Maass forms

introduction

Hyperbolic distribution problems and half-integral weight Maass forms
Automorphic forms correspond to representations that occur in L2(G/Γ). In the case when G is SL2, holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of G, while Maass wave forms correspond to (spherical vectors of) continuous series representations of G.

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

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

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)
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