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.

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(2s).

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

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 a unique pole of residue π at s = 1 (for all z in H). The constant term of the pole at s = 1 is described by the Kronecker limit formula.

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