Real analytic Eisenstein series

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.

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

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

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

There are several other slightly different definitions. Some authors omit the factor of ${\displaystyle 1/2}$, and some sum over all pairs of integers that are not both zero; this changes the function by a factor of ${\displaystyle \zeta (2s)}$, where ${\displaystyle \zeta }$ is the Riemann zeta function.

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

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

The function ${\displaystyle E(z,s)}$ is invariant under the action of ${\displaystyle \operatorname {SL} (2,\mathbb {Z} )}$ on ${\displaystyle z}$ 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 ${\displaystyle E(z,s)}$ is not a square-integrable function of ${\displaystyle z}$ with respect to the invariant Riemannian metric on ${\displaystyle {\mathcal {H}}}$.

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

The modified function

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

satisfies the functional equation

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

analogous to the functional equation for the Riemann zeta function.

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

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

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

where

${\displaystyle \Lambda (s)=\pi ^{-s/2}\Gamma {\biggl (}{\frac {s}{2}}{\biggr )}\zeta (s),\quad \sigma _{s}(m)=\sum _{d|m}d^{s},}$

and ${\displaystyle K_{s}(z)}$ are the modified Bessel functions

${\displaystyle {\begin{aligned}K_{s}(z)&={\frac {1}{2}}\int _{0}^{\infty }e^{-(z/2)(t+1/t)}t^{s-1}dt\\&\sim {\sqrt {\frac {\pi }{2z}}}e^{-z}\quad (z\rightarrow \infty ).\end{aligned}}}$

The Epstein zeta function ${\displaystyle \zeta _{Q}(s)}$ for a positive definite integral quadratic form ${\displaystyle Q(m,n)=cm^{2}+bmn+an^{2}}$, named after Paul Epstein, is defined by¹

${\displaystyle \zeta _{Q}(s)=\sum _{(m,n)\neq (0,0)}{1 \over Q(m,n)^{s}}.}$

It is essentially a special case of the real analytic Eisenstein series for a special value of ${\displaystyle z}$, since ${\displaystyle Q(m,n)=a|mz-n|^{2}\ }$ for

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

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