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.

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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