{"title":"Dirichlet l -函数在q方面的大偏差的上界","authors":"Louis-Pierre Arguin, Nathan Creighton","doi":"10.1016/j.jnt.2025.01.009","DOIUrl":null,"url":null,"abstract":"<div><div>We prove a result on the large deviations of the central values of even primitive Dirichlet <em>L</em>-functions with a given modulus. For <span><math><mi>V</mi><mo>∼</mo><mi>α</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></math></span> with <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span>, we show that<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mfrac><mi>#</mi><mrow><mo>{</mo><mi>χ</mi><mtext> even, primitive mod </mtext><mi>q</mi><mo>:</mo><mi>log</mi><mo></mo><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>χ</mi><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>></mo><mi>V</mi><mo>}</mo></mrow><mspace></mspace><mo>≪</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></mfrac></mrow></msup></mrow><mrow><msqrt><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></msqrt></mrow></mfrac><mo>.</mo></math></span></span></span> This yields the sharp upper bound for the fractional moments of central values of Dirichlet <em>L</em>-functions proved by Gao, upon noting that the number of even, primitive characters with modulus <em>q</em> is <span><math><mfrac><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. The proof is an adaptation to the <em>q</em>-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwiłł for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the <em>t</em>-aspect. We go further and get bounds on the case where <span><math><mi>V</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with <em>q</em>. The method involves a formula for the twisted mollified second moment of central values of Dirichlet <em>L</em>-functions, building on the work of Iwaniec and Sarnak.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"273 ","pages":"Pages 96-158"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Upper bounds on large deviations of Dirichlet L-functions in the q-aspect\",\"authors\":\"Louis-Pierre Arguin, Nathan Creighton\",\"doi\":\"10.1016/j.jnt.2025.01.009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We prove a result on the large deviations of the central values of even primitive Dirichlet <em>L</em>-functions with a given modulus. For <span><math><mi>V</mi><mo>∼</mo><mi>α</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></math></span> with <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span>, we show that<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mfrac><mi>#</mi><mrow><mo>{</mo><mi>χ</mi><mtext> even, primitive mod </mtext><mi>q</mi><mo>:</mo><mi>log</mi><mo></mo><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>χ</mi><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>></mo><mi>V</mi><mo>}</mo></mrow><mspace></mspace><mo>≪</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></mfrac></mrow></msup></mrow><mrow><msqrt><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></msqrt></mrow></mfrac><mo>.</mo></math></span></span></span> This yields the sharp upper bound for the fractional moments of central values of Dirichlet <em>L</em>-functions proved by Gao, upon noting that the number of even, primitive characters with modulus <em>q</em> is <span><math><mfrac><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. The proof is an adaptation to the <em>q</em>-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwiłł for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the <em>t</em>-aspect. We go further and get bounds on the case where <span><math><mi>V</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with <em>q</em>. The method involves a formula for the twisted mollified second moment of central values of Dirichlet <em>L</em>-functions, building on the work of Iwaniec and Sarnak.</div></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"273 \",\"pages\":\"Pages 96-158\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2025-02-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X25000289\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X25000289","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了具有给定模的偶原始狄利克雷l函数中心值的大偏差的一个结果。V∼α日志日志问0 & lt;α& lt; 1中,我们显示that1φ(q) #{χ甚至原始mod问:日志| L(χ,12)|祝辞V}≪e−V2log日志qlog日志q。这得到了由Gao证明的Dirichlet l -函数中心值的分数阶矩的明显上界,注意到模数为q的偶数原始字符的数目为φ(q)2+O(1)。该证明是对由Arguin, Bourgade和Radziwiłł为黎曼zeta函数的局部最大值而开发的递归格式的q方面的适应,并由Arguin和Bailey应用于t方面的大偏差。我们进一步求出V=o(log log)时的边界。这些边界并不会很明显,但是与中心极限定理估计的差异随着q的增长非常缓慢。该方法涉及Dirichlet l -函数中心值的扭曲缓和第二矩的公式,建立在Iwaniec和Sarnak的工作基础上。
Upper bounds on large deviations of Dirichlet L-functions in the q-aspect
We prove a result on the large deviations of the central values of even primitive Dirichlet L-functions with a given modulus. For with , we show that This yields the sharp upper bound for the fractional moments of central values of Dirichlet L-functions proved by Gao, upon noting that the number of even, primitive characters with modulus q is . The proof is an adaptation to the q-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwiłł for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the t-aspect. We go further and get bounds on the case where . These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with q. The method involves a formula for the twisted mollified second moment of central values of Dirichlet L-functions, building on the work of Iwaniec and Sarnak.
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