Analytic number theory and periods of automorphic forms

主要研究者提出了解析数论中关于模形式的傅立叶系数和l函数矩的几个课题。研究者将谱方法与公式相结合，这些公式将这些量与自同态形式的周期联系起来，以建立误差项上具有强界的渐近公式。在项目1中，研究者将为Hardy-Ramanujan配分函数建立一个新的渐近公式，从而提供一种计算大配分数的替代方法。在项目2中，研究者将使用弱质量形式的痕迹来构建模拟模形式的新示例，并为其系数建立渐近公式。在项目3中，研究者将建立l函数矩的渐近公式，并将这些结果应用于各种非消失和次凸问题。模形式和l函数是数论中最重要的对象。例如，模形式的傅里叶系数包含算术和组合信息，而l函数的中心值与椭圆曲线的不变量有关。拟议的研究将导致对这些对象的更好理解，并为新的应用铺平道路。其中一个应用程序涉及计算大分区数，这在科学计算和密码学中具有重要用途。建议的研究也将导致适合本科生，研究生和博士后研究员的问题。其中一些问题将由研究者建议的本科生通过NSF本科生研究经验项目进行研究。

The principal investigator proposes several projects in analytic number theory concerning Fourier coefficients of modular forms and moments of L-functions. The investigator will combine spectral methods with formulas which relate these quantities to periods of automorphic forms to establish asymptotic formulas with strong bounds on the error terms. In project 1 the investigator will establish a new asymptotic formula for the Hardy-Ramanujan partition function and thus provide an alternative approach to computing large partition numbers. In project 2 the investigator will use traces of weak Maass forms to construct new examples of mock modular forms and establish asymptotic formulas for their coefficients. In project 3 the investigator will establish asymptotic formulas for moments of L-functions and apply these results to various non-vanishing and subconvexity problems.Modular forms and L-functions are objects of fundamental importance in number theory. For example, the Fourier coefficients of modular forms contain arithmetic and combinatorial information, while central values of L-functions are related to invariants of elliptic curves. The proposed research will result in a greater understanding of these objects and pave the way for new applications. One such application involves computing large partition numbers, which have important uses in scientific computing and cryptography. The proposed research will also lead to problems suitable for undergraduate students, graduate students, and post-doctoral fellows. Some of these problems will be studied by undergraduates advised by the investigator through the NSF Research Experiences for Undergraduates program.