Maass forms

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

• used to estimate the Fourier coefficients of modular forms
• definition for prime p
  \(S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p\)
  
• generally defined as
  \(K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}\)
  
• http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/
• Kloosterman, H. D. On the representation of numbers in the form ax² + by² + cz² + dt², Acta Mathematica 49 (1926), pp. 407-464
•  

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• examples of harmonic Maass Forms

 

 

books

• Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
• 찾아볼 수학책
  
• http://gigapedia.info/1/Iwaniek
• http://gigapedia.info/1/Maass
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• [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
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

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  • http://blogsearch.google.com/blogsearch?q=
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articles

• 논문정리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

experts on the field

• http://arxiv.org/

 

 

TeX 

• Detexify2 - LaTeX symbol classifier