Real analytic Eisenstein series

Template:Short description In mathematics, a real analytic Eisenstein series is a special function of two variables that is used in the representation theory of SL(2, R) and, more broadly, in analytic number theory.

Definition

Let $\mathscr{H}$ be the upper half-plane. For $z\in \mathscr{H}$, the Eisenstein series $E(z,s)$ is defined by

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

for all $\mathrm{\Re }(s)>1$. The sum is over all pairs of coprime integers.

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; this changes the function by a factor of $\zeta (2s)$, where $\zeta$ is the Riemann zeta function.

Properties

As a function of z

Viewed as a function of $z=x+iy$, $E(z,s)$ is a real-analytic eigenfunction of the Laplace operator on $\mathscr{H}$ with eigenvalue $s(s-1)$. In other words, it satisfies the elliptic partial differential equation

$-y^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2})E(z,s)=s(1-s)E(z,s).$

The function $E(z,s)$ is invariant under the action of $\mathrm{SL}(2,\mathbb{Z})$ 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.

Note that $E(z,s)$ is not a square-integrable function of $z$ with respect to the invariant Riemannian metric on $\mathscr{H}$.

As a function of s

The Eisenstein series converges for $\mathrm{\Re }(s)>1$, but can be analytically continued to a meromorphic function of $s$ on the entire complex plane, with a unique pole of residue $3/\pi$ at $s=1$ in the half-plane $\mathrm{\Re }(s)>1/2$ (for all $z\in \mathscr{H}$) and infinitely many poles in the strip $0<\mathrm{\Re }(s)<1/2$ at $\rho /2$, where $\rho$ corresponds to a non-trivial zero of the Riemann zeta function. The constant term of the pole at $s=1$ is described by the Kronecker limit formula.

The modified function

$E^{\ast }(z,s)={\pi }^{-s}\mathrm{\Gamma }(s)\zeta (2s)E(z,s)$

satisfies the functional equation

$E^{\ast }(z,s)=E^{\ast }(z,1-s),$

analogous to the functional equation for the Riemann zeta function.

The scalar product of two different Eisenstein series $E(z,s)$ and $E(z,t)$ is given by the Maass-Selberg relations.

Fourier expansion

The above properties of the real analytic Eisenstein series, i.e. the functional equation for $E(z,s)$ and $E^{\ast }(z,s)$ using the Laplacian on $\mathscr{H}$, are shown from the fact that $E(z,s)$ has a Fourier expansion

$E(z,s)=y^s+\frac{\mathrm{\Lambda }(2s-1)}{\mathrm{\Lambda }(2s)}{y}^{1-s}+\frac{4}{\mathrm{\Lambda }(2s)}\sum _{n=1}^{\mathrm{\infty }}{n}^{s-1/2}{\sigma }_{1-2s}(n)\sqrt{y}{K}_{s-1/2}(2\pi ny)\cos (2\pi nx),$

where

$\mathrm{\Lambda }(s)={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{s}{2}\right)\zeta (s),{\sigma }_s(m)=\sum _{d|m}{d}^s,$

and $K_s(z)$ are the modified Bessel functions

$\begin{array}{cc}{K}_s(z)& =\frac{1}{2}{\int }_0^{\mathrm{\infty }}{e}^{-(z/2)(t+1/t)}{t}^{s-1}dt\\ & \sim \sqrt{\frac{\pi }{2z}}{e}^{-z}(z\to \mathrm{\infty }).\end{array}$

Epstein zeta function

The Epstein zeta function ${\zeta }_Q(s)$ for a positive definite integral quadratic form $Q(m,n)=c{m}^2+bmn+a{n}^2$, named after Paul Epstein, is defined byTemplate:Sfnp

${\zeta }_Q(s)=\sum _{(m,n)\ne (0,0)}\frac{1}{Q(m,n{)}^s}.$

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

$z=-\frac{b}{2a}+\frac{i\sqrt{-b^2+4ac}}{2a}.$

Generalizations

The real analytic Eisenstein series $E(z,s)$ is really the Eisenstein series associated to the modular group, the discrete subgroup $\mathrm{SL}(2,\mathbb{Z})$ of $\mathrm{SL}(2,ℝ)$. Selberg described generalizations to other discrete subgroups $\mathrm{\Gamma }$ of $\mathrm{SL}(2,ℝ)$, and used these to study the representation of $\mathrm{SL}(2,ℝ)$ on $L^2(\mathrm{SL}(2,ℝ)/\mathrm{\Gamma })$. Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.Template:Citation needed

See also

• Eisenstein series
• Kronecker limit formula
• Maass form

References

• J. Bernstein, Meromorphic continuation of Eisenstein series

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• A. Selberg, Discontinuous groups and harmonic analysis, Proc. Int. Congr. Math., 1962.

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