Hochschild homology, trace map and $\zeta$-cycles

TL;DR

This paper explores two spectral models for the zeros of the Riemann zeta function, connecting them to geometric and cohomological frameworks, with the Scaling Site as a key parameter space for understanding $ta$-cycles.

Contribution

It introduces novel spectral realizations of zeta zeros using Laplacian and sheaf cohomology, with the Scaling Site as a new geometric setting.

Findings

Spectral realization involving a Laplacian related to the prolate wave operator.

Spectral realization using sheaf cohomology focused on critical zeros.

The Scaling Site encodes the stability of $ta$-cycles through coverings.

Abstract

In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology. The novelty is that the base space is the Scaling Site playing the role of the parameter space for the $\zeta$-cycles and encoding their stability by coverings.