We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods, we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of Γ 0 (5)-cusp forms is presented.

Abstract. We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z; s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of E(z; s) as Re s = 1/2, Im s → ∞ and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z; s) when Re s > 1/2 near 1/2 and Im s → ∞, at least if we allow Re s → 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s ≥ 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

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, Z). Our results indicate that both Green's function Gs(z; w) and its Fourier coefficients Fn(z; s) have a Gaussian value distribution in the semiclassical limit when Re s = 1/2.

CONTENTS 1. Introduction 2. Preliminaries 3. Computations of F n (z; s) 4. Computations of G s (z; w) 5. Computations of Bessel Functions of Imaginary Order 6. Numerics 7. Results