Investigations on Low-Lying Zeros of L-Functions

米勒教授将研究L-函数零点在中心点附近的性质，以及与算术有趣序列和函数的数字有关的等分布问题。L-函数族中中心点附近的零点可以用经典紧群特征值在1附近的标度极限来模拟。对于椭圆曲线，Birch和Swinnerton-Dyer猜想将中心点的零点与有理解群的大小联系起来。因此，秩在Q（T）上的单参数椭圆曲线族应该为研究多个零点的影响提供一个可访问的实验室。除了研究特殊的L-函数族（重点是两个L-函数族和在中心点高度消失的L-函数族的扭曲零点如何表现），研究将包括为具有高重数特征值的集合开发随机矩阵理论模型，以及编写计算机程序和数值研究算法。这将通过分析与椭圆曲线相关的勒让德和（因为通过明确的公式，L函数的零上的和与其系数的素数上的和相关），分析Rankin-Selberg卷积的Satake参数，以及开发组合公式来处理随机矩阵理论计算来完成。由于零的行为与数论中的标准假设有关，这项工作提供了一个测试这种说法的机会（例如，算术级数中素数的剩余类对误差项的依赖性）。最后，米勒教授和A. Kontorovich最近证明了L-函数在临界线附近的值、随机矩阵系综和3x+1映射的迭代的特征多项式的值满足Benford数偏定律。这些连接将进一步探讨在其他系统的数论兴趣。零点的L-函数已涉及到算术问题，因为黎曼;标准的描述其分布意味着许多结果在数论中，与应用范围从最佳的错误项计数素数构造有效的算法在密码学。在过去的几十年里，高能核物理和随机矩阵理论也观察到了这种联系。因此，在这些主题之一的调查可以卓有成效地用于其他。研究数论和动力学系统中的数字偏差突出了这些问题的关键特征，并且是不同系统行为相似的另一个例子。检测和理解数字偏差的推导技术有着广泛的应用;这些方法已被用于测试数据完整性。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。

Professor Miller will study properties of zeros of L-functions near the central point, and equidistribution questions related to digits of arithmetically interesting sequences and functions. Zeros near the central point in families of L-functions are well modeled by scaling limits of eigenvalues near 1 of a classical compact group. For elliptic curves the Birch and Swinnerton-Dyer conjecture relates zeros at the central point to the size of the group of rational solutions. Thus one-parameter families of elliptic curves with rank over Q(T) should provide an accessible laboratory for investigating the effect of multiple zeros. In addition to studying special families of L-functions (with emphasis on how zeros of the twist of two families of L-functions and families of L-functions with high vanishing at the central point behave), the research will include developing random matrix theory models for ensembles with high multiplicity eigenvalues, as well as writing computer programs and algorithms for numerical investigations. This will be accomplished by analyzing Legendre sums related to elliptic curves (for by the explicit formula sums over zeros of an L-function are related to sums over primes of its coefficients), analysis of the Satake parameters of Rankin-Selberg convolutions, and developing combinatorial formulas to handle the random matrix theory calculations. As conjectures on the behavior of zeros are related to standard conjectures in number theory, this work provides an opportunity to test such claims (for example, the dependence of the error term on the residue class for primes in arithmetic progression). Finally, Professor Miller and A. Kontorovich recently proved that values of L-functions near the critical line, values of characteristic polynomials of random matrix ensembles and iterates of the 3x+1 map satisfy Benford's law for digit bias. These connections will be further explored in additional systems of number theoretic interest.Zeros of L-functions have been related to arithmetic problems since Riemann; standard conjectures on their distribution imply numerous results in number theory, with applications ranging from optimal error terms in counting primes to constructing efficient algorithms in cryptography. In the last few decades connections have been observed with high energy nuclear physics and random matrix theory as well. Thus investigations in one of these topics can be fruitfully used in the others. Studying digit bias in number theoretic and dynamic systems highlights key features of these problems, and is another example where different systems behave similarly. Deriving techniques to detect and understand digit bias have enormous applications; such methods have been used to test for data integrity. 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.