Selberg class - WikiHQ

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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 , who preferred not to use the word "axiom" that later authors have employed.

Definition

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

:<math>F(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}</math>

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

{p^{ns\right)</math>

and, for some θ < 1/2,

:<math>b_{p^n}=O(p^{n\theta}).\,</math>

Comments on definition

Without the condition <math>a_n = O(n^\varepsilon)</math>, there would be:

::<math>L(s+1/3,\chi_4)L(s-1/3,\chi_4)</math>

:which violates the Riemann hypothesis.

Without this functional equation we would have Dirichlet L-functions for any imprimitive character. If <math display=inline>\chi</math> is Dirichlet character induced by <math display=inline>\chi^\star</math>, then we have:

::<math>L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \chi^\star(p) p^{-s} \right)

</math>

:Despite this function satisfies any other axiom, from that additional factors follows that it has infinitely many zeros on <math>\operatorname{Re}(s)=0</math>. They are not symmetric with respect to <math>\operatorname{Re}(s)=\tfrac{1}{2}</math> and without functional equation proposed by Selberg it is hard to distinguish between trivial and nontrivial zeros of this function.

If function F satisfies functional equation with different gamma factors, then using Stirling formula for gamma function one can show:

::<math>\gamma_2(s) = C \cdot \gamma_1(s) \quad \text{where: } C \in \mathbb{R}</math>.

:However, by the multiplication formula the same gamma factor can be expressed in many different ways, involving different number of gamma functions with different constants. Despite this, Selberg proved that the sum <math display=inline>\sum_{i=1}^k\omega_i</math> is independent of the choice of the gamma factor formula.

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–Petersson conjecture holds, and which have a functional equation, but do not satisfy the Riemann hypothesis.

Having Euler product is essential, as notable counterexample given by Davenport and Heilbronn:

::<math>F(s) = \frac{1-\alpha i}{2}L(s,\chi_5) + \frac{1+\alpha i}{2}L(s,\overline{\chi}_5) \quad \text{for: } \alpha = \frac{\sqrt{10-\sqrt{5-2}{\sqrt{5}-1}</math>

:despite having analytic continuation, having periodic coefficients (thus satisfying Ramanujan conjecture) and satisfying functional equation have zeros lying outside critical line.

Properties

The Selberg class is closed under multiplication of functions: product of each two functions belonging to S are also in S. It is also easy to check that if F is in S, then function involved in functional equation:

:<math>\overline{F(\overline{s})} = \sum_{n=1}^{\infty}\frac{\overline{a_n{n^{s</math>

satisfies axioms and is also in S. If F is entire function in S, then <math display=inline>F(s + it)</math> for <math display=inline>t \in \mathbb{R}</math> is also in S.

From the Ramanujan conjecture, it follows that, for every <math>\epsilon > 0</math>:

:<math display=inline>\sum_{i=1}^{n}\vert a_i\vert = O(n^{1+\epsilon})</math>

then Dirichlet series defining function is absolutely convergent in the half-plane: <math display=inline>\operatorname{Re}(s) >1</math>.

Despite the unusual version of the Euler product in the axioms, by exponentiation of Dirichlet series, one can deduce that an is a multiplicative sequence and that

:<math>F_p(s)=\sum_{n=0}^\infty\frac{a_{p^n{p^{ns\text{ for Re}(s)>1.</math>

From <math display=inline>\theta < \tfrac{1}{2}</math> follows that for each factor of Euler product:

:<math>\log F_p = \sum_{n=1}^\infty \frac{b_{p^n{p^{ns</math>

is absolutely convergent in <math display=inline>\operatorname{Re}(s)>\tfrac{1}{2}</math>. Then <math>F_p(s)</math> is absolutely convergent and <math>F_p(s) \neq 0</math> in this region. In half-plane of absolute convergence of original Dirichlet series function is absolutely convergent product of non-vanishing factors, then for functions in Selberg class <math display=inline>F(s)\neq 0</math> in <math display=inline>\operatorname{Re}(s)>1</math>.

From the functional equation follows that every pole of the gamma factor γ(s) in <math display=inline>\text{Re}(s) < 0 </math> must be cancelled by a zero of F. Such zeros are called trivial zeros; the other zeros of F are called non-trivial zeros. All nontrivial zeros are located in the critical strip, <math display="inline">0 < \text{Re}(s) < 1</math>, and by the functional equation, the nontrivial zeros are symmetrical about the critical line, <math display="inline"> \text{Re}(s) = \frac{1}{2}</math>.

The real non-negative number

:<math>d_F=2\sum_{i=1}^k\omega_i</math>

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

:<math>d_{FG}=d_F+d_G.</math>

It can be shown that F = 1 is the only function in S whose degree is <math display=inline>d_F<1</math>. showed that the only cases of <math display=inline>d_F < 2</math> are the Dirichlet L-functions for primitive Dirichlet characters (including the Riemann zeta-function). Denoting the number of non-trivial zeros of F with by NF(T), Selberg showed that:

:<math>N_F(T)=d_F\frac{T\log(T+C)}{2\pi}+O(\log T).</math>

An explicit version of the result was proven by .

