A new mathematical object has been discovered by Ce Bian, a maths student at the University of Bristol. The news caused great excitement when it was announced at a workshop organised by the American Institute of Mathematics (AIM) and attended by 25 of the world’s leading analytic number theorists.
Dorian Goldfeld, Professor of Mathematics at Columbia University, USA, summarised 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.”
The work is a joint project between Bian and his supervisor, Andrew Booker, also at the University of Bristol. The two researchers 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 Riemann zeta-function is the granddaddy of all L-functions and it holds the secret of how the prime numbers are distributed. The Riemann Hypothesis, announced in 1859 and today the most important of all unsolved mathematical problems, is an example of something that should be true for every L-function.
A million-dollar prize has been offered by the Clay Mathematics Institute for the first correct proof of the Riemann Hypothesis. This discovery brings the prize a step closer.
During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results. Booker commented: “This work was made possible by a combination of theoretical advances and the power of modern computers.”
Michael Rubinstein from the University of Waterloo, Canada, tested the Riemann Hypothesis for this newly minted L-function. Rubenstein enthused: “Being able to explore this new L-function gives me the excitement that a biologist must feel when discovering a new mammal. The techniques developed by Bian and Booker open up whole new possibilities for experimenting with these powerful and mysterious functions.”
“It’s a big step towards our understanding the ‘world of L’, which is where most of the secrets of number theory are kept,” said Brian Conrey, Director of AIM and Professor of Number Theory at Bristol University, who himself won a prize earlier this year for his work on the Riemann Hypothesis.
There are two types of L-functions: algebraic and transcendental and these are classified according to their degree. Harold Stark of the University of California, San Diego, who, 30 years ago was the first to accurately calculate second degree transcendental L-functions joined in the praise. “It’s a big advance”, he said.