Abstract
Originally predicted by Ramanujan in 1916 for the discriminant function, the Ramanujan conjecture is a very deep statement concerning the size of the Fourier coefficients of cusp forms. The generalized Ramanujan conjecture expects that a generic unitary cuspidal representation of a reductive group over a global field should be locally tempered. While this conjecture is largely open to date, it is established for certain cases.
In this survey talk we explain some novel applications of the proven cases to explicit constructions of pseudorandom objects. These include (a) Ramanujan graphs and Ramanujan complexes, (b) points uniformly distributed on spheres, and (c) Golden Gate sets in quantum computing. The Ramanujan conjecture is closely tied to the Riemann Hypothesis, and the Ramanujan graphs/complexes can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. This reveals a very interesting connection between number theory and combinatorics.