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的性质，以及与算术感兴趣的序列和函数的数字有关的均匀分布问题。L函数族的中心点附近的零点可以用经典紧群的1附近的特征值的尺度极限来很好地模拟。对于椭圆曲线，Birch和Swinnerton-Dyer猜想将中心点的零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了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函数族的扭零点的行为)外，研究还包括建立具有高重数本征值的系综的随机矩阵理论模型，以及编写用于数值研究的计算机程序和算法。这将通过分析与椭圆曲线有关的勒让德和(因为通过显式公式L函数的零点上的和与其系数的素数上的和有关)，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来实现。由于关于零点行为的猜想与数论中的标准猜想有关，这项工作提供了一个检验这类断言的机会(例如，算术级数中素数的误差项对剩余类的依赖性)。最后，Miller教授和A.Kontorovich最近证明了临界线附近的L函数的值、随机矩阵集合的特征多项式的值以及3x+1映射的迭代满足本福德数字偏差定律。这些联系将在其他数论感兴趣的系统中进一步探索。L函数的零点自黎曼以来一直与算术问题有关；关于其分布的标准猜想蕴含着数论中的大量结果，其应用范围从计数素数的最佳误差项到构造密码学中的有效算法。在过去的几十年里，人们观察到了高能核物理和随机矩阵理论之间的联系。因此，对其中一个主题的调查可以在其他主题中卓有成效地使用。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，也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派生技术有着巨大的应用；这种方法已经被用于测试数据的完整性。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。

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.