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Partitions into distinct large parts. (English) Zbl 0824.11064

Let \(q_ m (n)\) denote the number of partitions of the positive integer \(n\) into distinct parts, each of which is at least \(m\). As to the relatively small values of \(m\), P. Erdős and the reviewer [Topics in classical number theory, Colloq. Budapest 1981, Colloq. Math. Soc. J. Bolyai 34, 397-450 (1984; Zbl 0548.10010)] observed that \(q_ m (n)\sim q_ 1 (n)/ 2^{m-1}\) for \(1\leq m\leq n^{1/5}\) and \(n\to\infty\). P. Erdős, J.-L. Nicolas and the reviewer [Number theory (Ulm, 1987), Lect. Notes Math. 1380, 19-30 (1989; Zbl 0679.10013)] proved that \(q_ m (n)\sim q_ 1 (n) \prod_{j=1}^{m-1} (1+ \exp (-\pi j(12n)^{- 1/2}) )^{-1}\) for \(m\leq n^{3/8- \varepsilon}\), \(\varepsilon>0\).
In the paper under review the authors give an asymptotic estimate of \(q_ m (n)\) as \(n\to \infty\), valid for \(m= o(n \log^{-9} n)\). However, the main term of the estimate for \(q_ m (n)\) in Theorem 1 involves a parameter which is not given explicitly in terms of \(m\) and \(n\). Theorem 2 yields an explicit asymptotic estimate for \(q_ m (n)\) which is valid for \(m\) relatively large compared with \(n^{1/2}\) and involves the inverse of the function \(x^{-2} \int_ x^ \infty y(1+ \exp (y))^{-1} dy\).
As a corollary, the authors obtain a family of asymptotic estimates for \(q_ m (n)\) in terms of elementary functions, each estimate being valid for large \(m\) in a specified interval of length at least \(n^{1/3}\). Finally, Theorem 3 refers to \(m=o (n^{1/3})\).

MSC:

11P82 Analytic theory of partitions
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