Topics in Number Theory

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Abstract for Granville's proposal.This award will support the research of the investigator and severalgraduate 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 leastmth 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 thedistribution 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 questionsis closely related to the seemingly unrelated area of `integral delay equations'' which are analyzed extensively in understanding howvarious biological mechanisms work. Thus results and conjectures about mean values of multiplicative functions can be ``translated'' intoresults 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-delayequations, and the investigator has been asked to present his ideas in a series of lectures at a major conference in that area, to stimulate furtherinteraction. 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 wasrecently ranked tenth in the US.