Analytic number theory and periods of automorphic forms

主要研究者提出了几个项目在解析数论有关傅立叶系数的模形式和时刻的L-函数。研究者将联合收割机谱方法与自守形式的周期公式相结合，以建立误差项具有强界的渐近公式。在项目1中，研究人员将建立一个新的渐近公式的Hardy-Ramanujan分区函数，从而提供一种替代方法来计算大分区数。在项目2中，研究者将使用弱Maass形式的迹来构造模拟模形式的新例子，并建立其系数的渐近公式。在项目3中，研究者将建立L-函数的矩的渐近公式，并将这些结果应用于各种非零和次凸性问题。模形式和L-函数是数论中具有根本重要性的对象。例如，模形式的傅立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物体的更好理解，并为新的应用铺平道路。一个这样的应用涉及计算大分区数，其在科学计算和密码学中具有重要用途。拟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人员通过NSF本科生研究经验计划建议的本科生进行研究。

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.