Timeline for Full asymptotics near 1 of the generating function for integer partitions
Current License: CC BY-SA 4.0
8 events
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| 5 hours ago | history | edited | RobPratt |
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| 9 hours ago | comment | added | Abdelmalek Abdesselam | @MarkWildon: thanks! | |
| 9 hours ago | comment | added | Mark Wildon | From Dedekind's functional equation for $P(e^{-2\pi s})$ namely $$ \log P(e^{-2\pi s}) - \log P(e^{-2\pi / s}) = \frac{\pi}{12}\left(\frac{1}{s}-s \right) + \frac{1}{2} \log s $$ we get $$\log P(e^{-t}) = \frac{\pi^2}{6t} - \frac{t}{24} + \frac{1}{2} \log t - \frac{1}{2} \log 2\pi + \log P(e^{-4\pi^2/t}). $$ The final term is $O(t^\infty)$. So the minus sign attached to $t/24$ by de Bruijn is correct. The functional equation is equivalent to the modularity of the Dedekind eta function so easily available in the literature. A comment since it's not a direct proof of de Bruijn. | |
| 9 hours ago | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| 10 hours ago | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| 10 hours ago | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| 10 hours ago | answer | added | Sean Eberhard | timeline score: 1 | |
| 11 hours ago | history | asked | Abdelmalek Abdesselam | CC BY-SA 4.0 |