Kloosterman Sums, Fourier Coefficients of Cusp Forms and Multiplicative Number Theory

Kloosterman 求和、尖点形式的傅里叶系数和乘法数论

• 批准号: 0070013
• 负责人: Roger Baker
• 金额: $ 7.45万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070013&HistoricalAwards=false
• 关键词: Kloosterman; Sums; Fourier; Coefficients; Cusp

Analytic number theory is the branch of pure mathematics in which one applies tools from analysis to study the distribution of the prime numbers and other interesting sequences such as the sequence of squarefree numbers. Tools from analysis include Fourier analysis, Fourier and Laplace transforms, automorphic forms and spectral theory, sieve methods, and character theory and exponential sums. A Kloosterman sum is a particular exponential sum that is frequently encountered in multiplicative number theory. While individual Kloosterman sums are known to satisfy general upper bounds, Kuznetsov proved a trace formula for GL_2 and used it to show that weighted sums of Kloosterman sums satisfy much sharper bounds. In 1982, Deshouillers and Iwaniec published a groundbreaking paper extending Kuznetsov's trace formula and combining it with the large sieve. Their resulting bounds on sums of Kloosterman sums is now know as 'Kloostermania" and has had a large number of applications to multiplicative number theory during the past decade. This is a proposal to re-examine the Kuznetsov trace formula and the Deshouillers-Iwaniec paper, with the goal of improving the Deshouillers-Iwaniec results.

解析数论是纯数学的分支，其中一个应用工具，从分析，以研究分布的素数和其他有趣的序列，如序列的squarefree号码。 分析工具包括傅立叶分析、傅立叶和拉普拉斯变换、自守形式和谱理论、筛法、特征理论和指数和。 Kloosterman和是乘法数论中经常遇到的一种特殊的指数和。 虽然个别Kloosterman和已知满足一般上界，库兹涅佐夫证明了GL_2的迹公式，并使用它来证明Kloosterman和的加权和满足更尖锐的界限。 在1982年，Deshouillers和Iwaniec发表了一篇突破性的论文，扩展了库兹涅佐夫的迹公式，并将其与大筛子相结合。 他们由此产生的界限总和Kloosterman和现在被称为“Kloostermania”，并已大量应用于乘法数论在过去的十年。 这是一个建议，重新审查库兹涅佐夫迹公式和Deshouillers-Iwaniec文件，目的是改善Deshouillers-Iwaniec结果。