- Tytuł:
- Growth of the product $∏^n_{j=1} (1-x^{a_j})$
- Autorzy:
-
Bell, J.
Borwein, P.
Richmond, L. - Powiązania:
- https://bibliotekanauki.pl/articles/1390697.pdf
- Data publikacji:
- 1998
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Opis:
-
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. - Źródło:
-
Acta Arithmetica; 1998, 86, 2; 155-170
0065-1036 - Pojawia się w:
- Acta Arithmetica
- Dostawca treści:
- Biblioteka Nauki
Artykuł