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Partitions sans petits sommants. (Partitions without small summands). (French) Zbl 0719.11067

A tribute to Paul Erdős, 121-152 (1990).
[For the entire collection see Zbl 0706.00007.]
Let p(n) denote the ordinary partition function, and denote by p(n,m) (resp. r(n,m)) the number of partitions of n into parts \(\leq m\) (resp. \(\geq m)\). In a previous paper [Number Theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. Janos Bolyai 51, 9-33 (1990; Zbl 0707.11072)] the authors have given an asymptotic formula for r(n,m) which is valid uniformly in the range \(m=o(n^{1/4})\). In the paper under review they extend this result by showing that the asymptotic estimate \[ r(n,m)\quad \sim \quad p(n)(m- 1)!(\frac{C}{2\sqrt{n}})^{m-1}\exp \{- (\frac{1}{8}C+\frac{1}{2C})\frac{m^ 2}{\sqrt{n}}\} \] holds with \(C=\pi \sqrt{2/3}\) uniformly in \(m\leq n^{1/3-\epsilon}\) as \(n\to \infty\), for any fixed \(\epsilon >0\). They also show that, for fixed \(\lambda >0\), log r(n,\(\lambda\sqrt{n})\sim g(\lambda)\sqrt{n}\) holds with a certain function g(\(\lambda\)) and investigate the analytic properties of g(\(\lambda\)). The latter result is analogous to a result of G. Szekeres [Q. J. Math., Oxf., II. Ser. 2, 85-108 (1951; Zbl 0042.041)] concerning the partition function p(n,m).

MSC:

11P82 Analytic theory of partitions