Srinivasa Ramanujan’s Contributions in Mathematics

Ramanujan

Srinivasa Ramanujan is considered to be one of the geniuses in the field of mathematics. He was born on 22nd December 1887, in a small village of Tamil Nadu during British rule in India. His birthday is celebrated as national mathematics day. In high school, he used to do very well in all subjects. In 1990, he started working on his mathematics in geometry and arithmetic series. Although he had no official training in mathematics, even then, he was able to solve problems that were considered unsolvable. He published his first paper in 1911. In January 1913, Ramanujan began a postal conversation with an English mathematician, G.H. Hardy at the University of Cambridge, England and wrote a letter after having seen a copy of his book Orders of infinity. He found Ramanujan’s work to be extraordinary and arranged for him to travel to Cambridge in 1914. As Ramanujan was an orthodox Brahmin, a vegetarian, his religion might have restricted him to travel. This difficulty of Ramanujan was solved partly by E H Neville, a colleague of Hardy. Hardy after analysing the works of Ramanujan, said,

Ramanujan had produced groundbreaking new theorems, including some that defeated me completely.I had never seen anything in the least like them before.’

At the age of 32, he died of Tuberculosis. In his short life span, he independently found 3900 results. He worked on real analysis, number theory, infinite series, and continued fractions. Some of his other works such as Ramanujan number, Ramanujan prime, Ramanujan theta function, partition formulae, mock theta function, and many more opened new areas for research in the field of mathematics. He worked out the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his theory of divergent series, in which he found a value for the sum of such series, using a technique he invented, that came to be called Ramanujan summation. In England, Ramanujan made further researches, especially in the partition of numbers, i.e, the number of ways in which a positive integer can be expressed as the sum of positive integers. Some of his results are still under research. His journal, Ramanujan Journal, was established to keep a record of all his notebooks and results, both published and unpublished, in the field of mathematics. As late as 2012, researchers studied even the small comments in his book, as they do not want to miss any results or identities given by him, that remained unsuspected until a century after his death. From his last letters in 1920 that he wrote to Hardy, it was evident that he was still working on new ideas and theorems of mathematics. In 1976, mathematicians found the ‘lost notebook’, that contained the works of Ramanujan from the last year of his life. Ramanujan devoted all his mathematical intelligence to his family goddess Namagir Thayar. He once said, “An equation for me has no meaning unless it expresses a thought of God.” Now, we will discuss in detail all his contributions to mathematics.

1. Infinite series of π

William Shanks, a 19th-century British mathematician tried calculating the value of infinite series of π. In 1873, he calculated the value of π to 707 decimal places. Ramanujan, in 1914, published ‘Modular equations and approximations to π’, which contained not only one, but 17 different series, that will converge very fastly to π, after calculating just fewer terms of the series.

 

Ramanujan estimating pi

2. Ramanujan number

The number 1729 is known as the Ramanujan number or Hardy-Ramanujan number. It is the smallest natural number that can be expressed as the sum of two cubes, in two different ways, i.e., 1729 = 13 + 123 = 93 + 103. There is a small story behind the discovery of this number. When Ramanujan was under treatment, G.H. Hardy once visited him in the hospital and had a conversation in which he mentioned,

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’

This is how the Ramanujan number came into existence. Later on, more properties of this number were discovered.

3. Ramanujan Prime

Ramanujan published a two-page paper on the proof of Bertrand’s postulate. At the end of the last page, he mentioned a result, π(x) – π(x/2) ≥ 1, 2, 3, 4, 5,….., for all x≥ 2, 11, 17, 29, 41,…. respectively, where π(x) is the prime counting function, equals to the number of primes equal or less than x. The nth Ramanujan prime number is the least integer {R}_{n}, for which there are at least n primes between x and x/2, for all x ≥ {R}_{n}. The first five Ramanujan primes are 2, 11, 17, 29, 41

4. Ramanujan Theta Function

Ramanujan theta function is the generalised form of the Jacob theta function. In particular, Jacobi triple product can be beautifully represented by the Ramanujan theta function. The Ramanujan theta function is given below.

ramanujan theta function

for |ab|<1.With the help of Ramanujan theta function, Jacobi triple product can be represented as,

jacobi triple product

5. Mock Theta Function

Ramanujan in his last letter to G.H. Hardy and in his ‘lost notebook’, gave the first example of mock theta function. A mock theta function is a mock modular form( the holomorphic part of a harmonic weak Maass form), of weight 1/2. His last letter to Hardy contained 17 examples of mock theta functions, and some more examples were mentioned in his ‘lost notebook.’ Ramanujan gave an order to his mock theta function. Before the attempts of Zwegers, the order of mock theta function was 3, 5, 6, 7, 8, 10.

