Selberg class

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In mathematics, the Selberg class S is an axiomatic definition of the 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 1991.

[edit] 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>

that satisfy four axioms:

(i) Analyticity: the function <math>(s-1)^mF(s)</math> is an entire function of s for some non-negative integer m.

(ii) Ramanujan conjecture: the elements show limited growth, so that <math>a_n < n^{r+\epsilon}</math> for some fixed positive real number r and any ε > 0.

(iii) Functional equation: one can factor the series into two parts, the first, called the Gamma factor, taking a simple form, and the other obeying the traditional functional equation. That is, one must be able to write

<math>F(s) = \gamma(s)\Phi(s)\,</math>

so that

<math>\Phi(s)=\overline{\Phi(1-\overline{s})}</math>

and the gamma factor γ taking the form

<math>\gamma(s)=e^{i\phi}Q^s

\prod_{i=1}^n \Gamma (\omega_is+\mu_i)</math>

with φ real, Q real and positive, the eigenvalues <math>\omega_i</math> real and positive, and the <math>\mu_i</math> complex with non-negative imaginary part. Here, Γ is the gamma function.

(iv) Euler product: The coefficients <math>a_n</math> are a multiplicative series, with <math>a_1=1</math> and <math>F(s)</math> can be written as a product over primes:

<math>F(s)=\prod_{p \in\mathbb{P}} F_p(s)\,</math>

when Re s > 1 and <math>\mathbb{P}</math> is the set of all primes, with <math>F_p(s)</math> being expressible as

<math>F_p(s)=\sum_{n=0}^\infty \frac{a_{p^n}}{p^{ns}}</math>

when Re s > 0. In addition, when re-written in the form

<math>\log F(s)=\sum_{n=1}^\infty b_n n^{-s}</math>,

one must have the condition that <math>b_n<n^{\theta+\epsilon}</math> for some <math>\theta<1/2</math> and every <math>\epsilon>0</math>.

[edit] Discussion

The condition that the real part of <math>\mu_i</math> be positive is because there are known L-functions that do not satisfy the Riemann hypothesis when <math>\mu_i</math> 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 <math>\theta<1/2</math> is important, as the <math>\theta=1/2</math> case includes the Dirichlet eta-function, which violates the Riemann hypothesis.

Note that for the case of automorphic L-functions, the <math>F_p(s)</math> are polynomials of degree independent of p.

In contrast to the Selberg class, Li's criterion provides a rather simple and direct requirement for the generalized Riemann hypothesis to hold. The impact of Li's criterion upon the Selberg class is uncertain.