• L-functions
• SelbergClass
• Definition

A Definition

This class is defined axiomatically and relies only on properties of the Dirichlet series for its specification.

Let (a_n:a_1=1,n\ge 1) be a sequence of complex numbers (called the Dirichlet coefficients). Then the function

is a member of the class \mathcal{S} if it satisfies the following four axioms:

• S1 Analyticity: There exists a non-negative integer m such that (s-1)^m L(s) is entire.

• S2 Ramanujan hypothesis: The Dirichlet coefficients satisfy a_n=O_\epsilon (n^\epsilon) for all \epsilon >0.

• S3 Functional equation: There exist k real constants \lambda_j>0, k complex constants \Re{\mu_j}\ge 0 and a positive constant Q such that if G(s)=\prod_{j=1}^k \Gamma(\lambda_j(s+\mu_j)) then, for some complex constant \epsilon with |\epsilon|=1 we have

  \Phi(s)=\epsilon\cdot\overline{\Phi(1-\bar{s})}
  where \Phi(s):= Q^s G(s) L(s).

• S4 Euler product: There exists Dirichlet coefficients (b_n) with b_n=0 other than possibly when n is a power of a rational prime, with b_n=O(n^\theta) for some \theta<\frac{1}{2}, such that

  L(s)= \prod_p L_p(s)
  where the product is over all finite rational primes p and where
  L_p(s)= \exp\big(\sum_{j=1}^\infty \frac{b_{p^j}}{p^{js}}\big).

Define the degree of an element F of \mathcal{S} as

where d_L=0 if the product of gamma factors is empty. The degree is well-defined, even though the specific form of the \Gammaa-factors G(s) is not. The degree is additive: if L,M\in\mathcal{S} then d_{L M}=d_L + d_M .

Properties of the Selberg Class

The set \mathcal{S}, with the usual functional product (or Dirichlet product of the corresponding Dirichlet series), is a commutative semigroup (i.e. associative) with identity, i.e. a monoid". Define a primitive element of \mathcal{S} as an F which satisfies L=L_1.L_2 implies L=L_1 or L=L_2.

Let \mathcal{S}(d) be the subclass of elements of \mathcal{S} with degree d. Then Conrey and Gosh ask whether \mathcal{S}(d) is empty other than when d\in\mathbb{N} ?

Examples of elements of \mathcal{S} are easy to find: The Riemann zeta function \zeta(s)\in\mathcal{S}(1) as is every shift L(s+i\alpha,\chi) of a Dirichlet L-function attached to a primitive character. The Dedekind zeta function of a number field of degree d is in \mathcal{S}(d). If it is holomorphic, the Artin L-function attached to an irreducible, non-trivial Galois representation of a normal and separable field extension of degree is in \mathcal{S}(d). An L-function attached to a holomorphic newform on a congruence subgroup of SL_2(\mathbb{Z}) is in \mathcal{S}(2).

Note that for all known examples of L\in\mathcal{S} one can find a gamma factor with each \lambda_i=\frac{1}{2}. If this is possible then the number Q and the \mu_i are uniquely determined.

A number of properties of elements of \mathcal{S} have been proved:

1. Two admissible gamma factors for a given member of \mathcal{S} differ at most by multiplication by a real constant.

2. If L\in\mathcal{S}, then L_p has no zero in \sigma>\frac{1}{2} and L has no zero for \sigma>1.

3. If L\in\mathcal{S} then either L(s)\equiv 1, the constant function with d_L=0, or d_L\ge 1.

4. If L\in\mathcal{S} and d_L=1, then Q\ge 1/\sqrt{\pi}.

5. If the gamma factor is specified with fixed parameters Q,\lambda_i,\mu_i and we consider the set of elements of \mathcal{S} having that gamma factor and satisfying the constraint d_L=1, then, by Bochner or Vigneras, the number of linearly independent (over \mathbb{C}) Dirichlet series in this set is bounded by

