On Jun 5, 2014, at 2:15 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice approximation.
Speaking of which, is there some heuristic reason the Hardy-Ramanujan asymptotic formula for the partition function
p(n) ~ exp(pi sqrt(2n/3)) / (4 sqrt(3) n).
or its general form
p(n) ~ A^sqrt(n) / (B*n) for some A, B in (0,oo)
ought to be true?
--Dan
A and B are related to the asymptotic behavior of an analytic function at a particular point where its first derivative vanishes, a “saddle point”. A is related to the location of the saddle, and B is related to the second derivative at that point. Here is the function: f(z) = 1/z^(n+1) \prod_{k=1}^\infty 1/(1-z^k) The infinite product is Euler’s generating function for partitions. By integrating f(z) on a closed contour that circles around the origin (and dividing out 2 pi i) you get p(n). For asymptotics you deform the contour so that it passes over the saddle point z*, f’(z*)=0, on the real axis between 0 and 1. For large n you find z* ~ exp(-sqrt(pi^2/(6n))). From that you get A, and with a bit more work (Taylor expanding log f(z) to second order about z*) you get B. The pi^2/6 comes from the sum 1+1/2^2+1/3^2+ … This asymptotic math is mirrored in a statistical analysis where you say partitions for large n have a typical “shape”, and you maximize the entropy with respect to that shape. -Veit