Topics in Number Theory

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Abstract for Granville's proposal. This award will support the research of the investigator and several graduate students on various topics in number theory. In the last few years the investigator, in collaboration with Soundararajan, has been revisiting the central topic of mean values of multiplicative functions, and strengthening known results as well as adding several new perspectives. In particular they have shown that such questions are in some sense equivalent to problems about integral delay equations. Granville proposes continuing this work, particularly looking at the range of values taken and to solve some thorny optimization problems. This, in turn, should give new results on known problems about values of (Dirichlet and Hasse-Weil) L-functions at 1, bounds on least mth power residues, and perhaps even distribution of zeros of L-functions. The investigator also proposes continuing his work, with Stark, on the connection between the abc-conjecture and Siegel zeros: On the one hand, looking further at Diophantine equations satisfied by modular functions; on the other hand, with Tucker, looking at whether the unproved abc-conjecture can be replaced by a weaker but feasibly provable hypothesis. The investigator, with collaborators, is looking at the finer aspects of the distribution of multiplicative functions. The functions turn up in many different areas in mathematics, physics and cryptography. The investigator recently showed how an understanding of such questions is closely related to the seemingly unrelated area of `integral delay equations'' which are analyzed extensively in understanding how various biological mechanisms work. Thus results and conjectures about mean values of multiplicative functions can be ``translated'' into results and conjectures about integral delay equations, so that the perspectives of one field can be brought to bear on the other, and vice-versa. This has already led to some new results and fresh questions in integral-delay equations, and the investigator has been asked to present his ideas in a series of lectures at a major conference in that area, to stimulate further interaction. In addition the investigator has been working with students, particularly from the Deep South, to get their PhDs, and obtain good positions in research, education, industry and government, with a training that will help them bring a fresh perspective to their future professions. This group was recently ranked tenth in the US.