The Ramanujan conjecture and its applications

Abstract

<jats:p>In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed.</jats:p> <jats:p>This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.</jats:p>

From Ramanujan graphs to Ramanujan complexes

“…Finally, another interesting hint to a connection with number theory: Ihara defined the notion of Zeta function of a k-regular graph X, and Sunada observed that X is Ramanujan if and only if this Zeta function satisfies 'the Riemann hypothesis'. We refer the reader to the survey [11] for more details.…”