We estimate the maximum of on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when is or when 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 is j. In contrast we show, under fairly general conditions, that the maximum is less than , 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 is asymptotically equal to the number of such partitions with an odd number of parts when satisfies these general conditions.
@article{bwmeta1.element.bwnjournal-article-aav86i2p155bwm,
author = {J. P. Bell and P. B. Borwein and L. B. Richmond},
title = {Growth of the product $$\prod$^n\_{j=1} (1-x^{a\_j})$
},
journal = {Acta Arithmetica},
volume = {84},
year = {1998},
pages = {155-170},
zbl = {0918.11054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav86i2p155bwm}
}
J. P. Bell; P. B. Borwein; L. B. Richmond. Growth of the product $∏^n_{j=1} (1-x^{a_j})$
. Acta Arithmetica, Tome 84 (1998) pp. 155-170. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav86i2p155bwm/
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