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

自20世纪70年代以来，研究人员已经成功地利用随机矩阵理论建模了L-函数零点的首项行为;最近，新的模型和算法，如L-函数比率猜想，已经超越了主项，并理解了低阶项，其中族的算术表面。继续以前的研究，PI和他的学生计划将这些模型扩展到许多其他L函数族，从数域到椭圆曲线。后者特别吸引人，因为它是唯一已知的零具有多重性的例子（根据伯奇和斯温纳顿-戴尔猜想，零在中心点的多重性等于有理解群的几何秩）。主要项目涉及建模这些零与一个新的随机矩阵模型（修改的Jacobi系综）相结合的算法的低阶项和离散化的L-函数的值在中心点。数值计算需要求解非线性Painleve VI微分方程的程序，这些程序将在Matlab和Sage中编写和提供。在相关的工作中，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族的行为方面提供强大的直觉。许多关于L-函数的中心点附近零点的行为的分析可以被改写为关于测度的等分布和收敛的问题。除了上述数论体系外，这些方法还可以应用于其他问题，如本福德数字偏差定律。许多数据集显示出强大而普遍的偏差，第一个数字是1的概率为30%，单调下降到第一个数字是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行为的公司纳税申报表）。PI将使用上述技术来确定哪些系统应该表现出这种行为，了解收敛速度（这在证明欺诈中至关重要），并推导出新的数据完整性测试。跨学科的中心问题之一是事件如何分布，无论是重核的能级，素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距定律支配着这些和其他现象;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。许多工作都试图理解黎曼zeta函数及其推广的L函数的零点之间的间距;人们早就知道数论中的许多重要问题（从素数的数量到有效的素数测试）都等价于这些零点的性质。椭圆曲线L-函数的零点是最重要的一类。这是唯一一个已知的具有多重零点的数论家族，因此理解它的行为对建模其他物理系统应该是有用的。该系统与许多其他系统一样，由非线性Painleve VI微分方程描述;除了构建模型外，PI及其同事还将生成并分发Matlab和Sage代码来求解这些方程。解决这些问题需要在复分析，傅立叶分析，数论和概率的工具和技术的发展。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，第一个数字是1的概率约为30%，第一个数字是9的概率约为5%）。PI还将研究涉及数据集前导位数分布的几个问题，特别是本福德行为的收敛速度，并确定哪些系统应该满足这一定律。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定律来定位公司税务欺诈。这些项目中的许多都有适合数值实验的组件;这些和易处理的特殊情况将与本科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.