{"title":"An Estimate for the Sum of a Dirichlet Series on an Arc of Bounded Slope","authors":"T. I. Belous, A. M. Gaisin, R. A. Gaisin","doi":"10.3103/s1066369x24700014","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The article considers the behavior of the sum of the Dirichlet series <span>\\(F(s) = \\sum\\limits_n {\\kern 1pt} {{a}_{n}}{{e}^{{{{\\lambda }_{n}}s}}},\\)</span> <span>\\(0 < {{\\lambda }_{n}} \\uparrow \\infty ,\\)</span> which converges absolutely in the left half-plane <span>\\({{\\Pi }_{0}}\\)</span>, on a curve arbitrarily approaching the imaginary axis—the boundary of this half-plane. We have obtained a solution to the following problem: under what additional conditions on <span>\\(\\gamma \\)</span> will the strengthened asymptotic relation the type of Pólya for the sum <i>F</i>(<i>s</i>) of the Dirichlet series be valid in the case when the argument <span>\\(s\\)</span> tends to the imaginary axis along <span>\\(\\gamma \\)</span> over a sufficiently massive set.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x24700014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
The article considers the behavior of the sum of the Dirichlet series \(F(s) = \sum\limits_n {\kern 1pt} {{a}_{n}}{{e}^{{{{\lambda }_{n}}s}}},\)\(0 < {{\lambda }_{n}} \uparrow \infty ,\) which converges absolutely in the left half-plane \({{\Pi }_{0}}\), on a curve arbitrarily approaching the imaginary axis—the boundary of this half-plane. We have obtained a solution to the following problem: under what additional conditions on \(\gamma \) will the strengthened asymptotic relation the type of Pólya for the sum F(s) of the Dirichlet series be valid in the case when the argument \(s\) tends to the imaginary axis along \(\gamma \) over a sufficiently massive set.
Abstract The article considers the behavior of the sum of the Dirichlet series \(F(s) = \sum\limits_n {\kern 1pt} {{a}_{n}}{{e}^{{{{\lambda }_{n}}s}},\)\(0 < {{\lambda }_{n}} \uparrow \infty ,\) 在左半平面 \({{\Pi }_{0}}\)上绝对收敛于任意接近虚轴的曲线--这个半平面的边界。我们得到了下面问题的一个解：当参数\(s\)在一个足够大的集合上沿着\(\gamma \)趋向于虚轴时，在\(\gamma \)上的加强渐近关系波利亚类型对于迪里希勒数列的和F(s)是有效的。
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