April 4, 2008
A new mathematical object has been discovered--the first example of a third-degree transcendental L-function. This may open up the way to solving the Riemann Hypothesis--unproven since 1859. The Riemann Hypothesis deals with the distribution of prime numbers and L-functions.
L-functions encode underlying connections among numerous areas of mathematics. There are two types of L-functions: algebraic and transcendental, and these are classified according to their degree. The Riemann zeta-function--a first-degree algebraic L-function--is significant because it may answer the question of how prime numbers are distributed.
Ce Bian and his adviser, Andrew Booker (University of Bristol), revealed their discovery at the March meeting of the American Institute of Mathematics (AIM). Booker said, "This work was made possible by a combination of theoretical advances and the power of modern computers." Bian reported that it took 10,000 hours of computer time to produce initial results.
Michael Rubinstein (University of Waterloo) confirmed the Riemann Hypothesis for the first few zeroes of the new L-function. Moreover, Rubinstein and William Stein (University of Washington) hope to chart all L-functions. "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," said Rubinstein.
"This breakthrough opens a door to the study of higher degree L-functions," said mathematician Dennis Hejhal (University of Minnesota). "It's a big advance," added Harold Stark (University of California, San Diego) who, 30 years ago, calculated second-degree transcendental L-functions. "I thought we were years away from doing this. The geometry of what you have to do and the scale of the computation are orders of magnitude harder," he said.
Source: Science Daily