The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or order of the $x_i$ ’s gives distinct solutions. If looking into the statistical mechanics for classical ideal gas in 3D, we meet with the same thing with $s=3N$, $N$ is the number of particles. But now the $3N$ squares problem is to count the number of the microstates in the so-called microscope ensemble. The following asymptotic expression of $r_{3N}(n)$ is experimentally validated, so it is physically proved:

$r_{3N}(n)\approx \frac{{\pi}^{3N/2}}{\Gamma (3N/2)} {{n}^{3N/2-1}}$, in thermodynamic limit $n/N=const.$ and $n \to \infty$ .

My question is: How to give an estimate of the error, and does anyone know such a formula in mathematical literature?

Ref.

S.C.Miline, New infinite families of exact sums of squares formulas, Jacobi elliptic functions and Ramanujan’s tau function, Proc. Natl. Acad. Sci. USA, 1996, 93:15004-15008, and references cited therein.

proved, that wasn't actually what you meant :-( Also now both $n$ and $N$ are "linearly indpendent" and $n/N$ is constant. :-( $\endgroup$ – Robin Chapman Sep 11 '10 at 11:40mathematicalquestion here? Are you still asserting that the assertion you claimed was "proved" is proved? If so please can you give a reference? $\endgroup$ – Robin Chapman Sep 11 '10 at 11:44