a3864253-8b58-4bbb-8a82-c30ebd76a10120250115091551501wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpAndrei-FlorinAlbişoruFaculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, ROMANIADorinGhişaDepartment of Mathematics, Glendon College, York University, Toronto, CANADAThe continuation of general Dirichlet series to meromorphic functions in the complex plane remains an outstanding problem. It has been completely solved only for Dirichlet L-series. A sufficient condition for the general case exists, however it is impossible to verify that it is fulfilled. We solve this problem here for another class of general Dirichlet series, namely those series which are obtained from infinite Blaschke products by a particular change of variable. This is a source of examples of general Dirichlet series with infinitely many poles. An interesting new case is now revealed, in which the singular points of the extended function form a continuum. We take a closer look at the case of Dirichlet series with natural boundary and give examples of such series. Some figures illustrate the theory.123020241230202492693996https://wseas.com/journals/mathematics/2024/b925106-048(2024).pdf10.37394/23206.2024.23.96https://wseas.com/journals/mathematics/2024/b925106-048(2024).pdfHardy, G.H. and Riesz, M., The General Theory of Dirichlet Series, Cambridge University Press, 1915. Valiron, G., General Theory of Dirichlet Series (Théorie générale des séries de Dirichlet), Mémoire des sciences mathématiques, 17, 1-56, 1926. 10.24033/asens.401Cahen, E., On the Riemann Zeta Function and the Analog Functions (Sur la fonction ζ(s) de Riemann et sur les fonctions analogues), Ann. École Norm., 3e series, t. 11, 1894. Cahen, E., On the Dirichlet Series (Sur les séries de Dirichlet), Compte rendue, t. 166, 1918. Landau, E., On the Convergence Problem of Dirichlet Series (Über convergenzproblem der Dirichletschen Reihen), Rendiconti di Palermo, t. 28, 1909. 10.1007/bf01457178Bohr, H., On the General Theory of Dirichlet Series (Zür Theorie der allgemeinen Dirichletschen reihen), 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.2018.88042Ghisa, D. and Horvat-Marc, A., Geometric Aspects of Quasi-periodic Property of Dirichlet Functions, Advances in Pure Mathematics, 8, 699-710, 2018. Cargo, G.T., The Boundary Behavior of Blaschke Products, Journal of Mathematical Analysis and Applications, 5, 1-16, 1962. 10.1007/bf01448042Speiser, A., The Geometry of Riemann Zeta Function (Geometrisches zür Riemannschen Zetafunction), Mathematische Annalen, 110, 514-521, 1934. 10.7546/giq-20-2019-131-160Ghisa, D., Fundamental Domains of Dirichlet Functions, in Geometry, Integrability and Quantization, Mladenov, Pulovand and Yoshioka etts., Sofia, 131-160, 2019. Albisoru, A.F. and Ghisa, D., Complex analytic Functions with Natural Boundary, WSEAS Transactions on Mathematics, 13 pages, 2024. 10.4236/apm.2017.71001Ghisa, D., The Geometry of the Mappings by General Dirichlet Series, Advances in Pure Mathematics, 7, 1-20, 2017. Ahlfors, L.V., Complex Analysis, International Series in Pure Mathematics, 1951. Granath, J., Finite Blaschke products and their properties, PhD Thesis, University of Stockholm, 2020. Ahlfors, L.V. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1974