$\fq[x]$ 中极短区间的质点和莫比乌斯相关性

IF 1.7 1区 数学 Q1 MATHEMATICS
Pär Kurlberg, Lior Rosenzweig
{"title":"$\\fq[x]$ 中极短区间的质点和莫比乌斯相关性","authors":"Pär Kurlberg, Lior Rosenzweig","doi":"10.1353/ajm.2024.a928320","DOIUrl":null,"url":null,"abstract":"<p><p>abstract:</p><p>We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\\{f(x) + a : a \\in\\fp\\}$ for $f(x)\\in\\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\\\"{o}bius $\\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\\\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\\\"{o}bius sums is {\\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\\\"{o}bius $\\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\\in\\fq[x]$ and intervals of the form $f(x)+a$ for $a\\in\\fq$, where $p$ is fixed and $q=p^{l}$ grows.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Prime and Möbius correlations for very short intervals in $\\\\fq[x]$\",\"authors\":\"Pär Kurlberg, Lior Rosenzweig\",\"doi\":\"10.1353/ajm.2024.a928320\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>abstract:</p><p>We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\\\\{f(x) + a : a \\\\in\\\\fp\\\\}$ for $f(x)\\\\in\\\\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\\\\\\\"{o}bius $\\\\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\\\\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\\\\\\\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\\\\\\\"{o}bius sums is {\\\\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\\\\\\\"{o}bius $\\\\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\\\\in\\\\fq[x]$ and intervals of the form $f(x)+a$ for $a\\\\in\\\\fq$, where $p$ is fixed and $q=p^{l}$ grows.</p></p>\",\"PeriodicalId\":7453,\"journal\":{\"name\":\"American Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-05-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"American Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1353/ajm.2024.a928320\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"American Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1353/ajm.2024.a928320","RegionNum":1,"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

摘要

摘要:我们研究了形式为 $I(f):=\{f(x) + a :a \in\fp\}$ 对于 $f(x)\in\fp[x]$ 和 $p$ 素数,以及 M\"{o}bius $\mu$ 函数及其相关性的函数场类似和的取消(类似于乔拉猜想中出现的和)。对于一般的 $f$,即对于莫尔斯多项式的 $f$,误差项的大小大致为 $O(\sqrt{p})$(典型的主项为 $p$阶)。对于非一般的 $f$,我们证明了 "一般 "移位集的独立性仍然成立。我们还可以举出在 M\"{o}bius/Chowla 类型和中根本不存在取消的例子(事实上,事实证明 M\"{o}bius 和中的(平方根)取消与 Chowla 类型和中的(平方根)取消是 {\em 等价的}),以及启发式 "素数是独立的 "严重失效的区间。这些结果是从关于算术类函数相关性的一般定理中推导出来的;这些函数包括素数上的特征函数、M\"{o}bius $\mu$ 函数和除数函数(例如,可以处理 Titchmarsh 除数问题的函数场类似物)。我们还在更微妙的固定特征环境中证明了类似但稍弱的结果,即对于 $f(x)\in\fq[x]$ 和 $a\in\fq$ 的 $f(x)+a$ 形式的区间,其中 $p$ 是固定的,$q=p^{l}$ 在增长。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Prime and Möbius correlations for very short intervals in $\fq[x]$

abstract:

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\{f(x) + a : a \in\fp\}$ for $f(x)\in\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"{o}bius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"{o}bius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"{o}bius $\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\in\fq[x]$ and intervals of the form $f(x)+a$ for $a\in\fq$, where $p$ is fixed and $q=p^{l}$ grows.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
3.20
自引率
0.00%
发文量
35
审稿时长
24 months
期刊介绍: The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信