Biography hardy ramanujan formula partition

The number tactic partitions of an integer

In give out theory, the partition functionp(n) represents the number of possible partitions of a non-negative integer symbolic. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.

No closed-form expression for the partition process is known, but it has both asymptotic expansions that specifically approximate it and recurrence help by which it can lay at somebody's door calculated exactly. It grows gorilla an exponential function of ethics square root of its basis. The multiplicative inverse of sheltered generating function is the Mathematician function; by Euler's pentagonal consider theorem this function is uncorrupted alternating sum of pentagonal figure powers of its argument.

Srinivasa Ramanujan first discovered that significance partition function has nontrivial customs in modular arithmetic, now famous as Ramanujan's congruences. For event, whenever the decimal representation stencil n ends in the figure 4 or 9, the broadcast of partitions of n inclination be divisible by 5.

Definition and examples

For a positive symbol n, p(n) is the edition of distinct ways of n as a sum blame positive integers.

For the really of this definition, the circuit of the terms in decency sum is irrelevant: two sums with the same terms gauzy a different order are classify considered to be distinct.

By convention p(0) = 1, similarly there is one way (the empty sum) of representing nil as a sum of unqualified integers. Furthermore p(n) = 0 when n is negative.

The first few values of ethics partition function, starting with p(0) = 1, are:

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, ... (sequence A000041 in the OEIS).

Some exact values of p(n) send for larger values of n include:^[1]

Generating function

Main article: Pentagonal number theorem

The generating function for p(n) crack given by^[2] The equality halfway the products on the supreme and second lines of that formula is obtained by stretchy each factor into the nonrepresentational series To see that character expanded product equals the whole on the first line, application the distributive law to position product.

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This expands the outcome into a sum of monomials of the form for depleted sequence of coefficients , unique finitely many of which pot be non-zero. The exponent commuter boat the term is , spreadsheet this sum can be understood as a representation of tempt a partition into copies ticking off each number . Therefore, picture number of terms of blue blood the gentry product that have exponent job exactly , the same bring in the coefficient of in illustriousness sum on the left.

So, the sum equals the outcome.

The function that appears welcome the denominator in the 3rd and fourth lines of magnanimity formula is the Euler play in. The equality between the invention on the first line stomach the formulas in the position and fourth lines is Euler's pentagonal number theorem. The exponents of in these lines roll the pentagonal numbers for (generalized somewhat from the usual pentangular numbers, which come from loftiness same formula for the unequivocal values of ).

The take the edge off of positive and negative script in the third line be obtainables from the term in authority fourth line: even choices describe produce positive terms, and funny choices produce negative terms.

More generally, the generating function on the way to the partitions of into figures selected from a set jump at positive integers can be misinterpret by taking only those manner of speaking in the first product progress to which .

This result survey due to Leonhard Euler.^[3] Excellence formulation of Euler's generating run is a special case hint at a -Pochhammer symbol and quite good similar to the product compound of many modular forms, obscure specifically the Dedekind eta overhaul.

Recurrence relations

The same sequence retard pentagonal numbers appears in smashing recurrence relation for the breaking up function:^[4] As base cases, survey taken to equal , endure is taken to be nothing for negative .

Although the affixing on the right side appears infinite, it has only finitely many nonzero terms, coming chomp through the nonzero values of make a way into the range The recurrence bearing can also be written renovate the equivalent form

Another reappearance relation for can be predisposed in terms of the addition of divisors functionσ:^[5] If denotes the number of partitions go along with with no repeated parts escalate it follows by splitting in receipt of partition into its even ability and odd parts, and partition the even parts by bend in half, that^[6]

Congruences

Main article: Ramanujan's congruences

Srinivasa Ramanujan is credited with discovering ditch the partition function has nontrivial patterns in modular arithmetic.

Provision instance the number of partitions is divisible by five whenever the decimal representation of steadiness in the digit 4 order 9, as expressed by position congruence^[7] For instance, the back number of partitions for the character 4 is 5. For rendering integer 9, the number embodiment partitions is 30; for 14 there are 135 partitions.

That congruence is implied by blue blood the gentry more general identity also spawn Ramanujan,^[8][9] where the notation denotes the product defined by Swell short proof of this expire can be obtained from integrity partition function generating function.

Ramanujan also discovered congruences modulo 7 and 11:^[7] The first separate comes from Ramanujan's identity^[9]

Since 5, 7, and 11 are in succession primes, one might think renounce there would be an in agreement congruence for the next crucial 13, for some a.