6. Any function in \mathcal{S} can be written as the product of primitive functions.

7. If L\in\mathcal{S} and d_L<2, then F is primitive.

Importance of each condition

• S1 Analyticity:

• S2 Ramanujan hypothesis:

• S3 Functional equation:

• S4: Euler product:

Selberg's conjectures

• SC1 Distribution regularity conjecture: For each L\in\mathcal{S} there is a non-negative integer n_L such that

  \sum_{p\le x} \frac{|a_p|^2}{p}= n_L \log \log x + O(1)
  as x\rightarrow\infty. If L is primitive then n_L=1.

• SC2 Degree conjecture: for every element L\in\mathcal{S}, the degree, d_L is always a non-negative integer.

• SC3 Orthonormality conjecture: If L,L'\in\mathcal{S} are distinct and primitive then

  \sum_{p\le x} \frac{a_p a'_p}{p}= O(1)
  where L(s)=\sum_{1\le n\le \infty} a_n n^{-s} and L'(s)=\sum_{1\le n\le \infty} a'_n n^{-s}.

• SC4 Closure under twisting conjecture: If \chi is a primitive Dirichlet character on \mathbb{Z} and for L\in\mathcal{S} with L(s)=\sum_{1\le n\le \infty} a_n n^{-s} we define

  L^\chi(s):= \sum_{n=1}^\infty \frac{a_n \chi(n)}{n^{s}}
  then, up to multiplication by a finite Euler product, L^\chi is also in \mathcal{S}.

If L=\prod_{1\le i\le k} L_i is the decomposition into primitive factors, then L^\chi=\prod_{1\le i\le k} L^\chi_i, and each L_i^\chi is primitive also.

• SC5 Riemann hypothesis for \mathcal{S} conjecture: For all L\in\mathcal{S} the zeros of L with non-zero imaginary part line on the line \sigma=\frac{1}{2}.

• SC6 Riemann zeta generator conjecture: If L\in\mathcal{S} has a pole at s=1 then there exists an L_1\in\mathcal{S} such that L(s)=\zeta(s)L_1(s). If the pole is of order m then shows that L(s)=\zeta(s)^m L_1(s) for some L_1.

Consequences of the Selberg definitions and conjectures

1. Each element can be factored uniquely into primitive elements.

2. If \dim L=1 and \lambda_1=\frac{1}{2}, or the \lambda_i are all rational, then L must be (a translate of) the Riemann zeta function or a Dirichlet L-function.

3. The Artin L-function attached to an irreducible non-trvial Galois representation of a number field extension is entire.

4. If \rho:Gal(K/k)\rightarrow GL(n,\mathbb{C}) is an irreducible representation of number fields, with Gal(K/k) solvable, then there is a cuspidal automorphic representation \pi of GL(n,\mathbb{A}_k) such that

This is a special case of Langland's reciprocity conjecture.

5. If L=L_1^{e_1}\cdots L_k^{e_k} is the decomposition of L\in\mathcal{S} into distinct primitives L_i, then n_L= e_1^2+\cdots + e_k^2.

6. If n_L=1 then L is primitive.

7. If L has a pole of order m at s=1 then \zeta(s)^m L_1(s)=L(s) for some L_1\in\mathcal{S}.

8. \mathcal{S} has unique factorization.

9. If K is a number field and \zeta_K(s) its Dedekind zeta function then \zeta_K(s) / \zeta(s) is an entire function of s.

10. Each L\in\mathcal{S} has no zeros on \sigma=1.

11. Farmer: The set of Langlands parameters cannot be just anywhere in the space it seems to occupy, but actually lives on sertain hyperplanes, maybe?

References

Selberg ..

Conrey Ghosh ..

Murty ..

Vingeras ..

..

Authorship

The primary author of this page was Kevin Broughan who takes full responsibility for any errors (unless and until they are fixed !).