Counterexamples for Türkelli's Modification on Malle's Conjecture

We give counterexamples for the modification on Malle’s Conjecture given by Türkelli. Türkelli’s modification on Malle’s conjecture is inspired by an analogue of Malle’s conjecture over a function field. As a consequence, our counterexamples demonstrate that the $b$ constant can differ between function fields and number fields. We also show that Klüners’ counterexamples give counterexamples for a natural extension of Malle’s conjecture to counting number fields by product of ramified primes. We then propose a refined version of Malle’s conjecture which implies a new conjectural value for the constant $b$ for number fields.