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

• 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.

• Functional equation does not have to be unique. By duplication formula for ${\displaystyle \Gamma }$ function new factors with different real constant can be produced. However, Selberg proven that the sum: ${\textstyle \sum _{i=1}^{k}\omega _{i}}$ is independent on choice of functional equation.

• 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.

Properties

Selberg class is closed to multiplication of functions, F and G are in the Selberg class, then so is their product.

From Ramanujan conjecture follows that for every ${\displaystyle \epsilon >0}$, ${\textstyle \sum _{i=1}^{n}\vert a_{i}\vert =O(n^{1+\epsilon })}$, hence Dirichlet series is absolutely convergent in half-plane ${\textstyle {\text{Re}}(s)>1}$.

Despite unusual version of Euler product in axioms, by exponentiation of Dirichlet series, one can deduce that the a_n is multiplicative function and that:

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

Real nonnegative number:

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

is called the degree (or dimension) of F. Since this sum is independent on choice of functional equation, it is well-defined for any function F. If F and G are in the Selberg class, then so is their product and:

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

It can be shown that F = 1 is the only function in S whose degree is ${\textstyle d_{F}<1}$. Kaczorowski and Perelli shown that only cases of ${\textstyle d_{F}<2}$ are Dirichlet L-functions for primitive Dirichlet characters (including the Riemann zeta-function).^[2]

From Euler product and Ramanujan conjecture follows that for ${\textstyle {\text{Re}}(s)>1}$ functions in Selberg class are non-vanishing. From functional equation every pole of gamma factor γ(s) in ${\textstyle {\text{Re}}(s)<0}$ must be cancelled by zero of F. That zeroes are called trivial zeroes, the other zeroes of F are called non-trivial zeroes. All nontrivial zeroes are located in critical strip: ${\textstyle 0<{\text{Re}}(s)<1}$ and by functional equation nontrivial zeroes are symmetrical with respect to axis: ${\textstyle {\text{Re}}(s)={\frac {1}{2}}}$. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by N_F(T),^[3] 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.^[4]

It was proven by Kaczorowski & Perelli that for F in Selberg class ${\textstyle F(1+it)\neq 0}$ for ${\displaystyle t\in \mathbb {R} }$ is equivalent to:

${\displaystyle \lim _{x\rightarrow \infty }{\frac {\sum _{p\leq x}\vert a_{p}\vert ^{2}}{\pi (x)}}=\kappa _{F}}$

where ${\displaystyle \kappa _{F}>0}$ is real number and ${\textstyle \pi }$ is prime-counting function. This result can be thought as generalization of prime number theorem for Selberg class.^[5]

Nagoshi & Steuding shown that function satisfying prime-number theorem condition have universality property for strip: ${\textstyle \sigma <Re(s)<1}$, where: ${\textstyle \sigma =\max \lbrace {\frac {1}{2}},1-{\frac {1}{d_{F}}}\rbrace }$. It generalizes universality property of zeta function and Dirichlet L-functions.^[6]

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, however uniqueness of such factorization is still open problem.

Examples

The prototypical example of an element in S is the Riemann zeta function.^[7] Also most of generalizations of zeta function like Dirichlet L-functions or Dedekind zeta functions belong to Selberg class.

Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters or Artin L-functions for irreducible representations.

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 ${\textstyle \tau (n)}$ is the Ramanujan tau function.^[8] This example can be considered as "normalized" or "shifted" L-function for original Ramanujan L-function defined:

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

whose coefficients satisfy inequality: ${\textstyle \vert \tau (n)\vert \leq n^{\frac {11}{2}}}$, have functional equation:

${\displaystyle ({\frac {1}{2\pi }})^{s}\Gamma (s)L(s)=({\frac {1}{2\pi }})^{12-s}\Gamma (12-s)L(12-s)}$

and is expected to have all nontrivial zeroes on line: ${\textstyle Re(s)=6}$.

All known examples are automorphic L-functions, and the reciprocals of F_p(s) are polynomials in p^−s of bounded degree.^[9]

Conjectures

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.

The first two Selberg conjectures are often colectively called Selberg orthogonality conjecture.

Other conjectures

It is conjectured that Selberg class is equal to class of automorphic L-functions.

It is conjectured that all reciprocals of factors of Euler product: F_p(s) are polynomials in p^−s of bounded degree.

It is conjectured that for any F in Selberg class ${\textstyle d_{F}}$ is nonnegative integer number. The best particular result due to Kaczorowski & Perelli shows it only for ${\textstyle d_{F}<2}$.

Consequences of the conjectures

Selberg orthogonality conjecture have numerous consequences for functions in Selberg class:

• Factorization of function F in S into primitive fuctions is unique.
• If ${\textstyle F=F_{1}^{e_{1}}\dots F_{n}^{e_{n}}}$ is factorization of F in S into primitive fuctions, then: ${\textstyle n_{F}=e_{1}^{2}+\dots +e_{n}^{2}}$. Particularly, this implies that ${\textstyle n_{F}=1}$ if and only if F is primitive function.^[10]
• The functions in S have no zeroes on ${\textstyle Re(s)=1}$. This implies that they satisfy generalization of prime number theorem and have universality property.
• 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.^[11]
• M. Ram Murty showed in (Murty 1994) that orthogonality conjecture imply the Artin conjecture. ^[12]
• L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive.^[13]

Generalized Riemann Hypothesis for S implies many different generalizations of original Riemann Hypothesis, the most notable being generalized Riemann hypothesis for Dirichlet L-functions and extended Riemann Hypothesis for Dedekind zeta functions, with multiple consequences in analytic number theory, algebraic number theory, class field theory and numerous branches of mathematics.

Combined with the Generalized Riemann hypothesis, different versions of orthogonality conjecture imply certain growth rates for the function and its logarithmic derivative.^[14][15][16]

If Selberg class equals class of automorphic L-functions, then Riemann hypothesis for S would be equivalent to Grand Riemann hypothesis.

See also

• List of zeta functions