Geodesic
On the question of representing entire functions by exponential series
Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 327-342 
A. F. Leont'ev. On the question of representing entire functions by exponential series. Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 327-342. http://geodesic.mathdoc.fr/item/SM_1977_33_3_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It was established by the author (RZhMat., 1966, 4B 107) that any entire function $F(z)$ of finite order may be represented in the whole plane by a Dirichlet series $$ F(z)=\sum_{k=1}^\infty A_ke^{|\lambda_k|z}. $$ It is established that for suitable choice of the sequence $\{\lambda_k\}$ the expression $\sum_{k=1}^\infty|A_k|e^{|\lambda_k|r}$ has, for large $r$, the upper bounds 1) $\exp r^{\rho+\varepsilon}$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$; 2) $\exp(\sigma+\varepsilon)r^\rho$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$ and finite type $\sigma$. Bibliography: 7 titles.

[1] A. F. Leontev, Ryady eksponent, izd-vo «Nauka», Moskva, 1976 | MR

[2] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956

[3] A. F. Leontev, “O predstavlenii proizvolnykh tselykh funktsii ryadami Dirikhle”, DAN SSSR, 165:4 (1965), 759–762 | MR

[4] V. I. Shevtsov, O predstavlenii tselykh funktsii ryadami po nekotoroi sisteme analiticheskikh funktsii, avtoreferat dissertatsii, Moskva, 1968

[5] V. P. Gromov, “O ryadakh po sisteme $\{f(\lambda_nz)\}$”, DAN SSSR, 144:1 (1962), 23–26 | MR | Zbl

[6] V. P. Gromov, “O roste funktsii, opredelyaemykh ryadami vida $\sum\limits_{k=1}^\infty d_nf(\lambda_nz)$”, Matem. sb., 67(109) (1965), 190–209 | MR | Zbl

[7] A. I. Markushevich, Teoriya analiticheskikh funktsii, Gostekhizdat, Moskva–Leningrad, 1950