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 H {\displaystyle {\mathcal {H}}} be the upper half-plane. For z ∈ H {\displaystyle z\in {\mathcal {H}}} , the Eisenstein series E ( z , s ) {\displaystyle E(z,s)} is defined by
E ( z , s ) = 1 2 ∑ ( m , n ) = 1 y s | m z + n | 2 s {\displaystyle E(z,s)={\frac {1}{2}}\sum _{(m,n)=1}{\frac {y^{s}}{|mz+n|^{2s}}}}
for all ℜ ( s ) > 1 {\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 1 / 2 {\displaystyle 1/2} , and some sum over all pairs of integers that are not both zero; this changes the function by a factor of ζ ( 2 s ) {\displaystyle \zeta (2s)} , where ζ {\displaystyle \zeta } is the Riemann zeta function.
Properties
As a function of z Viewed as a function of z = x + i y {\displaystyle z=x+iy} , E ( z , s ) {\displaystyle E(z,s)} is a real-analytic eigenfunction of the Laplace operator on H {\displaystyle {\mathcal {H}}} with eigenvalue s ( s − 1 ) {\displaystyle s(s-1)} . In other words, it satisfies the elliptic partial differential equation
− y 2 ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) E ( z , s ) = s ( 1 − s ) E ( z , s ) . {\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 E ( z , s ) {\displaystyle E(z,s)} is invariant under the action of SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} on z {\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 E ( z , s ) {\displaystyle E(z,s)} is not a square-integrable function of z {\displaystyle z} with respect to the invariant Riemannian metric on H {\displaystyle {\mathcal {H}}} .
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