The Ramanujan conjecture: from theory to applications

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 automorphic 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 (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. We shall also explain the connection between Ramanujan graphs/complexes and the Riemann Hypothesis satisfied by their associated zeta function.