The Size of (q; q)nforqon the Unit Circle Original Research Article

There is increasing interest inq-series with q=1. In analysis of these, all important role is played by the behaviour asn→∞ of(q; q)n=(1−q)(1−q2)…(1−qn). We show, for example, that for almost allqon the unit circlelog (q; q)n=O(log n)1+var epsiloniffvar epsilon>0. Moreover, ifq=exp(2πiτ) where the continued fraction ofτhas bounded partial quotients, then the above relation is valid withvar epsilon=0. This provides an interesting contrast to the well known geometric growth asn→∞ ofdouble vertical bar(q; q)ndouble vertical barL∞(q=1).