3f5d8cf0-6415-4227-84ee-945faf806e9d20250717124912458wseas:wseasmdt@crossref.orgMDT DepositEQUATIONS2732-99762944-914610.37394/232021https://wseas.com/journals/equations/562025562025510.37394/232021.2025.5https://wseas.com/journals/equations/2025.phpAndrei-FlorinAlbişoruFaculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, ROMANIADorinGhişaDepartment of Mathematics, Glendon College, York University, Toronto, CANADAThere 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.7172025717202548676https://wseas.com/journals/equations/2025/a12equations-005(2025).pdf10.37394/232021.2025.5.6https://wseas.com/journals/equations/2025/a12equations-005(2025).pdf10.37394/23206.2023.22.106Albisoru, A. F. and Ghisa, D. Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points, WSEAS Transactions on Mathematics, 22, 2023, 971-979. 10.1007/bf01448042Speiser, A., Geometrisches zür Riemannschen Zetafunction (The Geometry of Riemann Zeta Function), Mathematische Annalen, 110, 514-521, 1934. Arias-de-Reina, J., X-Ray of Riemann’s Zeta-Function, ArXiv:math/0309433, 42 p., 2003. 10.1007/bf01565437Hecke, E., Über die Bestimmung Dirichletscher Reihen durch ihre Functionalgleichung (On the Dirichlet Series Description by its Functional Equation), Math. Ann., 112, 664-699, 1936. 10.1088/0305-4470/9/8/006Zucker, I. J. and Robertson, M. M., Some properties of Dirichlet L-Series, J. Phys, A: Math. Gen., 9, 1207-1214, 1976. Cahen, E., Sur les séries de Dirichlet (On the Dirichlet Series), Compte rendue, t. 166, 1918. 10.1007/bf03029133Hadamard, J., Sur les séries de Dirichlet (On the Dirichlet Series), Rendiconti di Palermo, t.25, 1908. G.H. Hardy and M. Riesz, The General Theory of Dirichlet Series, Cambridge University Press, 1915. Valiron, G., Théorie générale des séries de Dirichlet (General Theory of Dirichlet Series), Mémoire des sciences mathématiques, 17, 1-56, 1926. 10.24033/asens.401Cahen, E., Sur la fonction ζ(s) de Riemann et sur les fonctions analogues (On the Riemann ζ(s) function and the analogue functions), Ann. École Norm., 3e series, t. 11, 1894. 10.1007/bf03018214Landau, E., Über convergenzproblem der Dirichletschen Reihen (On the Convergence Problem of the Dirichlet Series), Rendiconti di Palermo, t.28, 1909. 10.1007/bf01457178Bohr, H., Zür Theorie der allgemeinen Dirichletschen reihen (On the Theory of general Dirichlet series), Mathematische Annalen, 79, 136-156, 1913. Kojima, T., Note on the Convergence Abscissa of Dirichlet’s Series, Tohoku Mathematical Journal, 9, 28-37, 1916. 10.4236/apm.2017.71001Ghisa, D., The Geometry of the Mappings by General Dirichlet Series, Advances in Pure Mathematics, 7, 1-20, 2017. 10.1016/j.jmaa.2012.01.005Chalendar, I., Gorkin, P. and Partington, J. R., The group of invariants of an inner function with finite spectrum, Journal of Math. Anal. and Appl., 389, 1259-1267, 2012. Cargo, G. T., The Boundary Behavior of Blaschke Products, Journal of Mathematical Analysis and Applications, 5, 1-16, 1962. Granath, J., Finite Blaschke products and their properties, PhD Thesis, University of Stockholm, 2020. 10.37394/23206.2024.23.83Albisoru, A. F. and Ghisa, D., Complex Analytic Functions with Natural Boundary, WSEAS Transactions on Mathematics, 23, 802-814, 2024. Nehari, Z., Conformal Mapping, International Series in Pure and Applied mathematics, 1951. Graham, I. and Kohr, G., Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, Basel, 2003