Some Restricted Partition Functions

Abstract

We show that the supremum norm on the unit disk, {| q| ≤ 1}, of the nth partial product of ∏ ∞ k = 1, p[formula] k (1 − q k ) is asymptotic to p n/(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 α 1, ..., α n are integers none of which are divisible by p then ||Π n k = 1 (1 − q α k )|| {| q| = 1} ≥ p n/( p − 1) .