The central questions in this project concern how events are distributed in diverse systems, such as energy levels of heavy nuclei, leading digits in sets of data, and the prime numbers among the integers. Similar to the central limit theorem in probability and statistics, there seem to be 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 topics have immediate practical applications; for example, the Internal Revenue Service uses Benford's law to locate corporate tax fraud. Many of questions under study in this project have components that are amenable to numerical experimentation; these and tractable special cases will be investigated with undergraduate, graduate, and postdoctoral research assistants. The investigator will also continue work in mathematics education. In addition to providing professional development opportunities to students (such as arranging for them to referee for journals, contribute to Mathematical Reviews, write expository articles for journals, and co-organize special sessions at professional society meetings), the investigator will involve students in expanding the Math Riddles web page (mathriddles.williams.edu), a site that is used in junior high and high schools around the world to excite students about mathematics.
This research project studies a variety of problems on L-functions, additive number theory, and Benford's law. A central theme is an analysis of gaps between events. The main topic concerns zeros of L-functions; connections have been observed between these and high energy nuclear physics and random matrix theory (RMT). Among the questions under study are: n-level densities (main and lower order terms) for zeros of L-functions, alternatives to the Katz-Sarnak determinantal expansions that are more amenable for comparisons between number theory and RMT, determining the optimal test functions to bound excess rank, biases in second moments of Fourier coefficients of L-functions, modeling zeros near the central point through excised RMT ensembles, large gaps between zeros of L-functions, the density of states and behavior of the eigenvalues of structured random matrix ensembles, generalized Zeckendorf decompositions and the gaps between summands, generalized sum and difference sets, Ramsey theory for sets avoiding 3-term geometric progressions in finite fields and non-commutative settings, and Benford's law in fragmentation problems and fraud detection.
