Investigations on Low-Lying Zeros of L-Functions

点击翻译按钮获取中文摘要

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.