On the Northcott property of zeta functions over function fields

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 $1/2≤s≤1$, 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.

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