Written: May 11, 2016
I just finished reading Ken Ono's fascinating memoir "My Search for Ramanujan" (joint with Amir Aczel), where he narrates how both the life and mathematics of Srinivasa Ramanujan inspired him to become the world-class mathematician that he is today, after being a high-school "drop-out" and an unmotivated undergraduate.
While it is really nice that one can get inspired by a great mathematician like Ramanujan, one should also be honest, and ask, "Would Ramanujan have liked the math I do?". While I have no rigorous proof, of course, I am sure that, while he probably would have been polite about it, and been pleased that his work inspired fancy mathematicians to do fancy mathematics, he would, most probably not cared for it, if nothing else, because (like myself), he would not have understood it, and, (also like myself), would not have wanted to understand it.
Ono tells us that in the last year of his life, Ramanujan introduced the so-called "Mock-theta functions", and it
was an "enigma" to "make sense" out of his cryptic notes. This was, allegedly,
"accomplished" by Sander Zwegers, and further elaborated by
Ken Ono, his many students and postdocs, and others.
I am willing to believe that Zwegers' theory is a beautiful piece of fancy pure mathematics, but I am also
sure that this is not what Ramanujan had in mind, and he definitely would not have liked it.
Perhaps G.H. Hardy would have. For Ramanujan, a "mock theta function" was simply a q-series, and he did not know, and did not care, for the theory of modular functions and forms.
The true Ramanujan, before he was "stolen" by the Ken Ono fancy math gang, was an experimental mathematician, who dealt with (what are now called) q-series rather than "modular forms". He abhorred the abstract and loved the concrete, and the true heirs of Ramanujan's heritage are concrete mathematicians, like Frank Garvan, Mike Hirschhorn, Bruce Berndt (btw, I was taken aback by the dismissive tone in Ono's book towards Berndt's work as only doing "specific problems", while Ono (according to himself) looks at the "big picture"), and George Andrews (that while a great concretian, is an "abstract groupie", and, unlike myself, admires (vicariously) fancy stuff).
So, by all means, you fancy number theorists, it is OK to be inspired by Ramanujan (or by anyone else), but do not claim to have followed his footsteps.
It turns out that the fancy "modular machine" may be also viewed computationally and concretely, as brilliantly shown by Cristian-Silviu Radu, and thereby made much more appetizing, and this version Ramanujan would have probably liked. In any case, I like it!
People wonder what Ramanujan would have done with a computer. We will never know exactly, but we can try to emulate Ramanujan's concrete, manipulatorics, style, and teach it to our computers, and that would become the true Ramanujan heritage.