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

解析数论是纯数学的一个分支，它应用分析的工具来研究素数的分布和其他有趣的数列，如无平方数的数列。分析工具包括傅立叶分析，傅立叶和拉普拉斯变换，自同构形式和谱理论，筛法，特征理论和指数和。Kloosterman和是在乘法数论中经常遇到的一种特殊的指数和。虽然已知单个Kloosterman和满足一般上界，但库兹涅佐夫证明了GL_2的一个迹公式，并用它证明了Kloosterman和的加权和满足更大的上界。1982年，Deshouillers和Iwaniec发表了一篇开创性的论文，扩展了库兹涅佐夫的迹公式，并将其与大筛结合起来。他们得出的Kloosterman和的和的边界现在被称为“Kloostermania”，并且在过去十年中在乘法数论中有大量的应用。这是一个重新检验库兹涅佐夫迹公式和Deshouillers-Iwaniec论文的建议，目的是改进Deshouillers-Iwaniec的结果。

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.