{"title":"论厄尔多斯-胡利三角函数的易碎均值","authors":"B. Martin , G. Tenenbaum , J. Wetzer","doi":"10.1016/j.indag.2024.02.002","DOIUrl":null,"url":null,"abstract":"<div><p>For integer <span><math><mi>n</mi></math></span> and real <span><math><mi>u</mi></math></span>, define <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>|</mo><mrow><mo>{</mo><mi>d</mi><mo>:</mo><mi>d</mi><mo>∣</mo><mi>n</mi><mo>,</mo><mspace></mspace><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup><mo><</mo><mi>d</mi><mo>⩽</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>}</mo></mrow><mo>|</mo></mrow></mrow></math></span>. Then, the Erdős–Hooley Delta function is defined as <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>R</mi></mrow></msub><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> We provide uniform upper and lower bounds for the mean-value of <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> over friable integers, i.e. integers free of large prime factors.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000090/pdfft?md5=d2a0f3d37cb93941f7d1335c246fb3a7&pid=1-s2.0-S0019357724000090-main.pdf","citationCount":"0","resultStr":"{\"title\":\"On the friable mean-value of the Erdős–Hooley Delta function\",\"authors\":\"B. Martin , G. Tenenbaum , J. Wetzer\",\"doi\":\"10.1016/j.indag.2024.02.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For integer <span><math><mi>n</mi></math></span> and real <span><math><mi>u</mi></math></span>, define <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>|</mo><mrow><mo>{</mo><mi>d</mi><mo>:</mo><mi>d</mi><mo>∣</mo><mi>n</mi><mo>,</mo><mspace></mspace><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup><mo><</mo><mi>d</mi><mo>⩽</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>}</mo></mrow><mo>|</mo></mrow></mrow></math></span>. Then, the Erdős–Hooley Delta function is defined as <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>R</mi></mrow></msub><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> We provide uniform upper and lower bounds for the mean-value of <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> over friable integers, i.e. integers free of large prime factors.</p></div>\",\"PeriodicalId\":56126,\"journal\":{\"name\":\"Indagationes Mathematicae-New Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000090/pdfft?md5=d2a0f3d37cb93941f7d1335c246fb3a7&pid=1-s2.0-S0019357724000090-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae-New Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000090\",\"RegionNum\":4,\"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":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000090","RegionNum":4,"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
对于整数 n 和实数 u，定义 Δ(n,u)≔|{d:d∣n,eu<d⩽eu+1}||。然后，厄尔多斯-胡利Δ函数定义为Δ(n)≔maxu∈RΔ(n,u)。我们提供了Δ(n) 在易碎整数（即不含大素因子的整数）上均值的统一上界和下界。
On the friable mean-value of the Erdős–Hooley Delta function
For integer and real , define . Then, the Erdős–Hooley Delta function is defined as We provide uniform upper and lower bounds for the mean-value of over friable integers, i.e. integers free of large prime factors.
期刊介绍：
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.
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