L-functions are fundamental objects in analytic number theory which encode arithmetic information. The simplest examples are the Riemann zeta-function, Dirichlet L-functions, and the L-functions associated with modular forms. Understanding the statistical behavior of the values and zeros of these L-functions is the primary theme of Dr. Conrey.s research. He and his collaborators propose a variety of projects each involved with some aspect of L-functions. One involves a new formula with the divisor function d(n) which generalizes the classical Voronoi formula and has an application in the Nyman-Beurling approach to the Riemann Hypothesis; Conrey will develop further applications to moment formulae and to the question of understanding low degree L-functions. Conrey will use the Asymptotic Large Sieve (invented with Iwaniec and Soundararajan) to prove a conjecture about the mean square of all Dirichlet L-functions multiplied by an arbitrary Dirichlet polynomial. A third project involves Mazur.s conjecture that the symmetric power L-functions associated with an elliptic curve seldom vanish at their central point.
Professor Conrey's research is in the area of number theory. Modern Number Theory has surprisingly diverse applications, from enabling secure internet transactions, to the construction of optimal networks, and even to the question of cataloguing the various types of bodies in the study of low dimensional topology. One of the most successful tools invented by number theorists is the zeta-function. Its original purpose was to help with the study of prime numbers. Now it, and its analogues, are ubiquitous in number theory. However, there are still some very basic properties of zeta-functions which we do not understand, and which if we did would lead to much progress. The main question is Why do all of the zeros of zeta-functions occur on just one line? Professor Conrey's research is centered on the study of the zeros of zeta-functions. As part of this project, Professor Conrey will also continue his work with Math Teachers' Circles, which are a collection of 39 problem solving groups all across the country that involve professional mathematicians and Middle School math teachers working together to build communities of problem solvers.
L-functions are special functions whose properties are tied up with questions from number theory such as the distribution of prime numbers or determining when equations have solutions in whole number solutions. In this grant we developed a new tool that we call the "asymptotic large sieve" and that allows a new way to study the family of all "Dirichlet L-functions" whose properties tell us, for example, about primes in arithmetic progressions. Using this new tool we were able to prove that most of the zeros of these L-functions are in the complex plane on the vertical line which passes through the number 1/2. This is an approximation to the Riemann Hypothesis, long recognized as the fundamental problem in number theory, which asserts that all of the complex zeros of all L-functions are on this line. In addition we worked with students from the Morgan Hill Math program and teachers from the AIM MAth Teachers Circle program. We advised high school students Tara McLaughlin and Mark Holmstrom on a science fair project "Smooth Neighbors" and co-wrote and published a paper with them on this subject. We led several Math Teachers' Circles (MTC) sessions with the AIM MTC, the Austin MTC, and the Tulsa MTC. One of the AIM MTC sessions led to a publication "Sets, Planets, and Comets" (in the College Math Journal) that was joint with several of the teachers.
