On the Northcott property of Dedekind zeta functions

TL;DR

This paper investigates the Northcott property for complex values of Dedekind zeta functions, revealing nuanced behavior near trivial zeros using analytic and computational methods.

Contribution

It extends the understanding of the Northcott property to complex evaluations of Dedekind zeta functions, including delicate behaviors near trivial zeros.

Findings

Behavior near trivial zeros is more complex than in function fields.

Analytic and computer-assisted techniques are effective in studying these properties.

Surprising phenomena occur in the neighborhood of trivial zeros.

Abstract

The Northcott property for special values of Dedekind zeta functions and more general motivic $L$-functions was defined by Pazuki and Pengo. We investigate this property for any complex evaluation of Dedekind zeta functions. The results are more delicate and subtle than what was proven for the function field case in previous work of Li and the authors, since they include some surprising behavior in the neighborhood of the trivial zeros. The techniques include a mixture of analytic and computer assisted arguments.