Computations of Eisenstein series on Fuchsian groups

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 $\operatorname {Re} s=1/2$, $\operatorname {Im} s\to \infty$ and also, on non-arithmetic groups, a complex Gaussian limit distribution for $E(z;s)$ when $\operatorname {Re} s>1/2$ near $1/2$ and $\operatorname {Im} s\to \infty$, at least if we allow $\operatorname {Re} s\to 1/2$ at some rate. Furthermore, on non-arithmetic groups and for fixed $s$ with $\operatorname {Re} s \ge 1/2$ near $1/2$, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.