Geodesic
On the Maximum of the Modulus and the Maximal Term of Dirichlet Series
Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 437-443.

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For the Dirichlet series, we obtain a condition on the exponents for which the logarithms of the maximum of the modulus of its sum and of the maximal term are equivalent to the same convex function. 
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M. N. Sheremeta. On the Maximum of the Modulus and the Maximal Term of Dirichlet Series. Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 437-443. http://geodesic.mathdoc.fr/item/MZM_2003_73_3_a9/

[1] Sheremeta M. N., “O sootnosheniyakh mezhdu maksimalnym chlenom i maksimumom modulya tselogo ryada Dirikhle”, Matem. zametki, 51:5 (1992), 141–148 | MR | Zbl

[2] Sheremeta M. N., Fedynyak S. I., “O proizvodnoi ryada Dirikhle”, Sib. matem. zh., 39:1 (1998), 206–223 | MR | Zbl

[3] Sheremeta M. N., “O povedenii maksimuma modulya tselogo ryada Dirikhle vne isklyuchitelnogo mnozhestva”, Matem. zametki, 57:2 (1995), 283–296 | MR | Zbl