Indian Mathematician Takes Shot At Proving Riemann Hypothesis 160
First time accepted submitter jalfreize writes "Indian Mathematician Rohit Gupta (known by the moniker @fadesingh on twitter) has announced an online workshop which he intends to 'conclude by attacking an important problem in front of (the participants), in public view.' The problem is the Riemann Hypothesis, first proposed in 1859. Rohit outlines his approach based on quasicrystals first outlined by Freeman Dyson. His audacious plan, coupled with this recent news about quasicrystals, has kicked up a storm of interest in the Indian twitterverse."
Re:He had help: (Score:4, Informative)
I'm sure he was joking around when he said it.
(Just like I was joking around when posting that. But just to reassure you, Ramanujan was one of the greats of mathematics. And there is a long tradition of great Indian scientists doing mathematics going back centuries BC. My personal fave Indian scientist is J. C. Bose who was working wtih 60 Ghz radio waves in the late 1800s.)
What this means and how seriously is this (Score:5, Informative)
The Riemann Hypothesis is roughly the following: There's a function defined by zeta(s)= 1 + 1/1^s + 1/2^s + 1/3^s + 1/4^s... You can make this function make sense for any complex number as long as it has real part greater than 1. However, this series does not converge for s less than or equal to 1 1. However, it turns out that this function has what is called an "analytic continuation" http://en.wikipedia.org/wiki/Analytic_continuation [wikipedia.org]. Essentially it is possible to make a function on the complex plane that is smooth (in the sense of being infinitely differential), and agrees with this function everywhere. This function is known as the Riemman Zeta Function. The only caveat is that one cannot give a sensible definition for the value at s=1. (Essentially as s gets near 1, the value of the function goes to infinity).
It turns out that the behavior of zeta is deeply related to the prime numbers because of another way of writing the above series as a product over the prime numbers. So for example, a major triumph of 19th century math was showing that this function was not zero anywhere on the line with real part of s =1. This implied an approximate estimate for the size of the nth prime number called the prime number theorem. http://en.wikipedia.org/wiki/Prime_number_theorem [wikipedia.org].
The Riemann hypothesis is a much stronger claim about where the zeta function is zero. It turns out that it is very easy to show that the zeta function is zero at every negative even integer. These are the trivial zeros, There are other, more difficult to locate zeros. The hypothesis conjectures that these zeros all lie on the line with real part equal to 1/2. That is, every zero is of the form 1/2 + it where t is some real number. If this is true many nice things will follow.
Most people who have thought about this question believe that it is true. There's a lot of evidence for it, such as the fact that literally billions of zeros have been located on this line, and the fact that it can be shown in a certain sense that almost all the non-trivial zeros lie near the 1/2 line. We also know that in a certain sense a positive fraction of the non-trivial zeros need to lie on the line (one needs to be careful here with what this means since there are infinitely many such zeros).
There are a lot of current attempts to prove the Riemann Hypothesis, and some very serious mathematicians think that the quasicrystal approach might work. Right now there are a lot of different approaches, including some which connect the hypothesis to certain claims in quantum mechanics. However, at this point, despite the many attempts there are a lot of weaker claims that we can't prove that we'd expect to prove before the Riemann hypothesis. It turns out that all the non-trivial zeros need to have a real part strictly between 0 and 1. But we can't even prove what essentially amounts to the worst case scenario, that there are zeros arbitrarily near the 0 and 1 lines. I expect this to be dealt with well before the full Riemann hypothesis is proven. There are other weaker hypotheses that are implied by RH that one would also expect to be proven first. So far the quasicrystal approach sounds promising but has had very little in the way of actual fruit. But this may just be that it is a relatively new set of tools and they need to be carefully developed. Overall, I'd be surprised if this project works simply because even if a quasicrystal approach eventually proves the full result it will require so much stuff to happen before hand.
Re:What this means and how seriously is this (Score:4, Informative)
There's a function defined by zeta(s)= 1 + 1/1^s + 1/2^s + 1/3^s + 1/4^s...
Ok. Pretty basic mistake I made here. This series should not have the initial 1. Not sure why I wrote that. So one has zeta(s)= 1/1^s + 1/2^s + 1/3^s + 1/4^s...