S whose degree is less than 1.
If F and G are in the Selberg class, then so is their product and
A function F ≠ 1 in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2. If dF = 1, then F is primitive. Every function F ≠ 1 of S can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.
Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters. Assuming conjectures 1 and 2 below, L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive. [8]
In ( Selberg 1992), Selberg made conjectures concerning the functions in S:
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. [9]
M. Ram Murty showed in ( Murty 1994) 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. [10]
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. [11]
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. [1]
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 ωi real and positive, and the μi 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 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. [2] Another example, is the L-function of the modular discriminant Δ
where and τ(n) is the Ramanujan tau function. [3]
All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree. [4]
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. [5]
As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s). The other zeroes are referred to as the non-trivial zeroes of F. These will all be located in some strip 1 − A ≤ Re(s) ≤ A. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by NF(T), [6] Selberg showed that
Here, dF is called the degree (or dimension) of F. It is given by [7]
It can be shown that F = 1 is the only function in S whose degree is less than 1.
If F and G are in the Selberg class, then so is their product and
A function F ≠ 1 in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2. If dF = 1, then F is primitive. Every function F ≠ 1 of S can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.
Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters. Assuming conjectures 1 and 2 below, L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive. [8]
In ( Selberg 1992), Selberg made conjectures concerning the functions in S:
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. [9]
M. Ram Murty showed in ( Murty 1994) 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. [10]
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. [11]