{"title":"扭曲的加法除数问题","authors":"Alex Cowan","doi":"10.1016/j.jnt.2024.06.007","DOIUrl":null,"url":null,"abstract":"<div><p>We give asymptotics for shifted convolutions of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo><</mo><mi>X</mi></mrow></munder><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>u</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>χ</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>v</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow></msup></mrow></mfrac></math></span></span></span> for nonzero complex numbers <span><math><mi>u</mi><mo>,</mo><mi>v</mi></math></span> and nontrivial Dirichlet characters <span><math><mi>χ</mi><mo>,</mo><mi>ψ</mi></math></span>. We use the technique of <em>automorphic regularization</em> to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001586/pdfft?md5=03e4c4b87cd43372c6c4156f7d76d43a&pid=1-s2.0-S0022314X24001586-main.pdf","citationCount":"0","resultStr":"{\"title\":\"A twisted additive divisor problem\",\"authors\":\"Alex Cowan\",\"doi\":\"10.1016/j.jnt.2024.06.007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We give asymptotics for shifted convolutions of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo><</mo><mi>X</mi></mrow></munder><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>u</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>χ</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>v</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow></msup></mrow></mfrac></math></span></span></span> for nonzero complex numbers <span><math><mi>u</mi><mo>,</mo><mi>v</mi></math></span> and nontrivial Dirichlet characters <span><math><mi>χ</mi><mo>,</mo><mi>ψ</mi></math></span>. We use the technique of <em>automorphic regularization</em> to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001586/pdfft?md5=03e4c4b87cd43372c6c4156f7d76d43a&pid=1-s2.0-S0022314X24001586-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001586\",\"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/S0022314X24001586","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We give asymptotics for shifted convolutions of the form for nonzero complex numbers and nontrivial Dirichlet characters . We use the technique of automorphic regularization to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.