CAREER: Research in and Pathways to Analytic Number Theory

Primes are the multiplicative building blocks of integers, and understanding their properties is a central theme in number theory. One way to understand their distribution among the integers is through the study of the Riemann zeta-function, a pursuit that is foundational to the area of analytic number theory. In particular, a thorough understanding of the location of the so-called nontrivial zeros of this function would give very precise asymptotic formulas for the number of primes up to a given (large) integer. This is a central problem in all of mathematics with connections to other deep problems, such as the class number problem, originally studied by Gauss. The proposed work seeks to further explore the analytic properties of the Riemann zeta-function and, more generally, of L-functions, with an overarching goal to obtain new information regarding the zeros of these functions. In addition to the research objectives, the proposed work includes "Pathway Projects" to provide novel, comprehensive guides to areas of active research in analytic number theory, and an undergraduate educational program aimed at increasing the participation of historically underrepresented groups in STEM. The research objectives of this project are in analytic number theory and fall into three themes. The first theme concerns the vertical distribution of zeros of the Riemann zeta-function. In particular, limitations on the state-of-the-art methods used to detect fluctuations in gaps between non-trivial zeros will be determined. Moreover, a comprehensive guide to the problem of gaps between zeros of the Riemann zeta-function and its connection to several active areas of research in analytic number theory will be written. In the second theme, applications of the Chebotarev density theorem will be pursued. In particular, improved zero-density estimates for "most" L-functions within certain prescribed families will be proved. The third theme encompasses the mechanics and applications of the asymptotic large sieve. In particular, the asymptotic large sieve will be used to study the distribution of the zeros of various L-functions. A strengthening of the technique will be developed and subsequently applied to make new progress on calculating moments of certain L-functions. Finally, a comprehensive guide to the asymptotic large sieve, which is poised to be useful in various applications, will be written to make the technique more widely known and understood. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.