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

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

a ₁, . . ., a_{n}

is asymptotically equal to the number of such partitions with an odd number of parts when

a_{i}

satisfies these general conditions.

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J. 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 -

References

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