Attipat Krishnaswami Ramanujan (1929–1993)

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...

Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series

In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering. Norbert Wiener forged a powerful tool for electrical engineers by combining two distinct branches of...

Ramanujan-nagell Cubics

A well-known result of Beukers [3] on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity |x2 − 2n|. In this paper, we derive an inequality of the shape |x3 − 2n| ≥ x4/3, valid provided x3 6= 2n and (x, n) 6= (5, 7), and then discuss its implications for a variety of Diophantine problems.

Ramanujan Series Upside-down

We prove that there is a correspondence between Ramanujan-type formulas for 1/π and formulas for Dirichlet L-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.