Numerical Computations of Green's Function and Its Fourier Coefficients on PSL(2, ℤ)

Abstract

We present some examples of numerical investigations of the value distribution of Green's function and of its Fourier coefficients on the modular group PSL(2, (2, ℤ). Our results indicate that both Green's function G_s (z; w) and its Fourier coefficients F_n (z; s) have a Gaussian value distribution in the semiclassical limit when Re s = 1/2.

Keywords:

• Green's function
• resolvent kernel
• computational number theory
• quantum chaos
• Fourier coefficients
• spectral theory
• pseudo cusp forms
• Gaussian value distribution
• point scatterer