On the Northcott property of zeta functions over function fields

TL;DR

This paper investigates the Northcott property of zeta functions over function fields, identifying where it holds outside the critical strip and providing partial results within it, using advanced techniques in number theory.

Contribution

It determines the conditions under which the Northcott property holds for zeta functions over function fields, extending previous definitions to new regions.

Findings

Northcott property holds outside the critical strip for certain values.

Partial results obtained for real and complex values inside the critical strip.

Utilizes recent advances on the Shifted Moments Conjecture in the analysis.

Abstract

Pazuki and Pengo defined a Northcott property for special values of zeta functions of number fields and certain motivic $L$-functions. We determine the values for which the Northcott property holds over function fields with constant field $\mathbb{F}_q$ outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably $Re (s) < \frac{1}{2} - \frac{\log 2}{\log q}$ and for $s$ real such that $1/2 \leq s \leq 1$, and we obtain a partial result for complex $s$ in the case $1/2< Re(s)\leq 1$ using recent advances on the Shifted Moments Conjecture over function fields.