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| + | ==Maass form== | ||
| + | * $f(z+1)=f(z)$ and $\Delta f=\lambda f$ where $\lambda = s(1-s)$ and $\Re s \geq 1/2$ imply | ||
| + | $$ | ||
| + | f(x+iy)=\sum_{n\in \mathbb{Z}}a_n \sqrt{y}K_{s-1/2}(2\pi |n| y) e^{2\pi i n x} | ||
| + | $$ | ||
| + | where $K_{\nu}$ is the modified Bessel function of the second kind | ||
| + | * under the assumption that $f(x+iy)=f(-x+iy)$, we get | ||
| + | $$ | ||
| + | f(x+iy)=\sum_{n=1}^{\infty}a_n \sqrt{y}K_{s-1/2}(2\pi n y) \cos (2\pi i n x) | ||
| + | $$ | ||
| + | |||
==Eisenstein series== | ==Eisenstein series== | ||
| 41번째 줄: | 53번째 줄: | ||
==computational resource== | ==computational resource== | ||
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxMll2Y2lzeWg0UVE/edit | ||
* http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf | * http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf | ||
* http://www.ijpam.eu/contents/2009-54-2/12/12.pdf | * http://www.ijpam.eu/contents/2009-54-2/12/12.pdf | ||
2013년 4월 15일 (월) 10:24 판
introduction
- Hyperbolic distribution problems and half-integral weight Maass forms
- Automorphic forms correspond to representations that occur in $L_2(\Gamma\backslash G)$.
- In the case when $G$ is $SL_2$
- holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of $G$
- Maass wave forms correspond to (spherical vectors of) continuous series representations of G.
Maass form
- $f(z+1)=f(z)$ and $\Delta f=\lambda f$ where $\lambda = s(1-s)$ and $\Re s \geq 1/2$ imply
$$ f(x+iy)=\sum_{n\in \mathbb{Z}}a_n \sqrt{y}K_{s-1/2}(2\pi |n| y) e^{2\pi i n x} $$ where $K_{\nu}$ is the modified Bessel function of the second kind
- under the assumption that $f(x+iy)=f(-x+iy)$, we get
$$ f(x+iy)=\sum_{n=1}^{\infty}a_n \sqrt{y}K_{s-1/2}(2\pi n y) \cos (2\pi i n x) $$
Eisenstein series
Kloosterman sum
books
- Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
- Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29)
- Hans Maass, (pdf)
expositions
- Jianya Liu LECTURES ON MAASS FORMS
computational resource
- https://docs.google.com/file/d/0B8XXo8Tve1cxMll2Y2lzeWg0UVE/edit
- http://www.math.chalmers.se/~sj/forskning/level11_2a.pdf
- http://www.ijpam.eu/contents/2009-54-2/12/12.pdf
encyclopedia
- http://en.wikipedia.org/wiki/Kronecker_limit_formula
- http://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series