Selberg class
Encyclopedia
In mathematics
, the Selberg class S is an axiom
atic definition of a class of L-function
s. 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 form
s and the Riemann hypothesis
. The class was defined by Atle Selberg
in .
absolutely convergent for Re(s) > 1 that satisfy four axioms:
be positive is because there are known L-functions that do not satisfy the Riemann hypothesis
when
is zero or negative. Specifically, there are Maass cusp 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
is important, as the 
case includes the Dirichlet eta-function, which violates the Riemann hypothesis.
It is a consequence of 4. that the an are multiplicative
and that
where
and τ(n) is the Ramanujan tau function. Additionally, if F is in S and χ is a primitive Dirichlet character, then Fχ defined by
is also in S.
All known examples are automorphic L-function
s, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.
Here, dF is called the degree (or dimension) of F. It is given by
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 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 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Selberg class S is an axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic definition of a class of L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s. 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 form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s and the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
. The class was defined by Atle Selberg
Atle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...
in .
Definition
The formal definition of the class S is the set of all Dirichlet seriesabsolutely convergent for Re(s) > 1 that satisfy four axioms:
- AnalyticityAnalytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
: the function
is an entire functionEntire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
of finite order for some non-negative integer m;
- Ramanujan conjecture:
and 
for any ε > 0;
- Functional equationFunctional equation (L-function)In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural...
: there is a gamma factor of the form
where φ is real, Q real and positive, Γ is the gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
, the
real and positive, and the 
complex with non-negative real part, so that the function
satisfies
- Euler productEuler productIn number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
: F(s) can be written as a product over primes:
with
and, for some θ < 1/2,
Comments on definition
The condition that the real part of be positive is because there are known L-functions that do not satisfy the Riemann hypothesisRiemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
when
is zero or negative. Specifically, there are Maass cusp 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
is important, as the case includes the Dirichlet eta-function, which violates the Riemann hypothesis.It is a consequence of 4. that the an are multiplicative
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
and that
Examples
The prototypical example of an element in S is the Riemann zeta function. Another example, is the L-function of the modular discriminant Δwhere
and τ(n) is the Ramanujan tau function. Additionally, if F is in S and χ is a primitive Dirichlet character, then Fχ defined byis also in S.
All known examples are automorphic L-function
Automorphic L-function
In mathematics, an automorphic L-function is a function L of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comlplex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a...
s, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.
Basic properties
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 . Denoting the number of non-trivial zeroes of F with by NF(T), Selberg showed thatHere, dF is called the degree (or dimension) of F. It is given by
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 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 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
Cuspidal representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory...
automorphic representations that satisfy the Ramanujan conjecture are primitive.
Selberg's conjectures
In , Selberg made conjectures concerning the functions in S:- Conjecture 1: For all F in S, there is an integer nF such that
-
- and nF = 1 whenever F is primitive.
- Conjecture 2: For distinct primitive F, F′ ∈ S,
- Conjecture 3: If
- is a factorization of F into primitive functions and χ is a primitive Dirichlet character, then
- and the Fiχ are primitive.
- Riemann hypothesis for S: For all F in S, the non-trivial zeroes of F all lie on the line Re(s) = 1/2.
Consequences of the conjectures
Conjectures 1 and 2 imply that if F has a pole of order m at s = 1, then F(s)/ζ(s) is entire. In particular, they imply Dedekind's conjecture.M. Ram Murty
M. Ram Murty
Maruti Ram Pedaprolu Murty, FRSC is an Indo-Canadian mathematician, currently head of the Department of Mathematics and Statistics at Queen's University, where he holds a Queen's Research Chair in mathematics.-Career:...
showed in that conjectures 1 and 2 imply the Artin conjecture. In fact, Murty showed that Artin L-functions
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...
corresponding to irreducible representations of the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
: 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.