Geometric Aspects of Denseness Theorems for Dirichlet Functions

The first theorem related to the denseness of the image of a vertical line Re s = σ0, σ0 > 1 by the Riemann Zeta function has been proved by Harald Bohr in 1911. We argue that this theorem is not really a denseness theorem. Later Bohr and Courant proved similar theorems for the case 1/2 < Re s ≤ 1. Their results have been generalized to classes of Dirichlet functions and are at the origin of a burgeoning field in analytic number theory, namely the universality theory. The tools used in this theory are mainly of an arithmetic nature and do not allow a visualization of the phenomena involved. Our method is based on conformal mapping theory and is supported by computer generated illustrations. We generalize and refine Bohr and Courant results.