Topics in Number Theory

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This project involves research in the area of analytic number theory. An important class of problems in this area concerns L-functions, which encode arithmetic information. One example of an L-function is the Riemann zeta function, which controls the distribution of prime numbers. One of the investigator's main goals is to make progress on the value distribution and zeros of L-functions. Concurrently, one hopes to use such information to extract applications to problems in arithmetic. While these topics are of intrinsic interest in mathematics, progress in number theory has had important applications in cryptography and theoretical computer science. More specifically, the investigator will continue his investigations on moments and value distribution of L-functions, extending in particular his recent work with Radziwill. The investigator will also continue work on multiplicative functions, in collaboration with Granville, Harper, and Koukoulopoulos. Besides these two main projects, the investigator will work with coauthors on various problems at the interface of analysis, number theory, and combinatorics, and continue his work in training graduate students in this area.