Rohit Gupta, an Indian Mathematician, is trying to solve a difficult math problem and he intends to get you involved too! The problem the Riemann Hypothesis – is one of the toughest problems ever and no one knows a solution to it yet. It regards the distribution of prime numbers on the real number line, but more of that in just a bit. More interesting is the person himself and his goal. He intends to use Quasi-crystals (this year’s Nobel recipient in Chemistry, by the way) to solve this so-far intractable problem. The call is out for anyone who wants to join him anyone really, whether a mathematician or not!
He not only plans to attack the problem, he wants to do so in public. To do that he’ll conduct workshops dubbed KNK103, with KNK standing for Kali and Kaleidoscope’. But, he points out, this isn’t just a workshop it’s much more, it’s a mathematical expedition’.
The Riemann Hypothesis: The Beast and the Legend around it
The Riemann Hypothesis is probably the toughest Math problem known. We all know that prime numbers are scattered all over the number line, but we don’t know whether there is any pattern in this or not! The prime number theorem states that all prime numbers upto some integer n’ lie below a maximum line, given by different formulae.
The statement of the Riemann Hypothesis is easy enough to state all non-trivial zeroes of the Riemann Zeta function lie on the critical line whose real part is ½.What this cryptic statement means is difficult to explain. The Riemann zeta function is shown below, valid for only Real(s)>1. (Remember, s can be a complex number.)
The function has Analytic continuations, which are too technical to consider here. The analytic continuation has trivial zerosfor s=-2,-4 etc. There are also non-trivial zeros (i.e. where the value of the function vanishes) of the Riemann Zeta Functions (and its analytic continuation), apart from these zeros. The Riemann Hypothesis is that these zeros should lie on a line, whose real part is ½. Don’t worry if you’ve not understood much of this section, just keep reading.
Here’s a Mathematica demonstration of the Riemann Hypothesis
The Riemann Hypothesis is listed at number 8 on Hilbert’s list of 23 greatest unsolved problems, and has on its head a prize of US$ 1 million, announced by the Clay Mathematical Institute. The hypothesis has been tested for 1 trillion zeros, but no general proof exists, i.e. how do you know there isn’t a zero that doesn’t lie on the Re(s)= ½ line beyond the trillionth zero?
The solution to the Riemann Hypothesis is thought to be a key step in many problems in topology, number theory and even cryptography. It is rumored that internet security will be revolutionized if the problem is solved.
Role of Quasicrystals!!
Here comes the most interesting part: Quasicrystals. Rohit Gupta wants to use quasi-crystals to attack the problem. He plans to use the energy levels in these crystals and look into the pattern to see if it fits the pattern of the zeros of the Riemann zeta function on the critical Re(s)= ½ line.
The idea is not new and he admits to being inspired by a conversation between Freeman Dyson and Hugh Montgomery. They noticed that the energy levels of heavy atoms, like Erbium, have energy levels which follow the pattern of the zeros of the Riemann Zeta function. Here’s Marcus du Sautoy’s account:
They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar. It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.
Yes!! Quantum mechanical solutions might already have solved the Riemann Hypothesis, loosely speaking. Gupta mentions If the Riemann Hypothesis is true, then the zeros of the Riemann Zeta function form a quasi crystal.Freeman Dyson said of this approach:
Suppose that we find one of the quasi-crystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.
Who can participate in the grand project? How about you?
Gupta wants everyone to participate in this. He wants everyone from designers and copywriters to paranoid politicians and Polar bears with Polaroid Cameras (His own words)!
The introductory fee is Rs. 4900 (about $100) for lifetime. The workshops will be held online. There is no last date to register as the project is not time bound. He can be found on Twitter as @fadesingh and his gmail id is fadebox at gmail . com. You can register here.
I’ll use Freeman Dyson’s quote taken from Gupta’s page to finish this article: the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible.