Ramanujans and Hardy Partition Function

Hello Aliens, This blog is about the very famous Partition theory of numbers by Srinivasa Ramanajun and GH hardy.

The theory of partitions of numbers is an interesting branch of number theory. The concept of partitions was given by Leonard Euler in the 18th century. After Euler though, the theory of partition had been studied and discussed by many other prominent mathematicians like Gauss, Jacobi, Schur, McMahon, and Andrews, etc. but the joint work of Ramanujan with Prof. G.H. Hardy made a revolutionary change in the field of partition theory of numbers. Ramanujan and Hardy invented the circle method which gave the first approximations of the partition of numbers beyond 200.

A partition of a positive integer ‘n’ is a non-increasing sequence of positive integers, called parts, whose sum equals n. In simpler words, it means the number of ways in which a given number can be expressed as a sum of positive integers.

For example, p(4) = 5, i.e. there are five different ways that we can express the number 4. The partitions of the number 4 are:

3+1,

2+2,

2+1+1,

1+1+1+1

Here we can express 4 in the above-mentioned different terms including the number itself.

Lets take another example say 5. Now we can express it as-

4 + 1

3 + 2

3 + 1 + 1

2 + 2 + 1

2 + 1 + 1 + 1

1 + 1 + 1 + 1 + 1

That means we can write 5 as the sum of positive integers in 7 ways.

In general, we take p(0) as 1, which means there is only one way to represent zero as the sum of positive integers. Similarly, we can say for negative integers too.

We can write p(n) as :

p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+............

We can find the partition of small numbers after bit calculation but as the number increases the partition increases more exponentially. For example, you might be surprised that 10 has 42 partitions while 100 has more than 190 million partitions. Isn't it amazing? Both Ramanujan and Hardy derived an approximation asymptotic formula for the number of partitions. It is -

Srinivasa Ramanujan is credited with discovering that the partition function has nontrivial patterns in modular arithmetic. For example, the number of partitions is divisible by five whenever the decimal representation of ends in the digit 4 or 9, as expressed by the congruence.

For example, the number of partitions for the integer 4 is 5 that is divisible by five. For integer 9, the number of partitions is 30, which is divisible by 5 and for 14 there are 135 partitions, which is divisible by 5 too.

Ramanujan also discovered congruences modulo 7 and 11.

The study of Ramanujan type congruence is an interesting and popular research topic of number theory. Because of its great 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 recent times. More systematic study and a better understanding of partition theory will surely help in the advancement of mathematics with a new dimension.

Hope you have learned something new today. If you do so, then share this with others also. Till then have a good day.