The Hilbert-Polya Conjecture and the Prolate Spheroidal Operator

Abstract

One possible way of proving the famous and important Riemann hypothesis would be to realize the Riemann zeta zeroes as the eigenvalues of some self-adjoint operator, using self-adjointness to show that the non-trivial zeroes all lie on the critical line. This is known as the Hilbert-Polya conjecture. Interpreting such a self-adjoint operator as an observable of some quantum system, one can try to describe the Riemann zeta zeroes in physical terms, making the Hilbert-Polya conjecture an interesting border case between mathematics and physics.

Major work on the Hilbert-Polya conjecture was done by Alain Connes. In his framework, the prolate spheroidal differential operator plays an important but auxiliary role. Before Connes’ work, this second-order differential operator was mainly known for its use in signal analysis, specifically in Slepian’s work on signals which are both time-limited and band-limited. In a more recent article, Connes and Moscovici show that the relation between the prolate spheroidal operator and the Riemann zeta function might be deeper than it seems. They show that a certain self-adjoint realization of the prolate spheroidal operator on the entire real line has discrete spectrum that is asymptotically similar to the squares of the Riemann zeta zeroes. This surprising discovery allows them to construct an operator which approximately solves the Hilbert-Polya conjecture.

Though Connes and Moscovici’s article is wonderful and inspiring, it is tersely written. Hence, in this bachelor thesis, details have been provided to the reasoning of the article. In addition, the physical interpretations of the Riemann zeta zeroes, which support the Hilbert-Polya conjecture, have also been presented.