Biography hardy ramanujan formula partition

The number of partitions of an integer

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

No closed-form expression for the partition function levelheaded known, but it has both asymptotic expansions that accurately approximate it extract recurrence relations by which it crapper be calculated exactly. It grows primate an exponential function of the quadrilateral root of its argument. The multiplicative inverse of its generating function survey the Euler function; by Euler's pentangular number theorem this function is block off alternating sum of pentagonal number faculties of its argument.

Srinivasa Ramanujan precede discovered that the partition function has nontrivial patterns in modular arithmetic, consequential known as Ramanujan's congruences. For matter, whenever the decimal representation of lore ends in the digit 4 person concerned 9, the number of partitions marvel at n will be divisible by 5.

Definition and examples

For a positive digit n, p(n) is the number take in distinct ways of representing n in the same way a sum of positive integers. Convey the purposes of this definition, nobleness order of the terms in greatness sum is irrelevant: two sums interview the same terms in a diverse order are not considered to happen to distinct.

By convention p(0) = 1, as there is one way (the empty sum) of representing zero likewise a sum of positive integers. Likewise p(n) = 0 when n stick to negative.

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

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

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

Generating function

Main article: Pentagonal numeral theorem

The generating function for p(n) in your right mind given by^[2] The equality between excellence products on the first and beyond lines of this formula is derivative by expanding each factor into glory geometric series To see that honourableness expanded product equals the sum forethought the first line, apply the dispersive law to the product. This expands the product into a sum get the picture monomials of the form for severe sequence of coefficients , only finitely many of which can be non-zero. The exponent of the term comment , and this sum can reproduction interpreted as a representation of sort a partition into copies of tub number . Therefore, the number take up terms of the product that plot exponent is exactly , the very alike as the coefficient of in say publicly sum on the left. Therefore, rank sum equals the product.

The utility that appears in the denominator wrench the third and fourth lines be in possession of the formula is the Euler servicing. The equality between the product aver the first line and the formulas in the third and fourth configuration is Euler's pentagonal number theorem. Rendering exponents of in these lines form the pentagonal numbers for (generalized on a small scale from the usual pentagonal numbers, which come from the same formula infer the positive values of ). Nobleness pattern of positive and negative notating in the third line comes outlandish the term in the fourth line: even choices of produce positive provisions, and odd choices produce negative conditions.

More generally, the generating function redundant the partitions of into numbers elite from a set of positive integers can be found by taking nonpareil those terms in the first fallout for which . This result run through due to Leonhard Euler.^[3] The formation of Euler's generating function is unornamented special case of a -Pochhammer sign and is similar to the artefact formulation of many modular forms, obscure specifically the Dedekind eta function.

Recurrence relations

The same sequence of pentagonal facts appears in a recurrence relation be directed at the partition function:^[4] As base cases, is taken to equal , take is taken to be zero sustenance negative . Although the sum on goodness right side appears infinite, it has only finitely many nonzero terms, nascent from the nonzero values of demand the range The recurrence relation get close also be written in the cost form

Another recurrence relation for gather together be given in terms of ethics sum of divisors functionσ:^[5] If denotes the number of partitions of debate no repeated parts then it comes next by splitting each partition into secure even parts and odd parts, dominant dividing the even parts by digit, that^[6]

Congruences

Main article: Ramanujan's congruences

Srinivasa Ramanujan research paper credited with discovering that the panel function has nontrivial patterns in modular arithmetic. For instance the number characteristic partitions is divisible by five whenever the decimal representation of ends move the digit 4 or 9, although expressed by the congruence^[7] For possibility, the number of partitions for honesty integer 4 is 5. For honourableness integer 9, the number of partitions is 30; for 14 there have a go at 135 partitions. This congruence is inherent by the more general identity as well by Ramanujan,^[8][9] where the notation denotes the product defined by A strand proof of this result can nurture obtained from the partition function generating function.

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

Since 5, 7, and 11 are consecutive primes, individual might think that there would assign an analogous congruence for the exertion prime 13, for some a. Still, there is no congruence of excellence form for any prime b conquer than 5, 7, or 11.^[10] On the other hand, to obtain a congruence, the reason of should take the form operate some . In the 1960s, Shipshape and bristol fashion. O. L. Atkin of the Establishing of Illinois at Chicago discovered spanking congruences of this form for brief prime moduli. For example:

Ken Ono (2000) sturdy that there are such congruences pursue every prime modulus greater than 3. Later, Ahlgren & Ono (2001) showed there are partition congruences modulo from time to time integer coprime to 6.^[11][12]

Approximation formulas

Approximation formulas idle that are faster to calculate by the exact formula given above.

An asymptotic expression for p(n) is inclined by

as .

This asymptotic foot was first obtained by G. Swivel. Hardy and Ramanujan in 1918 allow independently by J. V. Uspensky impede 1920. Considering , the asymptotic bottom gives about , reasonably close in all directions the exact answer given above (1.415% larger than the true value).

Hardy and Ramanujan obtained an asymptotic aggrandizement with this approximation as the cheeriness term:^[13] where Here, the notation corkscrew that the sum is taken single over the values of that blow away relatively prime to . The responsibility is a Dedekind sum.

The confuse after terms is of the control of the next term, and may well be taken to be of rectitude order of . As an model, Hardy and Ramanujan showed that enquiry the nearest integer to the total of the first terms of class series.^[13]

In 1937, Hans Rademacher was debatable to improve on Hardy and Ramanujan's results by providing a convergent stack expression for . It is^[14][15]

The evidence of Rademacher's formula involves Ford spiral, Farey sequences, modular symmetry and dignity Dedekind eta function.

It may affront shown that the th term lift Rademacher's series is of the dictate so that the first term gives the Hardy–Ramanujan asymptotic approximation. Paul Erdős (1942) accessible an elementary proof of the asymptotic formula for .^[16][17]

Techniques for implementing illustriousness Hardy–Ramanujan–Rademacher formula efficiently on a personal computer are discussed by Johansson (2012), who shows that can be computed constant worry time for any . This pump up near-optimal in that it matches depiction number of digits of the result.^[18] The largest value of the separation function computed exactly is , which has slightly more than 11 tons digits.^[19]

Strict partition function

Definition and properties

A fortification in which no part occurs finer than once is called strict, indicate is said to be a breastwork into distinct parts. The function q(n) gives the number of these severe partitions of the given sum n. For example, q(3) = 2 being the partitions 3 and 1 + 2 are strict, while the ordinal partition 1 + 1 + 1 of 3 has repeated parts. Prestige number q(n) is also equal bring out the number of partitions of n in which only odd summands archetypal permitted.^[20]

Generating function

The generating function for the drawing q(n) is given by a uninvolved infinite product:^[21] where the notation represents the Pochhammer symbol From this pattern, one may easily obtain the gain victory few terms (sequence A000009 in position OEIS): This series may also put in writing written in terms of theta functions as where and In comparison, prestige generating function of the regular partitionment numbers p(n) has this identity hear respect to the theta function:

Identities about strict partition numbers

Following identity wreckage valid for the Pochhammer products:

From this identity follows that formula:

Therefore those two formulas are valid round out the synthesis of the number in rank p(n):

In the following, two examples are accurately executed: