Ritt-Sugimura Type Theorems

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/n7823-2870-5444-g Ritt-Sugimura Type Theorems Gaisin, A. M. , Gaisina, G. A. Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3. Abstract: At the end of the nineteenth century, E. Borel introduced the concept of the order of an entire function, and then a corresponding formula was obtained for calculating this quantity in terms of the coefficients of the Taylor expansion of this function. Later, J. Ritt extended this notion to entire functions represented by Dirichlet series with positive exponents. He also obtained a similar formula for this characteristic (\(R\)-order), which clearly depends on the coefficients and exponents of the Dirichlet series. In the works of A. M. Gaisin, this result was completely carried over to the case of a halfplane and also a bounded convex domain. In the latter case, the author deals with Dirichlet series with complex exponents, exponential series. In this article the relationship between the growth of the Dirichlet series and the expansion coefficients in terms of Ritt order (\(R\)-order) is studied. Cases when the series converges uniformly in the entire plane or only in a halfplane are considered separately. In both cases the necessary and sufficient conditions for the exponents are obtained, the fulfillment of which the corresponding formulas are correct, allowing to calculate this value through the series coefficients. All previously known results of this type were only of a sufficient character. In the case of a plane, we have shown accuracy of S. Tanaka's estimates for the \(R\)-order. Keywords: Dirichlet series, R-order, Ritt-Sugimura-Tanaka formula Language: Russian Download the full text For citation: Gaisin, A. M. and Gaisina, G. A. Ritt-Sugimura Type Theorems, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp. 47-57. DOI 10.46698/n7823-2870-5444-g + References 1. Bohr, H. Collected Mathematical Works, Copenhagen, 1952, 992 p. 2. Valiron, G. Sur l'Abscisse de Convergence des Series de Dirichlet, Bulletin de la Societe Mathematique de France, 1924, vol. 52, pp. 166-174. DOI: 10.24033/bsmf.1051. 3. Valiron, G. Entire Functions and Borel's directions, Proceedings of the National Academy of Sciences of the USA, 1934, vol. 20, pp. 211-215. DOI: 10.1073/pnas.20.3.211. 4. Kuniyeda, M. Uniform Convergence - Abscissa of General Dirichlet Series, Tohoku Mathematical Journal, 1916, vol. 9, pp. 7-27. 5. Ritt, J. On Certain Points in the Theory of Dirichlet Series, American Journal of Mathematics, 1928, vol. 50, no. 1, pp. 73-86. DOI: 10.2307/2370849. 6. Leont'ev, A. F. Ryady eksponent [Exponential Series], Moscow, Nauka, 1976, 536 p. (in Russian). 7. Gaisin, A. M. A Bound for the Growth in a Half-Strip of a Function Represented by a Dirichlet Series, Mathematics of the USSR-Sbornik, 1983, vol. 45, no. 3, pp. 411-422. DOI: 10.1070/SM1983v045n03ABEH001015. 8. Gaisin, A. M. Behavior of the Sum of a Series of Exponentials Near the Boundary of the Domain of Regularity, Mathematical Notes of the Academy of Sciences of the USSR, 1990, vol. 48, no. 3, pp. 904-910. DOI: 10.1007/BF01157432. 9. Leont'ev, A. F. Tselye funktsii. Ryady eksponent [Entire Functions. Exponential Series], Moscow, Nauka, 1983, 176 p. (in Russian). 10. Mandelbrojt, S. Series Adherentes. Regularisation des Suites. Applications, Gauthier-Villars, Paris, 1952, xiv+277 p. 11. Sugimura, K. Ubertragung Einiger Satze aus der Theorie der Ganzen Funktionen auf Dirichletsche Reihen, Mathematische Zeitschrift, 1929, vol. 29, pp. 264-277. DOI: 10.1007/BF01180529. 12. Tanaka C. Note on Dirichlet Series, V. On the Integral Functions Defined by Dirichlet Series, I, Tohoku Mathematical Journal, 1953, vol. 2, no. 3, pp. 67-78. DOI: 10.2748/tmj/1178245352. 13. Korobeynik Yu. F. Ryady eksponent s veshchestvennymi pokazatelyami [Exponential Series with Real Exponents], Rostov-on-Don, SFU, 2009, 84 p. (in Russian). 14. Korobeynik Yu. F. On Some Problems in the Theory of the Riemann's Zeta-Function, Ufa Mathematical Journal, 2015, vol. 7, no. 4, pp. 88-93. DOI: 10.13108/2015-7-4-88. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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