ScienceDaily (Mar. 17, 2008)
— A new mathematical object was revealed yesterday during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.
The news caused excitement at the AIM workshop attended by 25 of the world's leading analytic number theorists. The work is a joint project between Ce Bian and his adviser, Andrew Booker. Booker commented that, "This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.
"This breakthrough opens a door to the study of higher degree L-functions," said Dennis Hejhal, Professor of Mathematics at the University of Minnesota and Uppsala University.
"It's a big advance' added Harold Stark of the University of California, San Diego, who, 30 years ago was the first to accurately calculate 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."
There are two types of L-functions: algebraic and transcendental, and these are classified according to their degree. The Riemann zeta-function is the grand-daddy of all L-functions. It holds the secret to how the prime numbers are distributed, and is a first-degree algebraic L-function.
Jacobi and Madden have found a way to generate an infinite number of solutions for a puzzle known as 'Euler's Equation of degree four.'
The equation is part of a branch of mathematics called number theory. Number theory deals with the properties of numbers and the way they relate to each other. It is filled with problems that can be likened to numerical puzzles.
"It's like a puzzle: can you find four fourth powers that add up to another fourth power" Trying to answer that question is difficult because it is highly unlikely that someone would sit down and accidentally stumble upon something like that," said Madden, an associate professor of mathematics at The University of Arizona in Tucson.
Equations are puzzles that need certain solutions "plugged into them" in order to create a statement that obeys the rules of logic.
For example, think of the equation x + 2 = 4. Plugging "3" into the equation doesn't work, but if x = 2, then the equation is correct.