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I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about partitions).

The asymptotic formula always seems to be written as,

$ p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi \sqrt{\frac{2n}{3}}}, $

however I need to know the order of the omitted terms, (i.e. I need whatever the little-o of this expression is). Does anybody know what this is, and a reference for it? I haven't been able to find it online, and don't have access to a copy of Andrews 'Theory of Integer Partitions'.

Thank you.

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I believe the asymptotic nature of the Hardy-Ramanujan formula, notwithstanding its use to get exact values of $p(n)$, means that a "little-o" notation for omitted terms would be misplaced. –  hardmath Jul 5 '11 at 16:14
    
@Hardmath, I'm about to answer my own question (!), but also justify that there is a little-o representation, since in fact if f is asymptotically equivalent to g, then f = (1 + o(1))g... So as pointed out, I've now answered my original question... silly me. –  Owen88 Jul 5 '11 at 17:18
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Okay, I thought perhaps the exact convergent series given by Rademacher (1937) that refines the Hardy-Ramanujan formula (which forms the first term of the series) and its order of convergence might be of interest. G. Andrews has a chapter about this in his book Theory of Integer Partitions. –  hardmath Jul 5 '11 at 17:34
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1 Answer

The original paper

http://plms.oxfordjournals.org/content/s2-17/1/75.full.pdf

addresses this issue on p. 83.

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