RAMANUJAN FACTS AND THEORMS

Srinivasa Ramanujan was one of the world’s greatest mathematicians. His life story, with its humble and sometimes difficult beginnings, is as interesting in its own right as his astonishing work was.

1. When Ramanujan was thirteen, he could work out Loney’s Trigonometry exercises without any help!

By the age of 13, he had completed advanced trigonometry and discovered complex theorems on his own! 

2. As a young man, he failed to get a degree, as he did not clear his fine arts courses, although he always performed exceptionally well in mathematics. 

Who says failure should be the dead-end in a person's life! This man is an inspiration for many that temporary failure can't ever decide your future. Don't let minor disappointments come your way and keep following your passion. 

3. Because paper was expensive, poor Ramanujan often used to derive his results on a 'slate' to jot down results of his derivations.

After he died, people close to him found a treasure! This treasure was nothing materialistic but something which was even more precious! He left behind a 'notebook' with merely summaries and results in it, with little or no proofs - his personal notebook.

The first notebook had 351 pages with 16 organised chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, and his third notebook had 33 unorganized pages. The results in his notebooks inspired numerous papers by mathematicians! 

4. G.H. Hardy brought Ramanujam with him to England but unfortunately the English weather didn't suit him. He also reported of mild racism towards him. 

After getting inspired by his book Orders of Infinity, he wrote a letter to famous English mathematician G.H. Hardy (Who later became his mentor) in 1913. After a visit to India, G.H Hardy brought Ramanujam with him to England but unfortunately the English weather didn't suit him very well. