Ramanujan's congruences
Encyclopedia
In mathematics
, Ramanujan's congruences are some remarkable congruences for the partition function
p(n). The Indian mathematician Srinivasa Ramanujan
discovered the following
In his 1919 paper (Ramanujan, 1919), he gave proof for the first two congruences using the following identities (using q-Pochhammer symbol notation):
then stated that "It appears there are no equally simple properties for any moduli involving primes other than these" .
After Ramanujan died in 1920, G. H. Hardy
, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs Eisenstein series
.
In 1944, Freeman Dyson
defined the rank function and conjectured the existence of a "crank" function
for partitions that would provide a combinatorial proof
of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan successfully found such a function, and proved the celebrated result that the crank simultaneously “explains” the
three Ramanujan congruences modulo 5, 7 and 11.
Extending results of A. O. L. Atkin
, Ken Ono
in 2000 proved that there are
such Ramanujan congruences modulo every integer coprime to 6. For example,
his results give
Later Ken Ono
conjectured that the elusive crank also satisfies
exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg
in his 2005 paper Partition Congruences and the Andrews-Garvan-Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences
Paper of the Year prize.
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 by considering the Hausdorff dimension
of the following
function in the l-adic
topology:
.
It is seen to have dimension 0 only in the cases where l = 5,7 or 11 and since the partition function can be written as a linear combination of these functions this can be considered a formalization and proof of Ramanujan's observation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Ramanujan's congruences are some remarkable congruences for the partition function
Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
p(n). The Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
discovered the following
In his 1919 paper (Ramanujan, 1919), he gave proof for the first two congruences using the following identities (using q-Pochhammer symbol notation):
then stated that "It appears there are no equally simple properties for any moduli involving primes other than these" .
After Ramanujan died in 1920, G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
.
In 1944, Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
defined the rank function and conjectured the existence of a "crank" function
for partitions that would provide a combinatorial proof
Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan successfully found such a function, and proved the celebrated result that the crank simultaneously “explains” the
three Ramanujan congruences modulo 5, 7 and 11.
Extending results of A. O. L. Atkin
A. O. L. Atkin
Arthur Oliver Lonsdale Atkin , who published under the name A. O. L. Atkin, was a Professor Emeritus of mathematics at the University of Illinois at Chicago. As an undergraduate during World War II, he worked at Bletchley Park cracking German codes. He received his Ph.D...
, Ken Ono
Ken Ono
Ken Ono is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, and the fields of interest to Srinivasa Ramanujan...
in 2000 proved that there are
such Ramanujan congruences modulo every integer coprime to 6. For example,
his results give
Later Ken Ono
Ken Ono
Ken Ono is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, and the fields of interest to Srinivasa Ramanujan...
conjectured that the elusive crank also satisfies
exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg
Karl Mahlburg
Karl Mahlburg is an American mathematician whose research interests lie in the areas of modular forms, partitions, combinatorics and number theory....
in his 2005 paper Partition Congruences and the Andrews-Garvan-Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
The Proceedings of the National Academy of Sciences of the United States of America, usually referred to as PNAS, is the official journal of the United States National Academy of Sciences...
Paper of the Year prize.
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 by considering the Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
of the following
function in the l-adicP-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
topology:
.It is seen to have dimension 0 only in the cases where l = 5,7 or 11 and since the partition function can be written as a linear combination of these functions this can be considered a formalization and proof of Ramanujan's observation.
External links
- Dyson's rank, crank and adjoint. A list of references.