On the quantum density of states and partitioning an integer
Abstract
This paper exploits the connection between the quantum manyparticle density of states and the partitioning of an integer in number theory. For N bosons in a onedimensional harmonic oscillator potential, it is well known that the asymptotic ( N→∞) density of states is identical to the HardyRamanujan 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 powerlaw spectrum yields the partitioning formula for p^{s}( n), the latter being the number of partitions of n into a sum of sth 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 ErdosLehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.
 Publication:

Annals of Physics
 Pub Date:
 May 2004
 DOI:
 10.1016/j.aop.2003.12.004
 arXiv:
 arXiv:mathph/0309020
 Bibcode:
 2004AnPhy.311..204T
 Keywords:

 03.65.Sq;
 02.10.De;
 05.30.d;
 Semiclassical theories and applications;
 Algebraic structures and number theory;
 Quantum statistical mechanics;
 Mathematical Physics;
 Mathematics  Number Theory;
 Nuclear Theory
 EPrint:
 17 pages including figure captions. 6 figures. To be submitted to J. Phys. A: Math. Gen