On the number of prime factors with a given multiplicity over h-free and h-full numbers

Pub Date : 2024-09-23 DOI:10.1016/j.jnt.2024.08.007
Sourabhashis Das, Wentang Kuo, Yu-Ru Liu
{"title":"On the number of prime factors with a given multiplicity over h-free and h-full numbers","authors":"Sourabhashis Das,&nbsp;Wentang Kuo,&nbsp;Yu-Ru Liu","doi":"10.1016/j.jnt.2024.08.007","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>k</em> and <em>n</em> be natural numbers. Let <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em> with multiplicity <em>k</em> as studied by Elma and the third author <span><span>[5]</span></span>. We obtain asymptotic estimates for the first and the second moments of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> when restricted to the set of <em>h</em>-free and <em>h</em>-full numbers. We prove that <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-free numbers, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>&lt;</mo><mi>k</mi><mo>&lt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-free numbers and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mi>k</mi><mo>&gt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-full numbers.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X2400194X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Let k and n be natural numbers. Let ωk(n) denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of ωk(n) when restricted to the set of h-free and h-full numbers. We prove that ω1(n) has normal order loglogn over h-free numbers, ωh(n) has normal order loglogn over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ωk(n) with 1<k<h do not have normal order over h-free numbers and ωk(n) with k>h do not have normal order over h-full numbers.
分享
查看原文
关于在无h和满h数中具有给定倍数的质因数个数
设 k 和 n 都是自然数。让 ωk(n)表示乘数为 k 的 n 的不同质因数的个数,如 Elma 和第三作者所研究的那样[5]。我们得到了ωk(n)的第一矩和第二矩的渐近估计值,并将其限制在无 h 和满 h 的数集合中。我们证明ω1(n) 在 h 个无穷数上有正序 loglogn,ωh(n) 在 h 个满数上有正序 loglogn,而且它们都满足厄尔多斯-卡克定理。最后,我们证明含 1<k<h 的函数 ωk(n) 在无 h 数上没有正序,含 k>h 的函数 ωk(n) 在满 h 数上没有正序。
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
×
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