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Some general problems on the number of parts in partitions

L. B. Richmond — 1994

Acta Arithmetica

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

J. P. Bell; P. B. Borwein; L. B. Richmond — 1998

Acta Arithmetica

We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ 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...

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The moments of partitions, II

The moments of partitions, I

Some general problems on the number of parts in partitions

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

Advanced Search

Formula preview

Currently displaying 1 – 4 of 4

The moments of partitions, II

The moments of partitions, I

Some general problems on the number of parts in partitions

Growth of the product ∏ j = 1 n ( 1 - x a j )

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$