On the Northcott property of zeta functions over function fields

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 $1/2\le s\le 1$, and we obtain a partial result for complex $s$ in the case $1/2<Re(s)\le 1$ using recent advances on the Shifted Moments Conjecture over function fields.