{"title":"与某一狄利克列相关联的欧拉积段","authors":"Rajat Gupta , Aditi Savalia","doi":"10.1016/j.jnt.2024.06.003","DOIUrl":null,"url":null,"abstract":"<div><p>In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>|</mo><mi>n</mi></mrow></msub><msup><mrow><mi>d</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>. We obtain an exact identity relating the Dirichlet series <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>−</mo><mi>α</mi><mo>)</mo></math></span> and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A segment of Euler product associated to a certain Dirichlet series\",\"authors\":\"Rajat Gupta , Aditi Savalia\",\"doi\":\"10.1016/j.jnt.2024.06.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>|</mo><mi>n</mi></mrow></msub><msup><mrow><mi>d</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>. We obtain an exact identity relating the Dirichlet series <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>−</mo><mi>α</mi><mo>)</mo></math></span> and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-18\",\"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/S0022314X24001501\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001501","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A segment of Euler product associated to a certain Dirichlet series
In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function . We obtain an exact identity relating the Dirichlet series and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.
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