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

Tom 86 / 1998

J. Bell, P. Borwein, L. Richmond Acta Arithmetica 86 (1998), 155-170 DOI: 10.4064/aa-86-2-155-170

Streszczenie

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.

Autorzy

  • J. Bell
  • P. Borwein
  • L. Richmond

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