Discrete Ω-results for the Riemann zeta function

IF 1 3区数学 Q1 MATHEMATICS

Forum Mathematicum Pub Date : 2024-09-02 DOI:10.1515/forum-2023-0324

Paolo Minelli, Athanasios Sourmelidis

{"title":"Discrete Ω-results for the Riemann zeta function","authors":"Paolo Minelli, Athanasios Sourmelidis","doi":"10.1515/forum-2023-0324","DOIUrl":null,"url":null,"abstract":"We study lower bounds for the Riemann zeta function <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {\\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {\\zeta(s)} .","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"24 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0324","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We study lower bounds for the Riemann zeta function ζ ⁢ ( s )

{\zeta(s)}

along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s )

{\zeta(s)}

查看原文本刊更多论文

黎曼zeta函数的离散Ω结果

我们研究了黎曼zeta函数ζ ( s ) {\zeta(s)} 沿临界带右半部垂直算术级数的下界。我们证明，当 s 的虚部在给定区间内时，离散情况下获得的下界与连续情况下已知的下界重合，直至指数中的常数。我们的方法基于共振法的离散化，用于估计 ζ ( s ) {\zeta(s)} 的极值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Forum Mathematicum 数学-数学

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.