An adelic causality problem related to abelian L-functions

TL;DR

This paper links the Riemann Hypothesis for abelian L-functions to a causality condition in an adelic Lax-Phillips scattering framework, offering a novel perspective on a classical problem.

Contribution

It introduces an adelic causality problem associated with abelian L-functions, connecting it to the Riemann Hypothesis and extending previous work by Tate, Iwasawa, and Connes.

Findings

Causality in the scattering framework holds iff the Riemann Hypothesis is true for all abelian L-functions.

Provides an adelic variation of Nyman and Beurling's Hilbert space approach.

Links adelic aspects to classical number theory conjectures.

Abstract

I associate to a global field K a Lax-Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L-functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes.