Geodesic
On representation by Dirichlet series of functions analytic in a halfplane
Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 565-581 
A. F. Leont'ev. On representation by Dirichlet series of functions analytic in a halfplane. Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 565-581. http://geodesic.mathdoc.fr/item/SM_1971_14_4_a6/
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     title = {On representation by {Dirichlet} series of functions analytic in a~halfplane},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author has proved (RZhMat., 1969, 12B169) that every entire function can be represented by a Dirichlet series in the complex plane. In a more recent paper (Mat. Sb. (N.S.) 81(123) (1970), 552–579) he proved that if $D$ is a bounded open convex domain, then every function analytic in $D$ can be represented in $D$ by a Dirichlet series. This left open the question of the possible representation by Dirichlet series of functions analytic in an unbounded convex domain other than the entire plane, for example, a halfplane. Here it is proved that if $D$ is an unbounded open convex domain whose boundary consists of a finite number of line segments (for example, a halfplane, angle, or strip), then every function analytic in $D$ can be represented in $D$ by a Dirichlet series. Bibliography: 7 titles.

[1] A. F. Leontev, “Predstavlenie funktsii obobschennymi ryadami Dirikhle”, Uspekhi matem. nauk, XXIV:2(146) (1969), 97–164 | MR

[2] A. F. Leontev, “O predstavlenii analiticheskikh funktsii v otkrytoi oblasti ryadami Dirikhle”, Matem. sb., 81(123) (1970), 552–579 | MR | Zbl

[3] N. Aronszajn, “Sur les décompositions des fonctions analitiques uniformes et sur leurs application”, Acta Math., 65:1–2 (1935), 1–166 | DOI | MR

[4] M. V. Keldysh, “O priblizhenii golomorfnykh funktsii tselymi funktsiyami”, DAN SSSR, 47:4 (1945), 243–245

[5] G. Mittag-Leffer, “Sur la représentation analytique d'une branche uniforme d'une fonction monogene”, Acta Math., 29 (1905), 101–182 | DOI | MR

[6] M. M. Dzhrbashyan, Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, Moskva, 1966

[7] A. F. Leontev, Ryady polinomov Dirikhle i ikh obobscheniya, Trudy Matem. in-ta im. V. A. Steklova, XXXIX, 1951 | MR | Zbl