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.
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 extended 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 involved 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 required programs for solving non-linear Painleve VI differential equations, which were written and made available in Matlab and Sage. In related work, the PI explored 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 used the above techniques to determine which systems should exhibit such behavior, understand the rate of convergence (which is essential in proving fraud), and derived new tests for data integrity. 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, Additionally, the PI participated in numerous outreach events, from working with over 10 students each year on research projects to delivering numerous lectures to maintaining a mathematics riddles page which is used in schools throughout the world.
