{"title":"函数的分数阶矩和短间隔内两个平方和","authors":"Siegfred Baluyot, Steven M. Gonek","doi":"10.1112/mtk.70047","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math></math> if <span></span><math></math> is the sum of two perfect squares, and <span></span><math></math> otherwise. We study the variance of <span></span><math></math> in short intervals by relating the variance with the second moment of the generating function <span></span><math></math> along <span></span><math></math>. We develop a new method for estimating fractional moments of <span></span><math></math>-functions and apply it to the second moment of <span></span><math></math> to bound the variance of <span></span><math></math>. Our results are conditional on the Riemann hypothesis for the zeta-function and the Dirichlet <span></span><math></math>-function associated with the non-principal character modulo 4.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70047","citationCount":"0","resultStr":"{\"title\":\"Fractional moments of -functions and sums of two squares in short intervals\",\"authors\":\"Siegfred Baluyot, Steven M. Gonek\",\"doi\":\"10.1112/mtk.70047\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math></math> if <span></span><math></math> is the sum of two perfect squares, and <span></span><math></math> otherwise. We study the variance of <span></span><math></math> in short intervals by relating the variance with the second moment of the generating function <span></span><math></math> along <span></span><math></math>. We develop a new method for estimating fractional moments of <span></span><math></math>-functions and apply it to the second moment of <span></span><math></math> to bound the variance of <span></span><math></math>. Our results are conditional on the Riemann hypothesis for the zeta-function and the Dirichlet <span></span><math></math>-function associated with the non-principal character modulo 4.</p>\",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":\"71 4\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70047\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.70047\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.70047","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Fractional moments of -functions and sums of two squares in short intervals
Let if is the sum of two perfect squares, and otherwise. We study the variance of in short intervals by relating the variance with the second moment of the generating function along . We develop a new method for estimating fractional moments of -functions and apply it to the second moment of to bound the variance of . Our results are conditional on the Riemann hypothesis for the zeta-function and the Dirichlet -function associated with the non-principal character modulo 4.
期刊介绍:
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.