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.

Definition

Let ${\mathcal {H}}$ be the upper half-plane. For $z\in {\mathcal {H}}$, 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 $\Re (s)>1$. 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 ${\mathcal {H}}$ with eigenvalue $s(s-1)$. In other words, it satisfies the elliptic partial differential equation

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

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

As a function of s

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 $3/\pi$ at $s=1$ in the half-plane $\Re (s)>1/2$ (for all $z\in {\mathcal {H}}$) and infinitely many poles in the strip $0<\Re (s)<1/2$ 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

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

satisfies the functional equation

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

analogous to the functional equation for the Riemann zeta function.

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 $E^{*}(z,s)$ using the Laplacian on ${\mathcal {H}}$, are shown from the fact that $E(z,s)$ has a Fourier expansion

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

where

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

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

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

Epstein zeta function

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

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

It is essentially a special case of the real analytic Eisenstein series for a special value of $z$, 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 $\operatorname {SL} (2,\mathbb {Z} )$ of $\operatorname {SL} (2,\mathbb {R} )$. Selberg described generalizations to other discrete subgroups $\Gamma$ of $\operatorname {SL} (2,\mathbb {R} )$, and used these to study the representation of $\operatorname {SL} (2,\mathbb {R} )$ on $L^{2}(\operatorname {SL} (2,\mathbb {R} )/\Gamma )$. Langlands extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by Joseph Bernstein.

See also

• Eisenstein series
• Kronecker limit formula
• Maass form

References

• J. Bernstein, Meromorphic continuation of Eisenstein series

• Epstein, Paul (1903). "Zur Theorie allgemeiner Zetafunktionen" (PDF). Mathematische Annalen (in German). 56 (4): 615–644. doi:10.1007/BF01444309.

• A. Krieg (2001) [1994], "Epstein zeta-function", Encyclopedia of Mathematics, EMS Press

• Kubota, Tomio (1973). Elementary Theory of Eisenstein Series. Tokyo: Kodansha. ISBN 0-470-50920-1.

• Langlands, Robert P. (1976). On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Mathematics. Vol. 544. Springer Berlin, Heidelberg. ISBN 978-3-540-07872-2.

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

• Zagier, D. (1981). "Eisenstein series and the Riemann zeta-function". Automorphic Forms, Representation Theory and Arithmetic. Springer Berlin, Heidelberg. doi:10.1007/978-3-662-00734-1_10. ISBN 978-3-540-10697-5.