Real analytic Eisenstein series

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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.

Contents 1 Definition 2 Properties 3 Epstein zeta function 4 Generalizations 5 See also 6 References

[edit] Definition

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

[edit] Properties

The function E(z, s) has a pole of reside 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 function E(z, s) is invariant under the action of SL₂(Z) on z in the upper half plane (so as a function of z it is a non-analytic modular function).

The modified function

E ^* (z,s) = π ^{− s}Γ(s)ζ(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).

[edit] Epstein zeta function

The Epstein zeta function ζ_Q(s) 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

Q(m,n) = a | mz + n | ²

for

[edit] Generalizations

The real analytic Eisenstein series E(z, s) is really the Eisenstein series associated to the discrete group 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.

[edit] See also

• Eisenstein series

[edit] References

• D. Zagier, Eisenstein series and the Riemann zeta-function.
• T. Kubota, Elementary theory of Eisenstein series, ISBN 0-470-50920-1
• A. Selberg, Discontinuous groups and harmonic analysis, Proc. Int. Congr. Math., 1962.
• Langlands, On the functional equations satisfied by Eisenstein series, ISBN 0-387-07872-0
• J. Bernstein, Meromorphic continuation of Eisenstein series