Exact formulas for the partition function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:12:23Z http://mathoverflow.net/feeds/question/47611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function Exact formulas for the partition function? Frank Thorne 2010-11-28T20:58:31Z 2013-04-16T10:51:14Z <p>I am curious, what kind of exact formulas exist for the partition function $p(n)$?</p> <p>I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely messy transcendental function, and the approximation was so good that for large $n$ one could simply take the $k = 1$ term and truncate this to the nearest integer to get an exact formula.</p> <p>Reviewing the literature, it seems that I misremembered Rademacher's exact formula, which <em>is</em> of the above type but which requires more than one term. I am curious if there are other exact formulas, particularly of the type I mentioned? </p> <p>Also, if I am indeed wrong and no such formula has been proved, is some good reason why it would be naive to expect one?</p> <p>Thanks.</p> http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function/47706#47706 Answer by Martin Rubey for Exact formulas for the partition function? Martin Rubey 2010-11-29T18:31:37Z 2010-11-30T08:26:18Z <p>I'm not sure whether this should be a comment or an answer: it is curiously missing from all the links above that the generating function for integer partitions satisfies a reasonably nice (order four, homogeneous of degree four) algebraic differential equation:</p> <p><code>\begin{multline*} 4F^3 F'' + 5x F^3 F''' + x^2 F^3 F^{(\rm iv)} - 16F^2 F'^2 - 15x F^2 F' F'' + 20x^2 F^2 F' F'''\\ - 39x^2 F^2 F''^2 + 10x F F'^3 + 12x^2 F F'^2 F'' + 6x^2 F'^4 = 0 \end{multline*}</code></p> <p>There is actually also an order three differential equation, but it's not as nice.</p> <p>According to Don Zagier [The 1-2-3 of modular forms, Section 5.1, Proposition 15] already Ramanujan knew that every modular and every quasi-modular form on $\Gamma_1$ satisﬁes a third order algebraic diﬀerential equation. The equation above is found given the first 39 terms by </p> <pre> guessADE([partition n for n in 0..39], homogeneous==4) </pre> <p>from FriCAS in less than 0.01 seconds.</p> http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function/47725#47725 Answer by rlo for Exact formulas for the partition function? rlo 2010-11-29T21:33:49Z 2010-11-29T21:33:49Z <p>This doesn't really answer the question, so perhaps it would be better as a comment, but alas, I don't have the necessary reputation.</p> <p>Following up on Thomas Bloom's reference to the work of Bringmann and Ono, there is a paper of Folsom and Masri (Mathematische Annalen, available here: <a href="http://www.math.yale.edu/~alf8/Folsom-Masri-MathAnn07-10.pdf" rel="nofollow">http://www.math.yale.edu/~alf8/Folsom-Masri-MathAnn07-10.pdf</a>) which considers the main term one would get in an asymptotic formula arising from BO's Poincare series formula. In particular, they also consider the problem of the error arising from truncating the infinite sum at $O(n^{1/2})$, obtaining power savings over the best known results of $O(n^{-1/2+\epsilon})$ if one truncates at $\lfloor \sqrt{n/6} \rfloor$.</p> http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function/52687#52687 Answer by Frank Thorne for Exact formulas for the partition function? Frank Thorne 2011-01-20T21:19:09Z 2011-01-20T21:19:09Z <p>Jan Bruinier and Ken Ono just announced <a href="http://www.aimath.org/news/partition/brunier-ono" rel="nofollow">this</a>.</p> http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function/58181#58181 Answer by Jerome Malenfant for Exact formulas for the partition function? Jerome Malenfant 2011-03-11T18:11:08Z 2011-03-11T18:11:08Z <p>In answer to "What kind of exact formulas exist for the partition function", see <a href="http://arxiv.org/abs/1103.1585" rel="nofollow">http://arxiv.org/abs/1103.1585</a>, eqs.(10), (11), and (12).</p> http://mathoverflow.net/questions/47611/exact-formulas-for-the-partition-function/127690#127690 Answer by Abhiram for Exact formulas for the partition function? Abhiram 2013-04-16T10:09:54Z 2013-04-16T10:51:14Z <p>(r=1 to x)sigma{ ((n=0 to r-1)sigma{ (-1)^n * rCn * (r-n)^x }) / r! }</p> <p>where x is the no of elements. Just made this formula up.Hope it helps.</p>