Since its publication, Riemann’s paper has been the main focus of prime number theory and the main reason for the proof of something called the prime number theorem in 1896. Since then, several new proofs have been found, including elementary proofs by Selberg and Erdós. However, Riemann’s hypothesis about the roots of the zeta function has remained a mystery. Its mystery compounded by the fact that the R zeta function depends on a complex variable and it does not have a closed form expression – i.e. it can’t be expressed as a single formula that contains other standard functions.
If the Riemann Hypothesis is proven to be correct, it would allow mathematicians to better describe how the prime numbers are placed among whole numbers. The Riemann Hypothesis has been dubbed so important to the field of mathematics and so difficult to prove, that the Clay Mathematics Institute has offered $1,000,000 to the first person to prove it.
“On the critical line, the Riemann function depends only on the variable t”, said Fokas who spent nearly nine years wrestling with Lindelöf. “The Riemann Hypothesis can be verified with today’s computer for t up to order 10 to the power 13 which is a very large number, but still very small compared to infinity. This shows that we need to understand the behavior of the R zeta function when t is very large. This is where the Lindelöf Hypothesis comes in, which conjunctures that the R zeta function has a certain form as t gets very large.”
Fokas is a world expert in asymptotics, an applied mathematical domain which helps scientists answer questions about the behavior of functions when a parameter is very large. His work of Lindelöf may signify a breakthrough in understanding algorithmic complexity, a very important topic in computer science. Knowing the complexity of algorithms allows us to answer questions such as how long will a program run on an input? How much space will it take? Is the problem solvable?
“My approach was completely different from the usual approaches used,” Fokas said. “I first embed the R zeta function inside a bigger problem, namely I find that the R zeta function satisfies a very important problem in complex analysis called the Riemann-Hilbert problem. Then, I compute the large t behavior of this problem. Besides being conceptually novel, this approach is technically very hard due to the analysis of the aforementioned Riemann-Hilbert problem.”