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On the number of prime factors of summands of partitions. (English) Zbl 1108.11078

The three principal results of this paper show that “for almost all partitions of an integer the sequence of parts satisfies similar arithmetic properties as the sequence of natural numbers.”
For example, let \(\Phi\) be a non-decreasing function with \[ \lim_{N \to \infty} \Phi(N) = + \infty. \] For \(n \to \infty\), for all but \(o(p(n))\) partitions of \(n\), the number of parts \(\gamma_j\) with \[ | \omega(\gamma_j) - \log \log n| > \Phi(n)\sqrt{\log \log n} \] is \(o(\sqrt{n}\log n)\). Here the \(\gamma_j\) are the parts of a partition \(\gamma = (\gamma_1,\dots,\gamma_s)\) of \(n\), and \(\omega(m)\) is the number of distinct prime factors of \(m\).
The analytic and probabilistic methods are applications/extensions of the authors’ previous work [Acta Math. Hung. 110, No. 4, 323–335 (2006; Zbl 1121.11070)].

MSC:

11P82 Analytic theory of partitions

Citations:

Zbl 1121.11070
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References:

[1] C. Dartyge, A. Sárközy, M. Szalay, On the distribution of the summands of partitions in residue classes. Acta Math. Hungar. 109 (2005), 215-237. · Zbl 1119.11061
[2] P. Erdős, J. Lehner, The distribution of the number of summands in the partitions of a positive integer. Duke Math. Journal 8 (1941), 335-345. · Zbl 0025.10703
[3] M. Szalay, P. Turán, On some problems of the statistical theory of partitions with application to characters of the symmetric group II. Acta Math. Acad. Sci. Hungar. 29 (1977), 381-392. · Zbl 0371.10034
[4] M. Szalay, P. Turán, On some problems of the statistical theory of partitions with application to characters of the symmetric group III. Acta Math. Acad. Sci. Hungar. 32 (1978), 129-155. · Zbl 0391.10031
[5] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2e édition. Cours spécialisés no 1, Société mathématique de France (1995). · Zbl 0880.11001
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