6. Partition

Partition or integer partition of an integer ‘n’ is a way of writing ‘n’ as a sum of positive integers. Partitions that differ only in the order of summands are considered as the same partitions. Each summand in the partition is called a part. The number of partitions of an integer ‘n’ is denoted by p(n). For example, integer 4 has 5 partitions as given below.

4

1+1+1+1

1+2+1

1+3

2+2

Here partition 1+3 is the same as 3+1 and 1+2+1 is the same as 1+1+2 and p(4)=5. Partitions can also be visualised with the help of the Young diagram and Ferrers diagram.

7. Ramanujan Magic Squares

In his school days, he used to enjoy solving magic squares. Magic squares are the cells in 3 rows and 3 columns, filled with numbers starting from 1 to 9. The numbers in the cells are arranged in such a way that the sum of numbers in each row is equal to the sum of numbers in each column is equal to the sum of numbers in each diagonal. Ramanujan gave a general formula for solving the magic square of dimension 3×3,

Ramanujan magic square

where A, B, C and P, Q, R are in arithmetic progression. The following formula was also given by him.

Ramanujan magic square

8. Ramanujan Congruences

Ramanujan obtained three congruences when m is a whole number, p (5m+ 4) 0 (mod 5), p (7m+ 5) 0 (mod 7), p (11m+ 6) ≡ 0 (mod 11). Hardy and E.M. Wright wrote, 

he was first to led the conjecture and then to prove, three striking arithmetic properties associated with the moduli 5, 7 and 11.”
Values of p(n) for n ≡ 4 (mod 5), n ≡ 5 (mod 7), n ≡ 6 (mod 11) are given in the below tables repectively.

ramanujan congruences

9. Highly composite numbers

Composite numbers are the numbers that have factors other than 1 and the number itself. Ramanujan raised an interesting question that if ‘n’ is a composite number then what properties make a number highly composite. Ramanujan’s definition of Highly composite numbers,

A natural number is a highly composite number if d(m)< d(n) for all m < n.”

One of the highly composite numbers calculated by Ramanujan is 6746328388800. This number has 13 digits and its prime factorization is 26 x 34 x 52 x 72 x 111 x 131 x 171 x 191 x 231

He also published a paper on highly composite numbers in 1915. According to him, there were infinitely many highly composite numbers.

10. Symmetric Equation by Ramanujan

Ramanujan noticed symmetry in Diophantine’s equation, {x}^{y} = {y}^{x}. He proved that there exists only one integer solution to this equation, i.e., x=4, y=2, and an infinite number of rational solutions, for example, {(27/8)}^{(9/4)} = {(9/4)}^{(27/8)}.

11. Ramanujan-Nagell Equation

Ramanujan-Nagell Equation is the equation of type {2}^{n} – 7 = {x}^{2}. It is an example of Diophantine equation. In 1913, Ramanujan claimed that this equation had only 1²+7 = 2³, 3²+7 = {2}^{4}, 5²+7 = {2}^{5}, 11²+7 = {2}^{7}, 181²+7 = {2}^{15} integral solutions. This conjecture was later on proved by Trygve Navell and is widely used in coding theory.

12. On Certain Arithmetical Functions

Ramanujan published a paper “On certain arithmetic functions” in 1916, in which he discussed the properties of Fourier coefficients of modular forms. Though the concept of modular forms was not even developed then, he gave three fundamental conjectures. In 1936, after 20 years of his published paper, a Greman mathematician Erich Hecke developed the Hecke theory with the help of his first two conjectures. His last conjecture played a vital role in the Langlands program (a program that relates representation theory and algebraic number theory). “On certain arithmetical functions” by Ramanujan was very effective in creating a sensation in 2oth century mathematics.

