Selberg class

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]

Atle Selberg

Definition

The formal definition of the class S is the set of all Dirichlet series

${\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}$

absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them):

1. Analyticity: ${\displaystyle F(s)}$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s equals 1.
2. Ramanujan conjecture: a₁ = 1 and ${\displaystyle a_{n}\ll _{\varepsilon }n^{\varepsilon }}$ for any ε > 0;
3. Functional equation: there is a gamma factor of the form

   ${\displaystyle \gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})}$

   where Q is real and positive, Γ the gamma function, the ω_i real and positive, and the μ_i complex with non-negative real part, as well as a so-called root number

   ${\displaystyle \alpha \in \mathbb {C} ,\;|\alpha |=1}$,

   such that the function

   ${\displaystyle \Phi (s)=\gamma (s)F(s)\,}$

   satisfies

   ${\displaystyle \Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};}$

4. Euler product: For Re(s) > 1, F(s) can be written as a product over primes:

   ${\displaystyle F(s)=\prod _{p}F_{p}(s)}$

   with

   ${\displaystyle F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)}$

   and, for some θ < 1/2,

   ${\displaystyle b_{p^{n}}=O(p^{n\theta }).\,}$

Comments on definition

The condition that the real part of μ_i be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μ_i is negative. Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Petersson 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 ${\displaystyle (1-2^{-s})(1-2^{1-s})}$ whose zeros are not on the critical line.

Without the condition ${\displaystyle a_{n}\ll _{\varepsilon }n^{\varepsilon }}$ there would be ${\displaystyle L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})}$ which violates the Riemann hypothesis.

It is a consequence of 4. that the a_n are multiplicative and that

${\displaystyle F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>0.}$

Examples

The prototypical example of an element in S is the Riemann zeta function.^[2] Dirichlet L-functions associated with primitive characters modulo ${\displaystyle q\geq 2}$ belong to the Selberg class, too. Another example, is the L-function of the modular discriminant Δ

${\displaystyle L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}$

where ${\displaystyle a_{n}=\tau (n)/n^{11/2}}$ and τ(n) is the Ramanujan tau function.^[3]

All known examples are automorphic L-functions, and the reciprocals of F_p(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]

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 1 − A ≤ Re(s) ≤ A. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by N_F(T),^[6] Selberg showed that

${\displaystyle N_{F}(T)=d_{F}{\frac {T\log(T+C)}{2\pi }}+O(\log T).}$

An explicit version of the result was proven by Palojärvi.^[7] Here, d_F is called the degree (or dimension) of F. It is given by^[8]

${\displaystyle d_{F}=2\sum _{i=1}^{k}\omega _{i}.}$

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

${\displaystyle d_{FG}=d_{F}+d_{G}.}$

A function F ≠ 1 in S is called primitive if whenever it is written as F = F₁F₂, with F_i in S, then F = F₁ or F = F₂. If d_F = 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.^[9]

Selberg's conjectures

In (Selberg 1992), Selberg made conjectures concerning the functions in S:

• Conjecture 1: For all F in S, there is an integer n_F such that $${\displaystyle \sum _{p\leq x}{\frac {|a_{p}|^{2}}{p}}=n_{F}\log \log x+O(1)}$$ and n_F = 1 whenever F is primitive.
• Conjecture 2: For distinct primitive F, F′ ∈ S, $${\displaystyle \sum _{p\leq x}{\frac {a_{p}{\overline {a_{p}^{\prime }}}}{p}}=O(1).}$$
• Conjecture 3: If F is in S with primitive factorization $${\displaystyle F=\prod _{i=1}^{m}F_{i},}$$ χ is a primitive Dirichlet character, and the function $${\displaystyle F^{\chi }(s)=\sum _{n=1}^{\infty }{\frac {\chi (n)a_{n}}{n^{s}}}}$$ is also in S, then the functions F_i^χ are primitive elements of S (and consequently, they form the primitive factorization of F^χ).
• Generalized 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)^m is entire. In particular, they imply Dedekind's conjecture.^[10]

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.^[11]

Combined with the Generalized Riemann hypothesis, different versions of Conjectures 1 and 2 imply certain growth rates for the function and its logarithmic derivative.^[12][13][14]

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 n_F = 1.^[15]

See also

• List of zeta functions