{"title":"Bounds for smooth theta sums with rational parameters","authors":"Francesco Cellarosi , Tariq Osman","doi":"10.1016/j.jnt.2024.10.002","DOIUrl":null,"url":null,"abstract":"<div><div>We provide explicit families of pairs <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> such that for sufficiently regular <em>f</em>, there is a constant <em>C</em> for which the theta sum bound<span><span><span><math><mrow><mo>|</mo><munder><mo>∑</mo><mrow><mtext>n</mtext><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mi>f</mi><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mrow><mi>n</mi></mrow><mo>)</mo></mrow><mi>exp</mi><mo></mo><mo>{</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>(</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo>‖</mo><mrow><mi>n</mi></mrow><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi>β</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mi>x</mi><mo>+</mo><mrow><mi>α</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mo>}</mo><mo>|</mo></mrow><mspace></mspace><mo>≤</mo><mi>C</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> holds for every <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> and every <span><math><mi>N</mi><mo>∈</mo><mi>N</mi></math></span>. Central to the proof is realising that, for fixed <em>N</em>, the theta sum normalised by <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> agrees with an automorphic function <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> evaluated along a special curve known as a horocycle lift. The lift depends on the pair <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo></math></span>, and so the bound follows from showing that there are pairs such that <span><math><mo>|</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>|</mo></math></span> remains bounded along the entire horocycle lift.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 397-426"},"PeriodicalIF":0.6000,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24002178","RegionNum":3,"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
Abstract
We provide explicit families of pairs such that for sufficiently regular f, there is a constant C for which the theta sum bound holds for every and every . Central to the proof is realising that, for fixed N, the theta sum normalised by agrees with an automorphic function evaluated along a special curve known as a horocycle lift. The lift depends on the pair , and so the bound follows from showing that there are pairs such that remains bounded along the entire horocycle lift.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
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