Geodesic
Singular points of the sum of a Dirichlet series on the convergence line
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 54-69.

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We study the distribution of singular points of the sum of a Dirichlet series and obtain necessary and sufficient conditions for the sum of such a series to have at least one singular point on any segment of given length on the convergence line.
Keywords: singular point, invariant subspace, fundamental principle.
Mots-clés : convex domain 
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O. A. Krivosheeva; A. S. Krivosheev. Singular points of the sum of a Dirichlet series on the convergence line. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 54-69. http://geodesic.mathdoc.fr/item/FAA_2015_49_2_a5/

[1] A. F. Leontev, Ryady eksponent, Nauka, M., 1976 | MR

[2] J. Hadamard, “Essai sur l'etude des fonctions donnes par leur développement de Taylor”, J. Math. Pures Appl. Ser. (4), 4(8) (1892), 101–106

[3] E. Fabry, “Sur les points singuliers d'une function donnée par son développement en série et l'impossibilité du prolongement analytique dans des cas très généraux”, Ann. Ecole Norm. Sup. (3), 13 (1896), 367–399 | DOI | MR | Zbl

[4] P. Koosis, The Logarithmic Integral, v. 1, Cambridge University Press, Cambridge, 1997 | MR

[5] G. Pólya, “Untersuchungen über Lücken und Singularitäten von Potenzreihen”, Math. Z., 29:1 (1929), 549–640 | DOI | MR

[6] L. Biberbakh, Analiticheskoe prodolzhenie, Nauka, M., 1967 | MR

[7] W. Fuchs, “On the growth of functions of mean type”, Proc. Edinburgh Math. Soc., Ser. 2, 9, 1954, 53–70 | DOI | MR | Zbl

[8] P. Malliavin, “Sur la croissance radiale d'une function méromorphe”, Illinois J. Math., 1 (1957), 259–296 | DOI | MR | Zbl

[9] P. Koosis, The Logarithmic Integral, 2, Cambridge University Press, Cambridge, 2008 | MR

[10] G. Pólya, “Über die Existenz unendlich vieler singulärer Punkte auf der Konvergenzgeraden gewisser Dirichletscher Reihen”, Sitzungber. Preuss. Akad. Wiss., Phys.-Math. Kl., 1923, 45–50

[11] G. Pólya, “Eine Verallgemeinerung des Fabryschen Lückensatzes”, Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl., 1927, 187–195

[12] V. Bernstein, Leçons sur les progrèss récents de la théorie des séries de Dirichlet, Gauthier-Villars, Paris, 1933

[13] A. F. Leontev, Tselye funktsii. Ryady eksponent, M., Nauka, 1983 | MR

[14] O. A. Krivosheeva, “Osobye tochki summy ryada eksponentsialnykh monomov na granitse oblasti skhodimosti”, Algebra i analiz, 23:2 (2011), 162–205 | MR

[15] A. S. Krivosheev, “Fundamentalnyi printsip dlya invariantnykh podprostranstv v vypuklykh oblastyakh”, Izv. RAN, ser. matem., 68:2 (2004), 71–136 | DOI | MR | Zbl

[16] O. A. Krivosheeva, A. S. Krivosheev, “Kriterii spravedlivosti fundamentalnogo printsipa dlya invariantnykh podprostranstv v ogranichennykh vypuklykh oblastyakh kompleksnoi ploskosti”, Funkts. analiz i ego pril., 46:4 (2012), 14–30 | DOI | MR | Zbl

[17] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[18] P. Lelon, L. Gruman, Tselye funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1989 | MR

[19] O. A. Krivosheeva, “Oblast skhodimosti ryadov eksponentsialnykh monomov”, Ufimsk. matem. zh., 3:2 (2011), 43–56 | MR | Zbl

[20] V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR

[21] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. III: O rasprostranenii spektralnogo sinteza”, Matem. sb., 88(130):3 (1972), 331–352