Asymmetric distribution of extreme values of cubic L $L$ -functions at s = 1 $s=1$

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-09-24 DOI:10.1112/jlms.12996

Pranendu Darbar, Chantal David, Matilde Lalin, Allysa Lumley

{"title":"Asymmetric distribution of extreme values of cubic \n \n L\n $L$\n -functions at \n \n \n s\n =\n 1\n \n $s=1$","authors":"Pranendu Darbar, Chantal David, Matilde Lalin, Allysa Lumley","doi":"10.1112/jlms.12996","DOIUrl":null,"url":null,"abstract":"We investigate the distribution of values of cubic Dirichlet <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>-functions at <math>\n <semantics>\n <mrow>\n <mi>s</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$s=1$</annotation>\n </semantics></math>. Following ideas of Granville and Soundararajan for quadratic <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>-functions, we model the distribution of <math>\n <semantics>\n <mrow>\n <mi>L</mi>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>χ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$L(1,\\chi)$</annotation>\n </semantics></math> by the distribution of random Euler products <math>\n <semantics>\n <mrow>\n <mi>L</mi>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$L(1,\\mathbb {X})$</annotation>\n </semantics></math> for certain family of random variables <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>(</mo>\n <mi>p</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathbb {X}(p)$</annotation>\n </semantics></math> attached to each prime. We obtain a description of the proportion of <math>\n <semantics>\n <mrow>\n <mo>|</mo>\n <mi>L</mi>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>χ</mi>\n <mo>)</mo>\n <mo>|</mo>\n </mrow>\n <annotation>$|L(1,\\chi)|$</annotation>\n </semantics></math> that is larger or that is smaller than a given bound, and yield more light into the Littlewood bounds. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12996","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12996","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

We investigate the distribution of values of cubic Dirichlet $L$ -functions at $s = 1$ . Following ideas of Granville and Soundararajan for quadratic $L$ -functions, we model the distribution of $L (1, χ)$ by the distribution of random Euler products $L (1, X)$ for certain family of random variables $X (p)$ attached to each prime. We obtain a description of the proportion of $| L (1, χ) |$ that is larger or that is smaller than a given bound, and yield more light into the Littlewood bounds. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values.

Abstract Image

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

立方 L $L$ 函数极值在 s = 1 $s=1$ 时的非对称分布

我们研究了 s = 1 $s=1$ 时立方迪里夏特 L $L$ 函数值的分布。按照 Granville 和 Soundararajan 对二次 L $L$ - 函数的想法，我们通过附在每个素数上的特定随机变量 X ( p ) $\mathbb {X}(p)$ 的随机欧拉积 L ( 1 , X ) $L(1,\mathbb {X})$ 的分布来模拟 L ( 1 , χ ) $L(1,\chi)$ 的分布。我们得到了关于 | L ( 1 , χ ) | $|L(1,\chi)|$ 大于或小于给定边界的比例的描述，并为利特尔伍德边界提供更多启示。与二次情况不同，三次情况的下限和上限不对称，小值比大值更不可能出现。

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分钟内获得全文去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

期刊最新文献

Management of Cholesteatoma: Hearing Rehabilitation. Congenital Cholesteatoma. Evaluation of Cholesteatoma. Management of Cholesteatoma: Extension Beyond Middle Ear/Mastoid. Recidivism and Recurrence.