It was proven by that for F in the Selberg class, <math display=inline>F(1+it)\neq 0</math> for <math>t\in \mathbb{R}</math> is equivalent to

:<math>\lim_{x\rightarrow \infty}\frac{\sum_{p\leq x}\vert a_p\vert^2}{\pi(x)} = \kappa_F, </math>

where <math>\kappa_F > 0 </math> is a real number and <math display = inline>\pi</math> is the prime-counting function. This result can be thought of as a generalization of the prime number theorem. showed that functions satisfying the prime-number theorem condition have a universality property for the strip <math display=inline>\sigma < Re(s) < 1</math>, where <math display=inline>\sigma = \max \lbrace \frac{1}{2},1-\frac{1}{d_F} \rbrace</math>. It generalizes the universality property of the Riemann zeta function and Dirichlet L-functions.

A function <math>F \neq 1</math> in S is called primitive if, whenever it is written as <math>F = F_1 \cdot F_2</math>, with both of function in Selberg class, then <math>F = F_1</math> or <math>F = F_2</math>. As a consequence that degree is additive with respect to multiplication of functions and only function of degree <math>d_F < 1</math> is <math>F = 1</math>, every function can be written as a product of primitive functions. However, uniqueness of this factorization is still unproven.

Examples

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

Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters or some Artin L-functions for irreducible representations of Galois group of Galois extension of rational numbers (in general case of Artin L-function non-existence of poles violating analictity axiom is subject of Artin conjecture).

Another example is the L-function of the modular discriminant Δ,

:<math>L(s,\Delta)=\sum_{n=1}^\infty\frac{\tau(n)/n^{11/2{n^s},</math>

where <math display="inline">\tau(n)</math> is the Ramanujan tau function. This example can be considered a "normalized" or "shifted" L-function for the original Ramanujan L-function, defined as

:<math>L(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s},</math>

whose coefficients satisfy <math display=inline> \vert \tau(n)\vert \leq n^{\frac{11}{2</math>. It has the functional equation

:<math>( \frac{1}{2\pi} )^s\Gamma (s)L(s)=

(\frac{1}{2\pi})^{12-s}\Gamma(12-s)L(12-s)</math>

and is expected to have all nontrivial zeros on the line <math display=inline>Re(s)=6</math>.

All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.

Conjectures

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 <math display="block">\sum_{p\leq x}\frac{|a_p|^2}{p}=n_F\log\log x+O(1)</math> and nF = 1 whenever F is primitive.
Conjecture 2: For distinct primitive F, F′ ∈ S, <math display="block">\sum_{p\leq x} \frac{a_p\overline{a_p^\prime{p}=O(1).</math>
Conjecture 3: If F is in S with primitive factorization <math display="block">F = \prod_{i=1}^m F_i,</math> χ is a primitive Dirichlet character, and the function <math display="block"> F^\chi(s) = \sum_{n=1}^\infty\frac{\chi(n)a_n}{n^s}</math> is also in S, then the functions Fiχ are primitive elements of S (and consequently, they form the primitive factorization of Fχ).
Generalized Riemann hypothesis for S: For all F in S, the non-trivial zeros of F all lie on the line Re(s) = 1/2.

The first two Selberg conjectures are often collectively called the Selberg orthogonality conjecture.

Other conjectures

It is conjectured that Selberg class is equal to class of automorphic L-functions. Primitive functions are expected to be associated with irreducible automorphic representations.

It is conjectured that all reciprocals of factors Fp(s) of the Euler products are polynomials in p−s of bounded degree.

It is conjectured that, for any F in the Selberg class, <math display=inline>d_F</math> is a nonnegative integer. The best particular result due to Kaczorowski & Perelli shows this only for <math display=inline>d_F<2</math>.

Consequences of the conjectures

The Selberg orthogonality conjecture has numerous consequences for functions in the Selberg class:

The factorization of function F in S into primitive functions would be unique.
If <math display=inline> F = F_1^{e_1}\dots F_n^{e_n}</math> is a factorization of F in S into primitive functions, then <math display=inline>n_F = e_1^2 + \dots + e_n^2</math>. In particular, this implies that <math display=inline>n_F=1</math> if and only if F is a primitive function.
The functions in S have no zeros on <math display=inline>Re(s)=1</math>. This implies that they satisfy a generalization of the prime number theorem and have a 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.
M. Ram Murty showed in that the orthogonality conjecture implies the Artin conjecture.
In the same article Murty proven that ortogonality conjecture implies Langlands reciprocity for Artin L-functions of solvable extensions.
L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive.

The Generalized Riemann Hypothesis for S implies many different generalizations of the original Riemann Hypothesis, the most notable being the 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.

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

Notes

References

Reprinted in Collected Papers, vol 2, Springer-Verlag, Berlin (1991)