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Keywords:
Dirichlet series; order; abscissa of convergence
Dirichlet series; order; abscissa of convergence
Summary:
We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.
References:
[1] Blambert, M.: Sur la notion de type de l'ordre d'une fonction entière. Ann. Sci. Éc. Norm. Supér, III. Sér. 79 (1962), 353-375 French. DOI 10.24033/asens.1115 | MR 0179358 | Zbl 0111.27302
[2] Bohr, H.: Collected Mathematical Works. Copenhagen (1952), 992. Zbl 0049.00105
[3] Knopp, K.: Über de Konvergenzabszisse des Laplace-Integrals. Math. Z. 54 (1951), 291-296 German. DOI 10.1007/BF01574830 | MR 0042539
[4] Mandelbrojt, S.: Séries de Dirichlet. Principle et Méthodes, Paris, Gauthier-Villars (1969), 165 French. MR 0259079 | Zbl 0207.07201
[5] Izumi, S.: Integral functions defined by Dirichlet's series. Japanese Journ. of Math. 6 (1929), 199-204. DOI 10.4099/jjm1924.6.0_199
[6] Ritt, J. F.: On certain points in the theory of Dirichlet series. Amer. J. 50 (1928), 73-86. DOI 10.1021/ja01388a009 | MR 1506655
[7] Sugimura, K.: Übertragung einiger Sätze aus der Theorie der ganzen Funktionen auf Dirichletsche Reihen. M. Z. 29 (1928), 264-277 German. DOI 10.1007/BF01180529
[8] Tanaka, C.: Note on Dirichlet series. V: On the integral functions defined by Dirichlet series. (I.). Tôhoku Math. J., II. Ser. 5 (1953), 67-78. DOI 10.2748/tmj/1178245352 | MR 0057320 | Zbl 0053.37502
[9] Valiron, G.: Sur la croissance du module maximum des séries entières. S. M. F. Bull. 44 (1916), 45-64 French. MR 1504749
[10] Valiron, G.: Sur l'abscisse de convergence des séries de Dirichlet. S. M. F. Bull. 52 (1924), 166-174 French. MR 1504844
[11] Valiron, G.: Entire functions and Borel's directions. Proc. Natl. Acad. Sci. USA 20 (1934), 211-215. DOI 10.1073/pnas.20.3.211 | Zbl 0009.02503
[12] Valiron, G.: Théorie Générale des Séries de Dirichlet. Paris, Gauthier-Villars (Mémorial des sciences mathématiques, fasc. 17) (1926), French pp. 56.
[13] Yu, Ch.-Y.: Dirichlet series, Analytic functions of one complex variable. Contemp. Math., 48, Amer. Math. Soc., Providence, RI (1985), 201-216. MR 0838106
[14] Yu, Ch.-Y.: Sur les droites de Borel de certaines fonctions entières. Ann. Sci. Éc. Norm. Supér, III. Sér. 68 (1951), 65-104 French. DOI 10.24033/asens.986 | MR 0041223 | Zbl 0045.03802
[15] Yu, J.-R.: Some properties of random Dirichlet series. Acta Math. Sin. 21 (1978), 97-118 Chinese. MR 0507192 | Zbl 0386.60044
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