Ramanujan

On October the 22nd 2016, I watched a movie called “The Man Who Knew Infinity”, which memorizes an Indian mathematician Srinivasa Ramanujan (1887-1920). It describes what would happen if people from different cultures meet each other. There are some astonishingly good and definitely bad. I could also notice bright side and dark side of such a cultural confliction. The English is as easy as non-native speakers can understand, and the plot is traceable for people who are not familiar with mathematics.

 

Ramanujan proposed Ramanujan conjecture among prime p:

{\Delta(z) = e^{2\pi iz}\prod_{n=1}^{\infty}(1 – e^{2\pi inz})^{24} = \prod_{n=1}^{\infty}\tau(n) e^{2\pi inz},}

{L(s, \Delta) = \prod_{p}L_p(s, \Delta),}

{L_p(s, \Delta) = \frac{1}{1 - \tau(p)p^{-s} + p^{11 - 2s}},}

{|\tau(p)| \leq 2p^{\frac{11}{2}};}

the partition function; mock theta function; or an Euler product of degree 2. He was invited to University of Cambridge by Godfrey Harold Hardy (1877-1947) in 1914, stayed there until 1919. The bad nutritional condition during World War I forced him, as a vegetarian, to suffer serious illness. After a year since he returned India, he died in his young age, at 32 years old. Although he was said not to know complex analysis so well due to the failure in proving prime number formula, even Hardy misunderstood him for a certain extent and the partition function by Hardy and Ramanujan ended up in a mere asymptotic formula.

 

Ramanujan conjecture (1916) is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It is for weight 12, results in 11 dimensional algebraic manifold and relates to Riemann hypothesis. The Ramanujan conjecture has contributed to various aspects of modern mathematics, including ideas of mathematical spaces in EGA, SGA by Alexander Grothendieck (1928-2014), the solution of Ramanujan conjecture by Pierre Deligne (1944- ), the solution of Fermat’s Last Theorem (via the Euler product of degree 2) and Sato-Tate conjecture.

 

According to the developing field of mock theta functions nowadays, Ramanujan was still productive until his death. George Neville Watson (1886-1965) wrote a poem that describes the final moments of Ramanujan as:

“Pale, beyond porch and Portal,

Crowned with calm leaves, she stands

Who gathers all things mortal

With cold immortal hands.” 

 

 参考文献：ラマヌジャンζの衝撃（現代数学社）