13. On Fermat’s Last Theorem

In 2013, mathematicians found some evidence that revealed Ramanujan was working on Fermat’s last theorem. Pierre de Fermat mentioned that,

 if n is a whole number greater than 2, then there are no positive whole number triples x, y and z, such that xn + yn = zn.”

Ramanujan claimed that he had found an infinite family of whole numbers that will satisfy (approximately, not exactly) Fermat’s equation for n=3. He gave the example of the number 1729, which do not fits into the equation just by the mark of 1, for x=9, y=10, z=12. Ramanujan also worked on the equations of the form, y2 = x3 + ax + b. An elliptic curve is obtained, when the points (x,y) of this equation are plotted. These elliptic curves were of great significance and were used by Sir Andrew Wiles while he was proving Fermat’s last theorem in 1994.

14. Roger-Ramanujan Identities

In 1894, these identities were discovered and proved by Leonard James Rogers. Nearby 1913, Ramanujan rediscovered these identities. He had no proof but found Roger’s paper in 1917. Then they both united and gave a joint new proof.

roger ramanujan identities

15. Roger-Ramanujan Continued Fractions

Roger discovered continued fractions in 1894, which were later rediscovered by Ramanujan in 1912.

ramanujan continued fraction

Ramanujan found various results concerning R(q), for example, R({e}^{-2π}) is given below in the picture and he also calculated R({e}^{-2π√n}) for n= 4, 9, 16, 64

particular case of continued fraction

16. Ramanujan’s Master Theorem

Ramanujan’s Master Theorem provides an analytic expression for the Mellin transform of an analytical function. This theorem is used by Ramanujan to calculate definite integrals and infinite series. The result of the theorem is given in the picture below.

master theorem

17. Properties of Bernoulli Numbers

In 1904, Ramanujan independently studied and rediscovered Bernoulli numbers. In 1911, he wrote his first article on this topic. Bernoulli numbers {B}_{n} are the sequence of rational numbers, that appear in the Taylor series expansion of tangent and hyperbolic tangent functions. One of the properties that he discussed states that, the denominator of all Bernoulli numbers are divisible by six. Based on previous Bernoulli numbers, he also suggested a method to calculate Bernoulli numbers. According to the method proposed by him, if n is even but not equal to zero,

  1. Bn is a fraction and the numerator of Bn/n in its lowest terms is a prime number.
  2. The denominator of Bn contains each of the factors 2 and 3 once and only once.
  3. 2n(2n − 1)Bn/n is an integer and 2(2n − 1)Bn consequently is an odd integer.

formula for Bernoulli numbers

18. Euler Mascheroni Constant

Ramanujan calculated the Euler Mascheroni constant also known as the Euler constant, up to 15 decimal places. It is the limiting difference between the harmonic series and the natural logarithm. Later on, a value up to 50 decimal places was calculated and is equal to, 0.57721566490153286060651209008240243104215933593992…..

 

Euler constant

γ denotes the Euler constant

19. Ramanujan Summation

Ramanujan, in one of his books, stated that, if we add up all natural numbers starting from 1 up to infinity, then the sum will be a finite number, i.e., 1+2+3+……….+∞= -1⁄12

 

20. Ramanujan Puzzles

  • The first puzzle was to prove the equation of infinite nested radical. In 1911, Ramanujan sent the RHS of this equation to a mathematical journal as a puzzle. The puzzle and its solution are elaborated in the video below.

 

  • The next puzzle is to find the value of the Golden ratio(Φ), which is equal to the infinite continued fraction given in the picture below.

Golden ratio

 

The continued fraction in the black box is the same as that in the outer red box. Setting this equal to x, we get Φ = 1 + 1/x, which yields x2 – x – 1 = 0. The solutions of this quadratic equation are (1+√5)⁄2 and (1−√5)⁄2. Neglecting the negative solution, the value of Φ is (1+√5)⁄2

 

 

 

 

 

 

 

 

 

 

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