Computer Experimentation with Dirichlet Functions

There is a vast amount of literature about Dirichlet series, starting with the works of Cahen and followed by the works of Hardy and Riesz, Valiron, Landau, Bohr, Kojima, etc. These series are generalizations of the famous Euler series. Using his functional equation, Riemann extended the Euler series across the convergence line. The problem of extending general Dirichlet series using Riemann’s method appeared, and it has been successfully dealt with in the particular case of Dirichlet L-series, obtaining functions with properties similar to those of the Riemann Zeta function. However, until recently, no other class of Dirichlet series has been known, that can be continued as a meromorphic function in the whole complex plane. Moreover, the chance that Dirichlet series might exist, such that their continuation has several poles, appeared to be very small. Our discovery of Dirichlet functions generated by Blaschke products by a change of variable completely reversed this point of view. Now, it is known not only that a whole class of Dirichlet series exists with continuations, series that have infinitely many poles but also that they can have some essential singular points. In this paper, the behavior of a Dirichlet function in a neighborhood of an essential singular point is revealed, and the behavior is really surprising. The Dirichlet functions generated by finite Blaschke products are fit for computer experimentation since they are given by formulas that can be implemented with ease in computer programs. In this paper, we are dealing with such Dirichlet functions in a general context and indicate their zeros, poles, and branch points. We are looking for global mapping properties of these functions, describing in detail their fundamental domains. Computer graphics are offered, adding a new chapter to the study of Dirichlet functions, as well as in that of Blaschke products. Computer programs have been created that can deal with infinite Dirichlet series and with the remarkable properties of Dirichlet functions generated by them.