Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type
For entire Dirichlet series of the form \(F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }\) , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left(...
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| Datum: | 2001 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2001
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4303 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510434234204160 |
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| author | Salo, T. M. Skaskiv, O. B. Сало, Т. М. Скасків, О. Б. |
| author_facet | Salo, T. M. Skaskiv, O. B. Сало, Т. М. Скасків, О. Б. |
| author_sort | Salo, T. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:25:59Z |
| description | For entire Dirichlet series of the form \(F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }\) , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$ holds uniformly in \(y \in \mathbb{R}\;{as}\;{\sigma } \to + \infty\) outside a certain set E for which $$DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$$ where h(σ) is a positive continuous function increasing to +∞ on [0, +∞). |
| first_indexed | 2026-03-24T02:56:56Z |
| format | Article |
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| id | umjimathkievua-article-4303 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:56:56Z |
| publishDate | 2001 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/18/cf41ff2315fe5974378643e0fd209118.pdf |
| spelling | umjimathkievua-article-43032020-03-18T20:25:59Z Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type Цілі ряди Діріхле швидкого зростання і нові оцінки міри виняткових множин в теоремах типу Вімана - Валірона Salo, T. M. Skaskiv, O. B. Сало, Т. М. Скасків, О. Б. For entire Dirichlet series of the form \(F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }\) , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$ holds uniformly in \(y \in \mathbb{R}\;{as}\;{\sigma } \to + \infty\) outside a certain set E for which $$DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$$ where h(σ) is a positive continuous function increasing to +∞ on [0, +∞). Для цілих рядів Діріхле вигляду $F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }$ встановлено умови, при виконанні яких $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$ при ${\sigma } \to + \infty$ зовні деякої множини $E$ для якої $DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$, рівномірно по $y \in \mathbb{R}$, де $h(σ)$ — додатна неперервна зростаюча до $+ ∞$ на $[0, +∞)$ функція. Institute of Mathematics, NAS of Ukraine 2001-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4303 Ukrains’kyi Matematychnyi Zhurnal; Vol. 53 No. 6 (2001); 830-839 Український математичний журнал; Том 53 № 6 (2001); 830-839 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4303/5310 https://umj.imath.kiev.ua/index.php/umj/article/view/4303/5311 Copyright (c) 2001 Salo T. M.; Skaskiv O. B. |
| spellingShingle | Salo, T. M. Skaskiv, O. B. Сало, Т. М. Скасків, О. Б. Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title | Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title_alt | Цілі ряди Діріхле швидкого зростання і нові оцінки міри виняткових множин в теоремах типу Вімана - Валірона |
| title_full | Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title_fullStr | Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title_full_unstemmed | Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title_short | Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type |
| title_sort | entire dirichlet series of rapid growth and new estimates for the measure of exceptional sets in theorems of the wiman–valiron type |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4303 |
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