First example of a third degree transcendental L-function exhibited
holds the secret to how the prime numbers are distributed, and is a first-degree algebraic L-function. The Riemann Hypothesis, announced in 1859 — and today the most important of all unsolved math problems — is an example of something that should be true for every L-function. Michael Rubinstein from the University of Waterloo, a participant at the workshop, quickly tested and confirmed the Riemann Hypothesis for the first few zeros of this newly minted L-function.
Rubinstein, along with William Stein of the University of Washington, will direct a new initiative to chart all L-functions; this project has been recommended for funding by the National Science Foundation (NSF). “The techniques developed by Bian and Booker open up whole new possibilities for experimenting with these powerful and mysterious functions and are a key step towards making our group project a success.” Rubinstein added. “It’s a big step toward our understanding the ‘world of L’, which is where most of the secrets of number theory are kept,” said Brian Conrey, Director of AIM.
Dorian Goldfeld, professor of Mathematics at Columbia University summarized the excitement, saying “This discovery is analogous to finding planets in remote solar systems. We know they are out there, but the problem is to detect them and determine what they look like. It gives us a glimpse of new worlds.”
Background: L-functions and Modular Forms Project
L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. All branches of number theory have been touched by L-functions and modular forms. Besides containing deep information concerning the distribution of prime numbers and the structure of elliptic curves, they feature prominently in Andrew Wiles’ solution of the famous 350-year-old Fermat’s Last Theorem, and in the twentieth century classification of congruent numbers, a problem first posed by Arab mathematicians one thousand years ago. L-functions may have central importance, but mathematicians have only scratched the surface of these crucial and powerful functions.
Michael Rubinstein, University of Waterloo, and William Stein, University of Washington, are directing a a major new project involving a dozen researchers to systematically tabulate and study these functions. This project has been recommended for funding by the National Science Foundation (NSF). The work will fall into four categories: theoretical, algorithmic, experimental, and data gathering. The theoretical work will be stimulated by their goal of charting the world of L-functions and modular forms. Their experimental work will involve testing many key conjectures concerning these functions. The project will produce a large amount of training, with plans for three graduate student schools, an undergraduate research experience, and support for a score of postdocs and graduate students who will assist in research. It will result in the creation of a vast amount of data about a wide range of modular forms and L-functions, which will far surpass in range and depth anything computed before in this area. The data will be organized in a freely available online data archive, along with the actual programs which were used to generate these tables.
AIM says that by providing these tables and tools online, the researchers will guarantee that the usefulness of this project will extend far beyond the circle of researchers on this FRG. The archive will be a rich source of examples and tools for researchers working on L-functions and modular forms for years to come, and will allow for future updates and expansion.
