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 Fq outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably Re(s) < 1 2 − 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) ≤ 1 using recent advances on the Shifted Moments Conjecture over function fields.