In mathematics, the simplest real analytic Eisenstein series is a special function of two variables.
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.
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. 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)
The constant term of the pole at s = 1 is described by the Kronecker limit formula.
Scalar product of two different Eisenstein series E(z, s) and E(z, t) is given by the Maass-Selberg relations.
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 The Epstein zeta function ζQ(s) (Epstein 1903) for a positive definite integral quadratic form Q(m, n) = cm2 + bmn +an2 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.
Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.