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 $\mathrm{PSL}(2, \mathbb{Z})$. 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 $\operatorname{Re} s = 1/2$.