RUI: Low-Lying Zeros of L-functions and Problems in Additive Number Theory

Williams College, Williamstown MA

Investigators

Abstract

The PI plans on studying several projects on zeros of L-functions, as well as problems in additive number theory. The main research concerns the behavior of zeros near the central point. While these zeros have been known to be related to arithmetic problems since Riemann, in the last few decades connections have been observed with high energy nuclear physics and random matrix theory (RMT) as well. Thus investigations in one of these topics can be fruitfully used in the others. The additive number theory problems have a similar flavor; many of them concern density of states as well as gaps between events. Similar techniques are used in the analysis here as in some of the number theory and random matrix theory investigations. Specifically, the PI proposes to study: (1) the n-level densities of low-lying zeros of GL(2) L-functions (including level 1 Maass forms and increasing the support for holomorphic cuspidal newforms by deriving alternatives to the Katz-Sarnak determinantal expansions that are more amenable for comparing number theory and random matrix theory), (2) the n-level densities of Dirichlet and number field L-functions (which involves exploring and extending our understanding of the finer properties of the distribution of certain classes of primes), (3) finding random matrix theory analogues to Rankin-Selberg convolutions (to extend the predictive ability of RMT), (4) additional consequences of the L-function Ratios Conjecture and other arithmetic conjectures, (5) the density of states and behavior of the eigenvalues of structured random matrix ensembles (with special emphasis on the resulting combinatorics, which is frequently related to other problems of interest), (6) generalized Zeckendorf decompositions and the gaps between summands, (7) generalized sum and difference sets (especially phase transitions from different models of randomness, behavior in subsets of highly structured sets, and results in non-abelian cases), and finally (8) Benford's law of digit bias. The central questions in this proposal involve studying how events are distributed in diverse systems, such as energy levels of heavy nuclei, prime numbers and zeros of L-functions, leading digits in sets of data, and summands in generalized Zeckendorf decompositions. 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. Understanding these systems requires the development of tools and techniques in complex analysis, Fourier analysis, number theory and probability. Some of the projects have real world 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 with undergraduate research assistants. The PI will also continue his extensive work in math education. In addition to providing numerous mentoring opportunities to his students (such as arranging for them to referee for journals, write reviews for MathSciNet, write expository articles for journals, and co-organize AMS special sessions), the PI will also involve them with expanding his math riddles page (http://mathriddles.williams.edu/). This site is frequently one of the top hits when searching for math riddles, and is used in junior high and high schools around the world to excite students to mathematics.