# Growth of the product ${\prod}_{j=1}^{n}(1-{x}^{{a}_{j}})$

J. P. Bell; P. B. Borwein; L. B. Richmond

Acta Arithmetica (1998)

- Volume: 86, Issue: 2, page 155-170
- ISSN: 0065-1036

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topJ. P. Bell, P. B. Borwein, and L. B. Richmond. "Growth of the product $∏^n_{j=1} (1-x^{a_j})$." Acta Arithmetica 86.2 (1998): 155-170. <http://eudml.org/doc/207187>.

@article{J1998,

abstract = {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.},

author = {J. P. Bell, P. B. Borwein, L. B. Richmond},

journal = {Acta Arithmetica},

keywords = {supremum norm; partition; growth; power series; polynomials},

language = {eng},

number = {2},

pages = {155-170},

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

url = {http://eudml.org/doc/207187},

volume = {86},

year = {1998},

}

TY - JOUR

AU - J. P. Bell

AU - P. B. Borwein

AU - L. B. Richmond

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

JO - Acta Arithmetica

PY - 1998

VL - 86

IS - 2

SP - 155

EP - 170

AB - 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.

LA - eng

KW - supremum norm; partition; growth; power series; polynomials

UR - http://eudml.org/doc/207187

ER -

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