Hitting a prime by rolling a die with infinitely many faces

Alon and Malinovsky recently proved that it takes on average $2.42849\dots$ rolls of fair six-sided dice until the first time the total sum of all rolls arrives at a prime. Naturally, one may extend the scenario to dice with a different number of faces. In this paper, we prove that the expected stopping round in the game of Alon and Malinovsky is approximately $\log M$ when the number $M$ of die faces is sufficiently large.