RUI: Investigations of L-functions and Benford's Law

自20世纪70年代以来，研究人员利用随机矩阵理论成功地模拟了L函数零点的导项行为；最近又提出了新的模型和猜想，如L函数比猜想，以超越主项，理解低阶项，从而使家族的算术浮出水面。继续之前的研究，Pi和他的学生计划将这些模型推广到无数其他的L函数族，从数域情况到椭圆曲线。后者特别吸引人，因为它是已知的唯一具有重数的零点的例子(根据Birch和Swinnerton-Dyer猜想，中心点的零点的重数等于有理解群的几何秩次)。主要项目包括用一种新的随机矩阵模型(一种改进的雅可比系综)来模拟这些零点，该模型结合了低阶项的算法和L函数在中心点的离散化。数值计算需要求解非线性Painleve VI微分方程组的程序，该程序将编写并在MatLab和Sage中提供。在相关工作中，PI计划探索经典的随机矩阵集合，因为这些系统经常在理解子家族的行为方面提供强大的直觉。对L函数中心点附近零点行为的许多分析都可以转化为关于测度的均匀分布和收敛的问题。除了上述数论体系外，这些方法还可以应用于其他问题，如本福德数偏定律。许多数据集显示出强大而普遍的偏见，第一位数字为1的可能性为30%，单调下降到第一位数字为9的可能性为5%。本福德定律经常被用来测试数据的完整性(美国国税局用它来标记可能具有欺诈性的公司纳税申报单)。PI将使用上述技术来确定哪些系统应该表现出这种行为，了解收敛速度(这对于证明欺诈是至关重要的)，并推导出新的数据完整性测试。跨学科的核心问题之一是事件是如何分布的，无论是重核的能级、质数之间的间距还是银行的等待时间。与中心极限定理类似，似乎有一些普遍的间隔定律支配着这些现象和其他现象；因此，对其中一个主题的研究经常可以为其他主题提供有用的见解。许多拟议的工作试图理解Riemann Zeta函数及其推广的L函数的零点之间的间隔；众所周知，数论中的许多重要问题(从素数的计数到有效的素性检验)与这些零点的性质是等价的。要研究的最重要的族是椭圆曲线L函数的零点。这是已知的唯一具有零点和重数的数论族，因此了解它的行为应该有助于对其他物理系统进行建模。与许多其他系统一样，该系统由一个非线性Painleve VI微分方程式描述；除了构建一个模型外，PI和他的同事还将生成并分发用于求解这些方程的MatLab和Sage代码。解决这些问题需要发展复分析、傅立叶分析、数论和概率论的工具和技术。这些结果也适用于其他领域，特别是本福德的数字偏差定律(对于许多自然的数据集，第一位大约30%的时间是1，概率减少到第一位9大约5%的时间)。PI还将处理涉及数据集的前导数字分布的几个问题，特别是关于收敛到本福德行为的速度以及确定哪些系统应该满足这一定律。检测和理解数字偏差的推导技术有着巨大的应用；例如，美国国税局使用本福德定律来定位公司税务欺诈。这些项目中的许多都有适合进行数值实验的部分；这些容易处理的特殊情况将与本科生的研究助理一起进行调查。

Since the 1970s, researchers have successfully modeled the leading term behavior of zeros of L-functions using Random Matrix Theory; recently new models and conjectures, such as the L-functions Ratios Conjecture, have been advanced to go beyond the main term and understand the lower order terms, where the arithmetic of the families surface. Continuing previous research, the PI and his students plan on extending these models to numerous other families of L-functions, ranging from number field cases to elliptic curves. The latter is especially appealing, as it is the only known example with zeros with multiplicities (by the Birch and Swinnerton-Dyer conjecture, the multiplicity of the zero at the central point equals the geometric rank of the group of rational solutions). The main project involves modeling these zeros with a new random matrix model (a modified Jacobi ensemble) combining the arithmetic of the lower order terms and the discretization of the values of the L-functions at the central point. The numerical calculations require programs for solving non-linear Painleve VI differential equations, which will be written and made available in Matlab and Sage. In related work, the PI plans to explore classical random matrix ensembles, as these systems frequently provide powerful intuition in understanding the behavior of sub-families. Much of the analysis of the behavior of zeros near the central point of L-functions can be recast as questions about equidistribution and convergence of measures. In addition to the above number theoretic systems, these methods can also be applied to other problems, such as Benford's law of digit bias. Numerous data sets exhibit a powerful and universal bias, with the probability of the first digit being a 1 is 30%, dropping monotonically to a 5% chance of the first digit being a 9. Benford's law is frequently used to test for data integrity (the IRS uses it to flag corporate tax returns that are likely fraudulent). The PI will use the above techniques to determine which systems should exhibit such behavior, understand the rate of convergence (which is essential in proving fraud), and derive new tests for data integrity.One of the central questions across disciplines is how events are distributed, be it energy levels of heavy nuclei, spacings between prime numbers or waiting times at a bank. Similar to the Central Limit Theorem, there seem to be a few universal spacing laws that govern these and other phenomena; thus studies in one of these topics can frequently provide useful insights in the others. Much of the proposed work seeks to understand the spacings between zeros of the Riemann zeta function and its generalizations, L-functions; it has long been known that many important problems in number theory (ranging from counting the number of primes to efficient primality tests) are equivalent to properties of these zeros. The most important family to be studied are the zeros of elliptic curve L-functions. This is the only known number theory family with zeros with multiplicity, and thus understanding its behavior should be useful to model other physical systems. The system, like many others, is described by a non-linear Painleve VI differential equation; in addition to constructing a model the PI and his colleagues will generate and distribute Matlab and Sage code to solve these equations. Solving these problems requires the development of tools and techniques in complex analysis, Fourier analysis, number theory and probability. These results are applicable to other fields, in particular Benford's law of digit bias (for many natural sets of data, the first digit is 1 about 30% of the time, with the probability decreasing to a first digit of 9 about 5% of the time). The PI will also work on several problems involving the distribution of leading digits of data sets, especially on the rate of convergence to Benford behavior and determining which systems should satisfy this law. Deriving techniques to detect and understand digit bias have enormous applications; for example, the IRS uses Benford's law to locate corporate tax fraud. Many of these projects have components that are amenable to numerical experimentation; these and tractable special cases will be investigated in conjunction with undergraduate research assistants.