Maass forms
http://bomber0.myid.net/ (토론)님의 2010년 3월 5일 (금) 18:59 판
introduction
- Hyperbolic distribution problems and half-integral weight Maass forms
Eisenstein series
- z = x + iy in the upper half-plane
- Re(s) > 1
- definition
\(E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over|mz+n|^{2s}}\) - Maass form
\(\DeltaE(z,s)=-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)E(z,s) = s(1-s)E(z,s)\) - functional equation
\(E^{*}(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s) \)
\(E^{*}(z,s)=E^{*}(z,1-s)\) - a unique pole of residue 3/π at s = 1
Kloosterman sum
- Fourier coefficients
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- [1]http://en.wikipedia.org/wiki/Kronecker_limit_formula
- http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
- http://en.wikipedia.org/wiki/Kloosterman_sum
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- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
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- http://dx.doi.org/
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