From: ksbrown@ksbrown.seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Re: Additive Partitions of Number
Date: Tue, 22 Aug 1995 06:47:30 GMT

Charles Richmond <richmond@plano.net> wrote:
> Will someone please post the formula for the partitions of a number 
> that is due to the Indian mathematician Ramanujan? (This formula 
> calculates how many ways an integer can be expressed as the sum of 
> positive integers.) I heard of this on a Nova episode <The Man Who 
> Loved Numbers, but the formula was *not* really given. They did say 
> that the formula involved the constant PI though.

Hardy and Ramanujan did a lot of work with partitions around 1918, and
found that the log of p(n) is asymptotic to PI sqrt(2n/3), which led
to the approximate formula

                   e^(PI sqrt(2n/3))
         p(n) ~=   -----------------  
                      4n sqrt(3)

Subsequently, their celebrated "circle method" was used to obtain the
formula

            1      d  / e^(PI sqrt(2n/3 - 1/36)) \
p(n) = ----------- --( -------------------------  ) + O(e^(k sqrt(n))
       2PI sqrt(2) dn \     sqrt(n - 1/24)       /


where k < PI/6.  Further refinements led to a formula (with correction
terms) that gives p(200) to within 0.004 of the actual value
3972999029388.  Later still, Rademacher made more improvements along
these lines and was able to give an exact series expansion for p(n).
That formula is given in Abramowitz and Stegun's "Handbook of
Mathematical Functions".