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 SL₂(R) and in analytic number theory. It is closely related to the Epstein zeta function.

There are many generalizations associated to more complicated groups.

Definition

The Eisenstein series "E"("z", "s") for "z" in the upper half-plane is defined by

:$E(z,s)\; =\{1over\; 2\}sum\_\{(m,n)=1\}\{y^sover|mz+n|^\{2s$

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 1/2, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2"s").

Properties

Viewed as a function of "z", Eisenstein series is a real-analytic eigenfunction of the Laplace operator on the upper half plane with the eigenvalue "s"("s"-1), in other words, it satisfies the elliptic partial differential equation

: $y^2left(frac\{partial^2\}\{partial\; x^2\}+frac\{partial^2\}\{partial\; y^2\}\; ight)E(z,s)\; =\; s(s-1)E(z,s),$    where $z=x+yi.$

However, "E"("z", "s") is not a square-integrable function of "z" with respect to the invariant Riemannian metric.

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.

The Eisenstein series converges for Re("s")>1, but can be analytically continued to a meromorphic function of "s" on the whole complex plane, with a unique pole of residue 3/π at "s" = 1 (for all "z" in the upper half plane). The constant term of the pole at "s" = 1 is described by the Kronecker limit formula.

The modified function

:$E^*(z,s)\; =\; pi^\{-s\}Gamma(s)zeta(2s)E(z,s)$

satisfies the functional equation

:$E^*(z,s)\; =\; E^*(z,1-s)$

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.

Epstein zeta function

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

:$zeta\_Q(s)\; =\; sum\_\{(m,n)\; e\; (0,0)\}\; \{1over\; Q(m,n)^s\}.$

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

:$Q(m,n)\; =\; a|mz+n|^2$

for

:$z\; =\; \{-bover\; 2a\}\; +\; i\; \{sqrt\{4ac-b^2\}over\; 2a\}.$

Generalizations

The real analytic Eisenstein series "E"("z", "s") is really the Eisenstein series associated to the discrete subgroup SL₂(Z) of SL₂(R). Selberg described generalizations to other discrete subgroups Γ of SL₂(R), and used these to study the representation of SL₂(R) on L²(SL₂(R)/Γ). Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.

ee also

*Eisenstein series
*Maass wave form

References

*J. Bernstein, [http://www.math.uchicago.edu/~arinkin/langlands/Bernstein/ "Meromorphic continuation of Eisenstein series"]
*springer|id=E/e120130|title=Epstein zeta-function|author=A. Krieg
*T. Kubota, "Elementary theory of Eisenstein series", ISBN 0-470-50920-1
*Robert P. Langlands, [http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html "On the functional equations satisfied by Eisenstein series"] , ISBN 038707872X
*A. Selberg, "Discontinuous groups and harmonic analysis", Proc. Int. Congr. Math., 1962.
*D. Zagier, "Eisenstein series and the Riemann zeta-function".