Geodesic
Specification of asymptotic Pólya type estimate for Dirichlet series converging in half–plane
Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 12-20 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the asymptotic behavior of a Dirichlet series with positive exponents, converging in the left half–plane, on an arc of bounded slope ending on the convergence line. In the paper we obtain conditions under which the sum of the Dirichlet series satisfies an asymptotic equality of Pólya type on a set, the upper density of which is equal to one. In 2023 we obtained results related to dual cases. We showed that a Pólya type identity holds on an asymptotic set of positive upper density depending on the slope coefficient (Lipschitz constant) of the arc. In this paper, we prove a common theorem covering both of these cases, and we show that the asymptotic set has an upper density, which is equal to one.
Keywords: Dirichlet series, convergence half–plane, maximal term of series, curve of bounded slope, Pólya type identity. 
@article{UFA_2024_16_4_a1,
     author = {T. I. Belous and A. M. Gaisin and R. A. Gaisin},
     title = {Specification of asymptotic {P\'olya} type estimate for {Dirichlet} series converging in half{\textendash}plane},
     journal = {Ufa mathematical journal},
     pages = {12--20},
     year = {2024},
     volume = {16},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a1/}
}
TY  - JOUR
AU  - T. I. Belous
AU  - A. M. Gaisin
AU  - R. A. Gaisin
TI  - Specification of asymptotic Pólya type estimate for Dirichlet series converging in half–plane
JO  - Ufa mathematical journal
PY  - 2024
SP  - 12
EP  - 20
VL  - 16
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a1/
LA  - en
ID  - UFA_2024_16_4_a1
ER  - 
%0 Journal Article
%A T. I. Belous
%A A. M. Gaisin
%A R. A. Gaisin
%T Specification of asymptotic Pólya type estimate for Dirichlet series converging in half–plane
%J Ufa mathematical journal
%D 2024
%P 12-20
%V 16
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a1/
%G en
%F UFA_2024_16_4_a1
T. I. Belous; A. M. Gaisin; R. A. Gaisin. Specification of asymptotic Pólya type estimate for Dirichlet series converging in half–plane. Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 12-20. http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a1/

[1] A.M. Gaisin, “Behavior of the logarithm of the modulus value of the sum of a Dirichlet series converging in a half–plane”, Izv. Math, 45:1 (1995), 175–186 | DOI | MR

[2] O.B. Skaskiv, “On Wiman's theorem concerning the minimum modulus of a function analytic in the unit disk”, Math. USSR–Izv., 35:1 (1990), 165–182 | DOI | MR | Zbl

[3] A.M. Gaisin, R.A. Gaisin, T.I. Belous, “Regularity of the growth of Dirichlet series with respect to a strongly incomplete exponential system”, Sib. Math. J, 64:4 (2023), 854–863 | DOI | MR | Zbl

[4] T.I. Belous, A.M. Gaisin, R.A. Gaisin, “An estimate for the sum of a Dirichlet series on an arc of bounded slope”, Russ. Math, 68:1 (2024), 1–10 | DOI | MR | Zbl

[5] A.M. Gaisin, “Behavior of the sum of a Dirichlet series having a prescribed growth”, Math. Notes, 50:4 (1991), 1018–1024 | DOI | MR | Zbl | Zbl

[6] A.M. Gaisin, T.I. Belous, “Maximal term of Dirichlet series converging in half–plane: stability theorem”, Ufa Math. J., 14:3 (2022), 22–32 | MR | Zbl