High Quality Content by WIKIPEDIA articles! 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. 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 ? of SL2(R), and used these to study the representation of SL2(R) on L2(SL2(R)/?). Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.

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