# Real analytic Eisenstein series

In mathematics, the simplest **real analytic Eisenstein series** is a special function of two variables. It is used in the representation theory of SL(2,**R**) and in analytic number theory. It is closely related to the Epstein zeta function.

There are many generalizations associated to more complicated groups.

## DefinitionEdit

The Eisenstein series *E*(*z*, *s*) for *z* = *x* + *iy* in the upper half-plane is defined by

for Re(*s*) > 1, and by analytic continuation for other values of the complex number *s*. The sum is over all pairs of coprime integers.

**Warning**: there are several other slightly different definitions. Some authors omit the factor of ½, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2*s*).

## PropertiesEdit

### As a function on *z*Edit

Viewed as a function of *z*, *E*(*z*,*s*) is a real-analytic eigenfunction of the Laplace operator on **H** with the eigenvalue *s*(*s*-1). In other words, it satisfies the elliptic partial differential equation

- where

The function *E*(*z*, *s*) is invariant under the action of SL(2,**Z**) on *z* in the upper half plane by fractional linear transformations. Together with the previous property, this means that the Eisenstein series is a Maass form, a real-analytic analogue of a classical elliptic modular function.

**Warning**: *E*(*z*, *s*) is not a square-integrable function of *z* with respect to the invariant Riemannian metric on **H**.

### As a function on *s*Edit

The Eisenstein series converges for Re(*s*)>1, but can be analytically continued to a meromorphic function of *s* on the entire complex plane, with in the half-plane Re(*s*) 1/2 a unique pole of residue 3/π at *s* = 1 (for all *z* in **H**) and infinitely many poles in the strip 0 < Re(*s*) < 1/2 at where corresponds to a non-trivial zero of the Riemann zeta-function. The constant term of the pole at *s* = 1 is described by the Kronecker limit formula.

The modified function

satisfies the functional equation

analogous to the functional equation for the Riemann zeta function ζ(*s*).

Scalar product of two different Eisenstein series *E*(*z*, *s*) and *E*(*z*, *t*) is given by the Maass-Selberg relations.

### Fourier expansionEdit

The above properties of the real analytic Eisenstein series, i.e. the functional equation for E(z,s) and E^{*}(z,s) using Laplacian on **H**, are shown from the fact that E(z,s) has a Fourier expansion:

where

and modified Bessel functions

## Epstein zeta functionEdit

The **Epstein zeta function** ζ_{Q}(*s*) (Epstein 1903) for a positive definite integral quadratic form *Q*(*m*, *n*) = *cm*^{2} + *bmn* +*an*^{2} is defined by

It is essentially a special case of the real analytic Eisenstein series for a special value of *z*, since

for

This zeta function was named after Paul Epstein.

## GeneralizationsEdit

The real analytic Eisenstein series *E*(*z*, *s*) is really the Eisenstein series associated to the discrete subgroup SL(2,**Z**) of SL(2,**R**). Selberg described generalizations to other discrete subgroups Γ of SL(2,**R**), and used these to study the representation of SL(2,**R**) on L^{2}(SL(2,**R**)/Γ). Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.

## See alsoEdit

## ReferencesEdit

- J. Bernstein,
*Meromorphic continuation of Eisenstein series* - Epstein, P. (1903), "Zur Theorie allgemeiner Zetafunktionen I" (PDF),
*Math. Ann.*,**56**(4): 614–644, doi:10.1007/BF01444309. - A. Krieg (2001) [1994], "Epstein zeta-function",
*Encyclopedia of Mathematics*, EMS Press - Kubota, T. (1973),
*Elementary theory of Eisenstein series*, Tokyo: Kodansha, ISBN 0-470-50920-1. - Langlands, Robert P. (1976),
*On the functional equations satisfied by Eisenstein series*, Berlin: Springer-Verlag, ISBN 0-387-07872-X. - A. Selberg,
*Discontinuous groups and harmonic analysis*, Proc. Int. Congr. Math., 1962. - D. Zagier,
*Eisenstein series and the Riemann zeta-function*.