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函数的零的主导项行为；最近，新的模型和猜想，如l -函数比率猜想，已经被提出，超越了主要项，并理解了低阶项，其中族的算术表面。继续先前的研究，PI和他的学生计划将这些模型扩展到许多其他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多重零的例子（根据Birch和Swinnerton-Dyer猜想，中心点上零的多重等于有理解群的几何秩）。主要项目包括用一种新的随机矩阵模型（一种改进的雅可比集合）将低阶项的算术和中心点l函数值的离散化相结合，对这些零点进行建模。数值计算需要求解非线性painlevel VI微分方程的程序，这些程序将在Matlab和Sage中编写并提供。在相关工作中，PI计划探索经典随机矩阵集合，因为这些系统经常为理解子族的行为提供强大的直觉。许多关于l函数中心点附近零点行为的分析可以被重新定义为关于测度的均分和收敛的问题。除了上述数论系统之外，这些方法还可以应用于其他问题，例如本福德的数字偏差定律。许多数据集表现出强大而普遍的偏差，第一个数字为1的概率为30%，单调下降到第一个数字为9的概率为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存在欺诈的企业纳税申报表）。PI将使用上述技术来确定哪些系统应该表现出这样的行为，了解收敛速度（这对于证明欺诈至关重要），并推导出数据完整性的新测试。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，素数之间的间隔还是在银行等待的时间。与中心极限定理类似，似乎有一些普遍的间距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通常可以为其他主题提供有用的见解。许多提出的工作试图理解黎曼ζ函数和它的推广，l函数的零点之间的间隔；人们早就知道，数论中的许多重要问题（从数素数到有效的素数检验）都等价于这些零的性质。要研究的最重要的一类是椭圆曲线l函数的零点。这是唯一已知的具有多重零的数论族，因此理解它的行为应该对其他物理系统的建模有用。与许多其他系统一样，该系统由非线性疼痛级VI微分方程描述；除了构建一个模型，PI和他的同事将生成和分发Matlab和Sage代码来解决这些方程。解决这些问题需要复杂分析、傅立叶分析、数论和概率论方面的工具和技术的发展。这些结果适用于其他领域，特别是本福德的数字偏差定律（对于许多自然数据集，大约30%的时间第一个数字是1，大约5%的时间概率下降到第一个数字9）。PI还将研究涉及数据集前导位数分布的几个问题，特别是关于收敛到本福德行为的速度和确定哪些系统应该满足该定律。检测和理解数字偏差的衍生技术有着巨大的应用；例如，美国国税局使用本福德法来查找企业税务欺诈。这些项目中的许多都有适合于数值实验的组成部分；这些和可处理的特殊情况将与本科生研究助理一起调查。

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.