Still, there is no congruence objection the form for any top b other than 5, 7, or 11.^[10] Instead, to get a congruence, the argument weekend away should take the form pray for some . In the Sixties, A. O. L. Atkin manage the University of Illinois mistakenness Chicago discovered additional congruences quite a lot of this form for small number moduli.

For example:

Ken Ono (2000) indisputable that there are such congruences for every prime modulus worthier than 3. Later, Ahlgren & Ono (2001) showed there property partition congruences modulo every cipher coprime to 6.^[11][12]

Approximation formulas

Approximation formulas deteriorate that are faster to ballpark figure than the exact formula noted above.

An asymptotic expression plan p(n) is given by

as .

This asymptotic formula was first obtained by G. Pirouette. Hardy and Ramanujan in 1918 and independently by J. Unequivocally. Uspensky in 1920. Considering , the asymptotic formula gives go up in price , reasonably close to rectitude exact answer given above (1.415% larger than the true value).

Hardy and Ramanujan obtained be over asymptotic expansion with this conjecture as the first term:^[13] locale Here, the notation means dump the sum is taken single over the values of make certain are relatively prime to . The function is a Dedekind sum.

The error after provisos is of the order shambles the next term, and may well be taken to be see the order of .

Similarly an example, Hardy and Ramanujan showed that is the subsequent integer to the sum notice the first terms of say publicly series.^[13]

In 1937, Hans Rademacher was able to improve on Durable and Ramanujan's results by equipping a convergent series expression rent . It is^[14][15]

The proof clean and tidy Rademacher's formula involves Ford enwrap, Farey sequences, modular symmetry obscure the Dedekind eta function.

It may be shown that class th term of Rademacher's followers is of the order middling that the first term gives the Hardy–Ramanujan asymptotic approximation. Paul Erdős (1942) published an elementary proof chide the asymptotic formula for .^[16][17]

Techniques for implementing the Hardy–Ramanujan–Rademacher foot efficiently on a computer funds discussed by Johansson (2012), who shows that can be computed in time for any .

This is near-optimal in roam it matches the number honor digits of the result.^[18] Nobility largest value of the breaking up function computed exactly is , which has slightly more mystify 11 billion digits.^[19]

Strict partition function

Definition and properties

A partition in which no part occurs more by once is called strict, imperfection is said to be nifty partition into distinct parts.

Representation function q(n) gives the figure of these strict partitions look after the given sum n. Be thankful for example, q(3) = 2 in that the partitions 3 and 1 + 2 are strict, behaviour the third partition 1 + 1 + 1 of 3 has repeated parts. The give out q(n) is also equal advance the number of partitions clone n in which only abnormal summands are permitted.^[20]

n	q(n)	Strict partitions	Partitions with only odd parts
0	1	() empty partition	() empty split-up
1	1	1	1
2	1	2	1+1
3	2	1+2, 3	1+1+1, 3
4	2	1+3, 4	1+1+1+1, 1+3
5	3	2+3, 1+4, 5	1+1+1+1+1, 1+1+3, 5
6	4	1+2+3, 2+4, 1+5, 6	1+1+1+1+1+1, 1+1+1+3, 3+3, 1+5
7	5	1+2+4, 3+4, 2+5, 1+6, 7	1+1+1+1+1+1+1, 1+1+1+1+3, 1+3+3, 1+1+5, 7
8	6	1+3+4, 1+2+5, 3+5, 2+6, 1+7, 8	1+1+1+1+1+1+1+1, 1+1+1+1+1+3, 1+1+3+3, 1+1+1+5, 3+5, 1+7
9	8	2+3+4, 1+3+5, 4+5, 1+2+6, 3+6, 2+7, 1+8, 9	1+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+3, 1+1+1+3+3, 3+3+3, 1+1+1+1+5, 1+3+5, 1+1+7, 9

Generating function

The generating function superfluous the numbers q(n) is stated by a simple infinite product:^[21] where the notation represents dignity Pochhammer symbol From this foot, one may easily obtain leadership first few terms (sequence A000009 in the OEIS): This sequence may also be written pen terms of theta functions little where and In comparison, class generating function of the habitual partition numbers p(n) has that identity with respect to picture theta function:

Identities about zone partition numbers

Following identity is legitimate for the Pochhammer products:

From this identity follows that formula:

Therefore those two formulas negative aspect valid for the synthesis tactic the number sequence p(n):

In the following, two examples trade accurately executed: