Semiclassical analysis of distinct square partitions

We study the number P(n) of partitions of an integer n into sums of distinct squares and derive an integral representation of the function P(n). Using semiclassical and quantum statistical methods, we determine its asymptotic average part P-as(n), deriving higher-order contributions to the known leading-order expression [Tran et al.,Ann. Phys. (NY) 311, 204 (2004)], which yield a faster convergence to the average values of the exact P(n). From the Fourier spectrum of P(n) we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean triples (m, p, q) of integers with m(2) + p(2) = q(2) play an important role in the oscillations of P(n). We analyze the oscillating part delta P(n) = P(n) - P-as(n) in the spirit of semiclassical periodic orbit theory [M. Brack and R. K. Bhaduri: Semiclassical Physics (Westview, Boulder, 2003)]. A semiclassical trace formula is derived which accurately reproduces the exact delta P(n) for n greater than or similar to 500 using ten pairs of orbits. For n greater than or similar to 4000 only two pairs of orbits with the frequencies 4 and 5, belonging to the lowest Pythagorean triple (3,4,5), are relevant and create the prominent beating pattern in the oscillations. For n greater than or similar to 100 000 the beat fades away and the oscillations are given by just one pair of orbits with frequency 4