Selberg class

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.

Definition

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

:$F(s)=sum\_\{n=1\}^infty\; frac\{a\_n\}\{n^s\}$

that satisfy four axioms:

:"(i)" Analyticity: the function $(s-1)^mF(s)$ is an entire function of "s" for some non-negative integer "m".

:"(ii)" Ramanujan conjecture: the elements show limited growth, so that for some fixed positive real number "r" and any ε > 0.

:"(iii)" Functional equation: there is a gamma factor of the form

::$gamma(s)=e^\{iphi\}Q^sprod\_\{i=1\}^n\; Gamma\; (omega\_is+mu\_i)$:where φ is real, "Q" real and positive, Γ is the gamma function, the "eigenvalues" $omega\_i$ real and positive, and the $mu\_i$ complex with non-negative imaginary part, so that the function

::$Phi(s)\; =\; gamma(s)\; F(s),$

:satisfies

::$Phi(s)=overline\{Phi(1-overline\{s\})\}.$

:"(iv)" Euler product: The coefficients $a\_n$ are a multiplicative series, with $a\_1=1$ and $F(s)$ can be written as a product over primes:

::$F(s)=prod\_\{p\; inmathbb\{P\; F\_p(s),$

:when Re "s" > 1 and $mathbb\{P\}$ is the set of all primes, with $F\_p(s)$ being expressible as

::$F\_p(s)=sum\_\{n=0\}^infty\; frac\{a\_\{p^n\{p^\{ns$

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

::$log\; F(s)=sum\_\{n=1\}^infty\; b\_n\; n^\{-s\}$,

:one must have the condition that

Discussion

The condition that the real part of $mu\_i$ be positive is because there are known "L"-functions that do not satisfy the Riemann hypothesis when $mu\_i$ 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 is important, as the $heta=1/2$ case includes the Dirichlet eta-function, which violates the Riemann hypothesis.

Note that for the case of automorphic "L"-functions, the $F\_p(s)$ are polynomials of degree independent of "p".

References

* Atle Selberg, "Old and new conjectures and results about a class of Dirichlet series", Collected Papers, vol 2, Springer-Verlag, Berlin (1991)

* J. Brian Conrey and Amit Ghosh, " [http://arxiv.org/abs/math.NT/9204217 On the Selberg class of Dirichlet series: small degrees] ", Duke Math. J. 72 no.3 (1993) pp. 673-693

* M. Ram Murty, " [http://arxiv.org/abs/math?papernum=9407219 Selberg's Conjectures and Artin L-functions] ", Bull. Amer. Math. Soc. 31 (1994), 1-14.