The Riemann zeta function, the Laplacian, and how to recover Lebesgue integration from the pole of a zeta function (Part 1)

Abstract: 

The Riemann zeta function is most famous for the, as yet undtermined, zeros of its analytic continuation. Much less focus is given to the residue of the simple pole of the Riemann zeta function at z=1 and the relation to the harmonic series.

The first part, of this two part talk, introduces a generalised notion of the Riemann zeta function using compact operators on a separable Hilbert space. We show how the residue of the zeta function of a compact operator can be identified with the Dixmier trace, a non-normal trace used as the foundation for the "noncommutative integral" in Alain Connes' theory of Noncommutative Geometry. We highlight the contributions which Fedor Sukochev (UNSW) and his collaborators made to the area, and the role a joint paper between Fedor, myself and Aleksandr Sedaev (Vorenzh, Russia) has played in the categorisation of zeta functions. 

The second part, given in the Analysis Seminar on Wednesday 8th April, considers specific zeta functions associated to the Laplacian of a compact Riemannian manifold. For example, the zeta function associated to the Laplacian on the circle is just a multiple of the Riemann Zeta Function. We introduce zeta functions weighted by bounded operators and show how, in recent work, we solved a problem concerning the "noncommutative integral" that has been open for 20 years.

Namely, we recover the Lebesgue integral of any bounded (and then any) integrable function as the residue of a zeta function. If time permits, we will introduce the integral on the "noncommutative torus" and show, using the same technique, that it can be recovered from the zeta functions associated to the "noncommutative Laplacian".