Ramanujan's congruences

In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The Indian mathematician Srinivasa Ramanujan discovered the following

\[ p(5k+4)\equiv 0 \pmod 5 \]

\[ p(7k+5)\equiv 0 \pmod 7 \]

\[ p(11k+6)\equiv 0 \pmod {11}. \]

In his 1919 paper (Ramanujan, 1919), he gave proof for the first two congruences using the following identities (using q-Pochhammer symbol notation):

\[ \sum_{k=0}^\infty p(5k+4)q^k=5\frac{(q^5)_\infty^5}{(q)_\infty^6} \]
\[ \sum_{k=0}^\infty p(7k+5)q^k=7\frac{(q^7)_\infty^3}{(q)_\infty^4}+49q\frac{(q^7)_\infty^7}{(q)_\infty^8}. \]

then stated that "It appears there are no equally simple properties for any moduli involving primes other than these" [1].

After Ramanujan died in 1920, G. H. Hardy, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921).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

\[ p(4063467631k+30064597)\equiv 0\pmod{31}. \]

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 [2] by considering the Hausdorff dimension of the following P function in the l-adic topology:

\[ P_l(b;z) := \sum_{n=0}^\infty p\left(\frac{l^bn+1}{25}\right)q^{\frac{n}{24}}. \]

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[3] this can be considered a formalization and proof of Ramanujan's observation.
See also

Tau-function, for which there are other so-called Ramanujan congruences

References

K. Ono, Distribution of the partition function modulo m, Annals of Mathematics 151, (2000), 293–307.
S. Ramanujan, Some properties of p(n); the number of partitions of n, Proc. Cambridge Philos. Soc. 19 (1919), 207–210.

^ S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147–153.
^ Ken Ono, Amanda Folsom, & Zach Kent (2011). l-adic properties of the partition function American Institute of Mathematics.
^ Ken Ono & Jan Bruinier (2011). AN ALGEBRAIC FORMULA FOR THE PARTITION FUNCTION American Institute of Mathematics.

External links

"Abstract". Proceedings of the National Academy of Science. Retrieved 2007-03-31.
Karl Mahlburg (2005). "Partition Congruences and the Andrews–Garvan–Dyson Crank" (PDF). Proceedings of the National Academy of Sciences 102 (43): 15373–76. doi:10.1073/pnas.0506702102. PMC 1266116. PMID 16217020.
Dyson's rank, crank and adjoint. A list of references.