{"title":"Bilinear sums with GL(2) coefficients and the exponent of distribution of d3","authors":"Prahlad Sharma","doi":"10.1112/plms.12589","DOIUrl":null,"url":null,"abstract":"We obtain the exponent of distribution <mjx-container aria-label=\"1 divided by 2 plus 1 divided by 30\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"7,9\" data-semantic-content=\"3\" data-semantic- data-semantic-role=\"addition\" data-semantic-speech=\"1 divided by 2 plus 1 divided by 30\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,+\" data-semantic-parent=\"8\" data-semantic-role=\"addition\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/af700ab1-c439-4943-bca5-0381bd669efc/plms12589-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"7,9\" data-semantic-content=\"3\" data-semantic-role=\"addition\" data-semantic-speech=\"1 divided by 2 plus 1 divided by 30\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"operator\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,+\" data-semantic-parent=\"8\" data-semantic-role=\"addition\" data-semantic-type=\"operator\">+</mo><mrow data-semantic-=\"\" data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic-parent=\"8\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"operator\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">30</mn></mrow></mrow>$1/2+1/30$</annotation></semantics></math></mjx-assistive-mml></mjx-container> for the ternary divisor function <mjx-container aria-label=\"d 3\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"d 3\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/dab932c8-c2bf-41b1-bdff-665fc63b9655/plms12589-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"d 3\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></msub>$d_3$</annotation></semantics></math></mjx-assistive-mml></mjx-container> to square-free and prime power moduli, improving the previous results of Fouvry–Kowalski–Michel, Heath-Brown and Friedlander–Iwaniec. The key input is certain estimates on bilinear sums with <mjx-container aria-label=\"upper G upper L left parenthesis 2 right parenthesis\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,7\" data-semantic-content=\"8\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"upper G upper L left parenthesis 2 right parenthesis\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"1,5\" data-semantic-content=\"6,1\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"7\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"7\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3\" data-semantic-content=\"2,4\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b8845506-b90d-441a-96e3-2a4a7fc9497b/plms12589-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,7\" data-semantic-content=\"8\" data-semantic-role=\"implicit\" data-semantic-speech=\"upper G upper L left parenthesis 2 right parenthesis\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">G</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"1,5\" data-semantic-content=\"6,1\" data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"7\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">L</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"7\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"3\" data-semantic-content=\"2,4\" data-semantic-parent=\"7\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow></mrow>$GL(2)$</annotation></semantics></math></mjx-assistive-mml></mjx-container> coefficients obtained using the delta symbol approach.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"54 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12589","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We obtain the exponent of distribution for the ternary divisor function to square-free and prime power moduli, improving the previous results of Fouvry–Kowalski–Michel, Heath-Brown and Friedlander–Iwaniec. The key input is certain estimates on bilinear sums with coefficients obtained using the delta symbol approach.
期刊介绍：
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