We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.
Cahen, E. (1894) Sur la function ζ(s) de Riemann et sur des fonctions analogues, Ann. Ecole Norm., 3e serie, t. 11.
Cahen, E. (1918) Sur les séries de Dirichlet, Compte rendue, t. 166.
Hadamard, J. (1896) Sur la distribution des zéros de la function ζ(s) de Riemann et ses cons équences arithmé tiques, Bull. Soc. math., t. 24.
Hadamard, J. (1908) Sur les s éries de Dirichlet, Rendiconti di Palermo, t. 25.
Landau, E. (1907) über die Multiplication Dirichletscher Reihen, rendiconti di Palermo, t. 24.
Landau, E. (1909) über das Konvergenzproblem der Dirichletschen Reihen, Rendiconti di Palermo, t. 28.
Landau, E. (1917) Neuer Beweis eines Haptsatzes aus der Theorie der Dirichletschen Reihen, Leipziger Ber., t. 69.
Landau, E. (1921) über die gleichmassige Konvergenz Dirichletscher Reihen, Math. Zeitschrift, t. 11.
Bohr, H. (1913) über die gleichmassige Konvergenz Dirichletscher Reihen, J. fuer Math, t. 143.
Bohr, H. (1913) Einige Bemerkungen über das konvergenzproblem Dirichletscher Reihen, Rendicnti di Palermo, t. 37.
Bohr, H. (1913) Darstellung der gleichmassigen Konvergenzabszisse einer Dirichletschen reihe ∑ n=1 ∞ a n /n s als Funktionen der Koeffizienten der Reihe. Archiv der Mathematik und Physik, 21, 326-330.
Bohr, H. (1913) Zür Theorie der allgemeinen Dirichletschen Reihen. Mathematische Annalen, 79, 136-156.
Hardi, G.-H. (1911) The Multiplication of Dirichlet’s Series. Vol. 10, London Mathematical Society, London.
Hardy, G.H. and Riesz, M. (1915) The General Theory of Dirichlet’s Series. Cambridge University Press, Cambridge.
Kojima, T. (1914) On the Convergence Abscissa of General Dirichlet’s Series. Tohoku Mathematical Journal, 6, 140-142.
Kojima, T. (1916) Note on the Convergence Abscissa of Dirichlet’s Series. Tohoku Mathematical Journal, 9, 28-37.
Kuniyeda, M. (1916) Uniform Convergence Abscissa of General Dirichlet’s Series. Tohoku J., t. 9, 7-27.
Valiron, G. (1926) Théorie générale des séries de Dirichlet. Mémorial des sciences mathématiques, fascicule, 17, 1-56.
Yu, J. (1963) Borel’s Line of Entire Functions Represented by Laplace-Stieltjes Transformations. Acta Mathematica Sinica, 13, 471-484.
Kong, Y.Y. and Daochun, S. (2008) On Type-Function and Growth of Laplace-Stieltjes Transformations Convergent in the Right Half Plane, 2008. Advances in Mathematics, 37, 197-205.
Luo, X. and Kong, Y.Y. (2012) On the Order and Type of Laplace-Stieltjes Transforms of Slow Growth. Acta Mathematica Sinica, 32A, 601-607.
Kong, Y.Y. and Yang, Y. (2014) On the Growth Properties of the Laplace-Stieltjes Transform. Complex Variables and Elliptic Equations, 59, 553-563. https://doi.org/10.1080/17476933.2013.766174
Singhal, C. and Srivastava, G.S. (2015) On the Approximation of an Analytic Function Represented by Laplace-Stieltjes Transformation. Analysis in Theory and Applications, 31, 407-420.
Srivastava, G.S. and Singhal, C. (2015) On the Generalized Order and the Generalized Type of Laplace-Stieltjes Transformation Convergent in the Right Half Plane. Global Journal of Pure and Applied Mathematics, 11, 469-477.
Kamthan, P.K. and Shing Gautam, S.K. (1975) Bases in a Certain Space of Functions Analytic in the Half Plane. Indian Journal of Pure and Applied Mathematics, 6, 1066-1075.
Khoi, L.H. (1998) Holomorphic Dirichlet Series in the Half Plane. Vietnam Journal of Mathematics, 26, 259-271.
Defant, A., Garcia, D., Maestre, M. and Perez-Garcia, D. (2008) Bohr’s Strip for Vector Valued Dirichlet Series. Mathematische Annalen, 342, 533-555. https://doi.org/10.1007/s00208-008-0246-z
Bonet, J. (2015) Abscissas of Weak Convergence of Vector Valued Dirichlet Series. arXiv: 1502.00418v1.
Srivastava, B.L. (1983) A Study of Spaces of Certain Classes of Vector Valued Dirichlet Series. Thesis, Indian Institute of Technology Kanpur, Kalyanpur.
Srivastava, G.S. and Sharma, A. (2011) Bases in the Space of Vector Valued Analytic Dirichlet Series. International Journal of Pure and Applied Mathematics, 7, 993-1000.
Srivastava, G.S. and Sharma, A. (2012) Spaces of Entire Functions Represented by Vector Valued Dirichlet Series of Slow Growth. Acta Universitatis Sapientiae, Mathematica, 4, 154-167.
Sharma, A. and Srivastava, G.S. (2016) Coefficient Multipliers on Spaces of Vector Valued Entire Dirichlet Series.
Ghisa, D. (2013) Fundamental Domains and the Riemann Hypothesis. Lambert Academic Publishing, Saarbrücken.
Ghisa, D. (2014) On the Generalized Riemann Hypothesis, Complex Analysis and Potential Theory with Applications. Cambridge Scientific Publishers, Cambridge, 77-94.
Ghisa, D. (2016) On the Generalized Riemann Hypothesis II. International Journal of Scientific and Innovative Mathematical Research, 4, 46-55.
Ghisa, D. (2016) Fundamental Domains and Analytic Continuation of General Dirichlet Series. British Journal of Mathematics & Computer Science, 9, 94-111.
Andreian-Cazacu, C. and Ghisa, D. (2009) Global Mapping Properties of Analytic Functions. Proceedings of 7th ISAAC Congress, London, 13-18 July 2009, 3-12.
Andreian-Cazacu, C. and Ghisa, D. (2011) Fundamental Domains of Gamma and Zeta Functions. International Journal of Mathematics and Mathematical Sciences, 2011, Article ID: 985323. https://doi.org/10.1155/2011/985323
Arias-de-Reina, J. (2003) X-Ray of Riemann’s Zeta-Function. ArXiv: math/0309433, 42 p.
Wegert, E. (2012) Visual Complex Functions. Springer, Basel. https://doi.org/10.1007/978-3-0348-0180-5
Wegert, E. (2016) Visual Exploration of Complex Functions. In: Qian, T. and Rodino, L.G., Eds., Mathematical Analysis, Probability and Applications—Plenary Lectures, Vol. 177, Springer International Publishing, 253-279. https://doi.org/10.1007/978-3-319-41945-9_10
Barza, I., Ghisa, D. and Muscutar, F.A. (2014) On the Location of the Zeros of the Derivatives of Dirichlet L-Functions. Annals of the University of Bucharest (Mathematics Series), 5, 21-31.
Cao-Huu, T., Ghisa, D. and Muscutar, F.A. (2016) Multiple Solutions of Riermann Type of Functional Equations. British Journal of Mathematics and Computer Science, 17, 1-12. https://doi.org/10.9734/BJMCS/2016/26322