17,914 research outputs found
Power partitions and saddle-point method
For , denote by the number of partitions of an integer
into -th powers. In this note, we apply the saddle-point method to
provide a new proof for the well-known asymptotic expansion of . This
approach turns out to significantly simplify those of Wright (1934), Vaughan
(2015) and Gafni (2016).Comment: An error in the proof of Lemma 2.3 has been correcte
On the number of summands in a random prime partition
Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012We 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.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
Gentile statistics and restricted partitions
In a recent paper (Tran et al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions pks (n), which is the number of partitions of an integer n into the summand of sth powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well-known asymptotic results for s = 1 apart from yielding more general results for arbitrary values of s
On the Quantum Density of States and Partitioning an Integer
This paper exploits the connection between the quantum many-particle density
of states and the partitioning of an integer in number theory. For bosons
in a one dimensional harmonic oscillator potential, it is well known that the
asymptotic (N -> infinity) density of states is identical to the
Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of
integers. We show that the same statistical mechanics technique for the density
of states of bosons in a power-law spectrum yields the partitioning formula for
p^s(n), the latter being the number of partitions of n into a sum of s-th
powers of a set of integers. By making an appropriate modification of the
statistical technique, we are also able to obtain d^s(n) for distinct
partitions. We find that the distinct square partitions d^2(n) show pronounced
oscillations as a function of n about the smooth curve derived by us. The
origin of these oscillations from the quantum point of view is discussed. After
deriving the Erdos-Lehner formula for restricted partitions for the case
by our method, we generalize it to obtain a new formula for distinct restricted
partitions.Comment: 17 pages including figure captions. 6 figures. To be submitted to J.
Phys. A: Math. Ge
POWER PARTITIONS AND SADDLE-POINT METHOD
For k 1, denote by p k (n) the number of partitions of an integer n into k-th powers. In this note, we apply the saddle-point method to provide a new proof for the well-known asymptotic expansion of p k (n). This approach turns out to significantly simplify those of Wright (1934), Vaughan (2015) and Gafni (2016)
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