Page 27 - Ramanujan Yatra
P. 27
Ramanujan
YATRA
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3 which are 5, 4+1 and 3+2, respectively. Hence, we have seen that the number of partitions into odd parts of the number 5 is equal to the number of partitions into the distinct parts.
The generating function for the partition function is generally given by, ∑∞ p (n) qn = 1
n=0 (a; q)∞
where (a; q)∞ = ∏∞n=0 (1 - aqn), |q| < 1
For any positive integer n, (a; q)∞= (1-a) (1-aq) (1-aq2)
The three brilliant congruences established by Ramanujan for the partition function p(n), are as follows: for n > or = O,
p (5n+4) = O(mod 5),
p (7n+4) = O(mod 7), p (11n+6) = O(mod 11).
The study of Ramanujan-type congruence has emerged as an interesting and popular area of research in number theory. Because of its scope for applications in different areas like probability and particle physics, especially in quantum field theory, the theory of partitions has become one of the richest research areas of mathematics in modern times. Ramanujan’s congruences on p (n) has encouraged several mathematicians to take up the studies on arithmetic properties of many other partition functions like t-core partition, Frobenius partition, l-regular partition, over partition, broken k diamond partition, k dots bracelet partitions etc. According to Bruce C Berndt, the Professor of Mathematics at the University of Illinois, USA “the theory of modular forms is where Ramanujan’s ideas have been most influential”.
Together with Hardy, he began the powerful “Circle Method” to provide an exact formula for p(n), the number of integer positions of n (e.g., p(5)=7, where the seven partitions are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1). Circle Method has played an important role in subsequent developments of analytic number theory.
These works were mentioned in Ramanujan’s nomination as a Fellow of
