Growth of the product $∏^n_{j=1} (1-x^{a_j})$

J. Bell¹ ; P. Borwein¹ ; L. Richmond¹

¹

Acta Arithmetica, Tome 86 (1998) no. 2, pp. 155-170

Résumé

We estimate the maximum of $∏^n_{j=1} |1 - x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_j$ is j.    In contrast we show, under fairly general conditions, that the maximum is less than $2^n/n^r$, where r is an arbitrary positive number. One consequence is that the number of partitions of m with an even number of parts chosen from $a₁,...,a_n$ is asymptotically equal to the number of such partitions with an odd number of parts when $a_i$ satisfies these general conditions.

Détail

DOI : 10.4064/aa-86-2-155-170