Srinivasa Ramanujan, one of India’s greatest mathematical geniuses, was born in a small village in the southwest of Madras, India. Though he passed away at a very young age on 26th April 1920 in a short span of just 32 years, he successfully made some significant contributions to the advancement of mathematics. Some of his most significant contributions to the world of mathematics are elucidated here.
Ramanujan made mathematics interesting in the field of infinite summation by inventing the Ramanujan summation. This technique can be used for assigning a value to divergent infinite series. It is fundamentally a property of the partial sums, instead of a property of the entire sum. This method for summation of numbers points to the fact that ‘S’= -1/12, where
S = 1+2+3+4+5+6+7+……..
Though the result is astonishing,, some complex analytics like string theory and quantum field theory use it extensively to derive equations.
Highly Composite Numbers
A highly composite number is essentially a positive integer which has more divisors than any smaller positive integer. Ramanujan created this term in 1915. There is an infinite number of highly composite numbers starting from 1, 2, 4, 6, 12, 24, 36, 48, 60…and so on. The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12… and so on. In 1915, Ramanujan enumerated 102 highly composite numbers up to 6746328388800. Later in 1983 and 1988, Robin & Nicholas updated this list.
Ramanujan’s Master Theorem
Ramanujan’s master theorem is associated with Analytic functions and Mellin transforms. A Mellin transform is an integral transform that can be considered as the multiplicative version of the two-sided Laplace transform while an analytic function is a function that is locally specified by a convergent power series. Ramanujan’s master theorem is chiefly a technique that provides an analytic expression for the Mellin transform of an analytic function. According to this,
If a complex-valued function f(x) is expanded as –
f(x) = ∑ φ(k)/k! (-x)^k …for k ranging from 0 to infinity,
Then, the Mellin transform of f(x) is given by –
Integral[x^(s-1)f(x)]dx = Γ(s)φ(-s)
[integration limit is 0 to infinity] , Γ(s) represents gamma function. Ramanujan used this theorem widely to compute definite integrals and infinite series. This theorem is also of great importance in quantum mechanics.
Hardy Ramanujan Number
When G.H. Hardy came to see Ramanujan in a taxi numbered 1729, G.H Hardy stated: “Ramanujan said that 1729 seemed to be a very dull number and hoped it doesn’t turn out to be an unfavorable omen”.
But, due to his love for numbers, Ramanujan found something special about this number as well and said: “it is a very interesting number, 1729 is the smallest number which can be written in the form of the sum of cubes of two numbers in two ways, i.e. 1729=1³+12³=9³+10³ “.Since then the number 1729 is called Hardy-Ramanujan’s number.
Additionally, he also made substantial contributions to the analytical theory of numbers, elliptic functions, continued fractions, hypergeometric series, infinite series, and many other mathematical fields. He also provided solutions to some of the mathematical problems that were once considered unsolvable.
H Hardy, the famous English mathematician says about his friend and colleague –
“The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems…to orders unheard of, whose mastery of continued fraction was… beyond that of any mathematician in the world, who had found for himself the functional equation of zeta function and the dominant terms of many of the most famous problems in analytical theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of complex variable was…”
Despite never receiving any formal training in pure mathematics, Ramanujan still made noteworthy contributions to various fields of mathematics making this self-taught scholar one of the most determined, intellectual and inspiring mathematicians to admire!