Denominator in Average Precision at K

Question

Question 1:

I've seen slightly different definitions for "Average Precision @ K" in the context of recommender system measures. One definition I've seen is:

AP@K is the sum of precision@K for different values of K divided by the total number of relevant items in the top K results.

But according to the ml_metrics implementation, the denominator is the minimum of k and the number of relevant items.

Which is correct? (Or how do you decide which one to choose?)

Question 2:

What do you do if the total number of relevant items is 0? The denominator would be zero. In that case, do you just set the AP@k to 0, 1, or something else? Dependng on whether I choose 0 or 1, it would have a drasically different effect on Mean Average Precision (the average of AP@ks over all queries).

Question 3:

How is AP@K different from average precision in the sense of the area under the precision-recall curve (one implementation being the sklearn one here)? I get that with AP@K, you limit the number of retrieved items to look at, but is that the only difference?