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

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series 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":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12996","RegionNum":2,"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 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分钟内获得全文求助全文

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.