Investigations on Low-Lying Zeros of L-Functions

米勒教授将研究l函数在中心点附近的零点性质，以及与算术上有趣的数列和函数的数字相关的等分布问题。用经典紧群的特征值在1附近的标度极限很好地模拟了l函数族中点附近的零点。对于椭圆曲线，Birch和Swinnerton-Dyer猜想将中心点上的零与有理解群的大小联系起来。因此，秩在Q(T)以上的椭圆曲线的单参数族应该为研究多重零的影响提供一个可接近的实验室。除了研究特殊的l -函数族（重点是两个l -函数族和在中心点具有高消失的l -函数族的扭转的零如何表现），研究将包括为具有高多重特征值的系综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来完成（因为通过显式公式，l函数的零上和与其系数的素数上和有关），分析Rankin-Selberg卷积的Satake参数，以及开发处理随机矩阵理论计算的组合公式。由于关于零的行为的猜想与数论中的标准猜想有关，因此这项工作提供了一个测试这些断言的机会（例如，对于等差数列中的素数，误差项对剩余类的依赖）。最后，Miller教授和A. Kontorovich最近证明了l函数在临界线附近的值、随机矩阵集合的特征多项式的值以及3x+1映射的迭代满足Benford的数字偏差定律。这些联系将在数论兴趣的其他系统中进一步探讨。自黎曼以来，l函数的零点一直与算术问题有关；关于它们分布的标准猜想在数论中蕴涵着许多结果，其应用范围从计数素数的最优误差项到构造密码学中的高效算法。在过去的几十年里，在高能核物理和随机矩阵理论中也观察到这种联系。因此，对其中一个主题的研究可以有效地应用于其他主题。研究数论和动态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衍生技术有着巨大的应用；这些方法已被用于测试数据的完整性。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。

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.