Some Restricted Partition Functions Original Research Article

We show that the supremum norm on the unit disk, {q ≤ 1}, of the nth partial product of ∏∞k = 1, p[formula]k(1 − qk) is asymptotic to pn/(p − 1) for p = 2, 3, 5, 7, 11, and 13 (but not for any p > 15). This, for these primes, is an asymptotically best possible result since if α1, ..., αn are integers none of which are divisible by p then Πnk = 1(1 − qαk){q = 1} ≥ pn/(p − 1).