THE SIZE OF (q; q)n FOR q ON THE UNIT CIRCLE

There is increasing interest in q series with jqj = 1. In analysis of these, an important role is played by the behaviour as n!1 of (q; q)n = (1 q)(1 q):::(1 q): We show, for example, that for almost all q on the unit circle log j(q; q)nj = O(logn) i¤ " > 0. Moreover, if q = exp(2 i ) where the continued fraction of has bounded partial quotients, then the above relation is valid with " = 0. This provides an interesting contrast to the well known geometric growth as n!1 of k (q; q)n kL1(jqj=1) : 1. Statement of Results There are a growing number of applications of q series with jqj = 1 and q 6= 1 in number theory, Pade approximation, continued fractions, ... [3-7], [15-17], [1920], [22-24]. In analysis of a continued fraction of Ramanujan [16], the author was confronted with the need to analyse the behaviour as n!1 of (1.1) (q; q)n := (1 q)(1 q):::(1 q) for q on the unit circle. Obviously, the size of (q; q)nwill play an important role in the development of q-series for jqj = 1. To …rst order, the answer to this question is provided by an old identity: (1.2) 1 X n=0 z (q; q)n = exp 1 X n=1 z n(1 qn) ! : Hardy and Littlewood showed [10] that this identity remains valid even for jqj = 1, that is both power series above have the same radius of convergence. Thus lim inf n!1 j(q; q)nj = lim inf n!1 j1 qnj : It follows easily from the elementary theory of diophantine approximation, that if (1.3) q = exp(2 i ); 2 [0; 1) then for almost all 2 [0; 1] (and in fact except for in a set of Hausdor¤dimension 0 and logarithmic dimension 2 [14]), lim inf n!1 j(q; q)nj = 1: Date : September 17, 1998. 1991 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. q-series, diophantine approximation. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1