Since this is year end holiday time, I got a fair share of my time to read up on several random topics. Amidst the many sad out comings in the past couple of weeks starting from the shocking events from Newtown Connecticut elementary school shootout, then on had a few more disturbing news all over the world, particularly this one from the other side of globe. A rather non disturbing news is on the new claim on proving/establishing the so called Ramanujan’s deathbed conjecture. I was doing a bit of follow up read up on this to first understand the premise of the problem in news.
OK, the subject deals with what is known as mock theta functions (Note: There is no credible Wikipedia entry to it and hence I cannot add one here. Also remember that this mock theta functions are not the same as Ramanujan’s theta functions; Just to not confuse these subtle mathematical terms!). These are not quite the Riemann Zeta functions, which are well known, but more similar to the Jacobi theta functions. The mathematical prodigy Srinivasa Ramanujan while on deathbed had shared some quick tidbits to the famous Cambridge mathematician (and I should say a hard core cricket enthusiasts! If you guys have not read his memoir A mathematician’s Apology already, step out and grab it soon. A fabulous quick read that!) G.H.Hardy who was paying a visit to his now famous student. After their famous meet up (or may be during their meeting itself), somewhere in 1920, before his death, Ramanujan wrote a letter to Hardy in Cambridge in which he introduced as many as seventeen new functions. These functions, as per Ramanujan, had shared some distinct similarities with the theta functions (The Zeta functions were itself had been studied before before by mathematicians like Jacobi, Riemann and several others. Now of course the connection between Jacobi theta functions and Riemann Zeta functions is already established!).
The Jacobi theta function itself is pretty complicated to lesser mortals like us. It looks harmless for a starter, but the properties, its various avatars and connections to many real world applications is what makes it monstrous. And then there are also its generalized cousin Riemann Zeta functions , which as well, appear as a simple and elegant looking form for , for higher changes the form beyond what we can fathom (For example, it is not even known whether for larger such a number is transcendental!). I remember playing with (I think it was 2000 or 2001) a proof of using some integral calculus in two variables, which again turns out to be rather well known as easy. There are other ways to prove as well for , but it stops being that simplistic there!) Anyway, Jacobi’s theta function has the form , which in words can be roughly stated as some form of elliptic form of exponentials.
Much like Fermat did for his famous last theorem, Ramanujan too didn’t give much hints beyond listing and stating that they are elegant. For example, he didn’t reveal where they come from or how to formulate them or for that matter what use having them. Like many of his other famous mathematical findings, Ramanjunan, a self taught genius, made a quick listing of these. Rest of the curiosity has enthralled and haunted the mathematical minds of coming generations, all over the world. A brilliant spark from the great genius helped to stir an intellectual stir for almost 10 years!
The latest is that, two Wisconsin mathematician, Ken Ono and University of Cologne professor Kathrin Bringmann came up with a framework explaining what mock theta functions are and how to derive them. They connect modular forms or more specifically the mass forms to Ramanujans’ functions. If you recall, the modular forms also played a big part in the proof of the other famous problem of this kind Fermat’s last theorem.
It will take a while for me to understand (if at all I manage to digest it) all the details of this new claim. I must say, it is likely that I will fail to comprehend it in full . As always, I will be curious to read it up and grab as much as I can. As and Indian, the connection to India is always an interesting pursuit!
I leave you with a nice documentary video on Ramanjuan that I found in youtube. If you have not see this already, check this out. Pretty good documentary, unfortunately tarred by a lot of useless and unrelated comments.