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On the number of summands in a random prime partition

Abstract

We study the length (number of summands) in partitions of an integer into primes, both in the restricted (unequal summands) and unrestricted case. It is shown how one can obtain asymptotic expansions for the mean and variance (and potentially higher moments), which is in contrast to the fact that there is no asymptotic formula for the number of such partitions in terms of elementary functions. Building on ideas of Hwang, we also prove a central limit theorem in the restricted case. The technique also generalizes to partitions into powers of primes, or even more generally, the values of a polynomial at the prime numbers.

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Correspondence to Dimbinaina Ralaivaosaona.

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This project is supported by the German Academic Exchange Service (DAAD), in association with the African Institute for Mathematical Sciences (AIMS). Code No. A/09/04406.

Communicated by Christian Krattenthaler.

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Ralaivaosaona, D. On the number of summands in a random prime partition. Monatsh Math 166, 505–524 (2012). https://doi.org/10.1007/s00605-011-0337-x

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Keywords

  • Asymptotic expansions
  • Limit distribution
  • Mellin transform
  • Prime partitions
  • Saddle point method

Mathematics Subject Classification (2010)

  • Primary 05A17
  • Secondary 11P82