L-Functions

99780392这位研究者和他的合作者提议研究解析数论中的一些问题。具体地说，他们试图证明大筛不等式的一个渐近公式；这将适用于Dirichlet L-函数的零点之间的小间隙以及Dirichlet L-函数的六阶矩和八阶矩的平均值公式。此外，他们还想证明存在无穷多个狄里克莱特L函数--与没有正实零点的实本原特征标有关的函数。这一断言并不适用于任何L函数的无穷族。此外，他们还将研究黎曼假说的巴拉扎德-赛亚斯标准。例如，这个标准是否等同于黎曼假设？数论是数学的一个分支，处理涉及整数的问题。数论在编码、网络、数据压缩和密码学中的应用将数论推向了现代数学的前沿。L函数为研究数论的许多问题提供了一个重要的工具，然而我们对L函数的基本性质的理解是不完整的。这位研究人员和他的合作者试图填补我们对L函数基础知识的一些空白。

99780392The investigator and his collaborators propose to study some problems in analytic number theory. Specifically, they seek to prove an asymptotic formula for the large sieve inequality; this would have applications to small gaps between zeros of Dirichlet L-functions and to mean value formulae for sixth and eighth moments of Dirichlet L-functions. Further, they want to show that there are infinitely many Dirichlet L-functions associated with real primitive characters that have no positive real zeros. This assertion is not known to hold for any infinite family of L-functions. Also, they will investigate the Balazard-Saias criterion for the Riemann Hypothesis. For example, is this criterion equivalent to the Riemann Hypothesis? Number theory is a branch of mathematics that deals with problems involving whole numbers. Applications of number theory to coding, networking, data compression, and cryptography have thrust number theory into the forefront of modern mathematics. L-functions provide an important tool for investigating many questions of number theory, yet our understanding of the fundamental properties of L-functions is incomplete. The investigator and his collaborators seek to fill in some of the gaps in our basic knowledge of L-functions.