Partition function (number theory): Difference between revisions

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where each number ''i'' appears ''a''_''i'' times. This is precisely the definition of a partition of ''n'', so our product is the desired generating function. More generally, the generating function for the partitions of ''n'' into numbers from a set ''A'' can be found by taking only those terms in the product where ''k'' is an element of ''A''. This result is due to [[Leonhard Euler|Euler]].

The formulation of theEuler's generating function is a special case of a [[q-series]] and is similar to the product formulation of many [[modular form]]s, givingand some idea ofspecifically the connection[[Dedekind betweeneta the twofunction]]. It can also be used in conjunction with the [[pentagonal number theorem]] to derive a recurrence for the partition function stating that

p(<i>k</i>) &minus; p(''k'' &minus; 1) &minus; p(''k'' &minus; 2) + p(''k'' &minus; 5) + p(''k'' &minus; 7) &minus; p(''k'' &minus; 12) &minus; ... = 0,

where the sum is taken over all [[polygonal number|pentagonal numbers]] of the form ½''n''(3''n'' &minus; 1), including those where ''n'' < 0, and the terms continue to alternate +, +, &minus;, &minus;, +, +, ...

Euler's generating function is a special case of a [[q-series]] and is closely related to the [[Dedekind eta function]].

==Table of values==

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*p(10000) = 36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144

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==Rademacher's series==

An [[asymptotic]] expression for ''p''(''n'') is given by

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\pi i s(m,k) - 2\pi inm/k \right)</math>.

Here, the notation <math>(m,n)=1</math> implies that the sum should occur only over the values of ''m'' that are relatively prime to ''n''. The function <math>s(m,k)</math> is a [[Dedekind sum]]. The proof of Rademachers formula is interesting in that it involves [[Ford circle]], [[Farey sequence]]s, [[modular group|modular symmetry]] and the [[Dedekind eta function]] in a central way.

==ReferncesCongruences==

For any number that ends in 4 or 9, the number of partitions is always divisible by 5. Similar congruences can be found for 7, 11, ... rank ... crank ... Freeman Dyson ... Recently in the news: a general theory of congruences by Karl Mahlburg. ...

==References==

* Tom M. Apostol, ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 ''(See chapter 5)''.

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