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An effective bound for the partition function. (English) Zbl 1451.11112

Let \(p(n)\) denote the number of partitions of \(n\). In [Adv. Math. 246, 198–219 (2013; Zbl 1304.11021)], J. H. Bruinier and K. Ono derive a formula of \(p(n)\) as a finite sum of algebraic numbers which lie in the usual discriminant \(1-24n\) ring class field. The main result in the paper under review is a simplified estimate of the formula Bruinier and Ono of \(p(n)\) with an explicit bound on the error term. They prove that \[p(n)=M(n)+E(n),\] for all \(n\in\mathbb{N}\), where the main term \(M(n)\) is obtained from the formula Bruinier and Ono, and the error term \(E(n)\) satsify \[|E(n)|\le 5.6044\times 10^{23}\cdot \frac{H(1-24 n)}{24n-1},\] where \(H(D)\) is the Hurwitz-Kronecker class number.

MSC:

11P82 Analytic theory of partitions

Citations:

Zbl 1304.11021
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References:

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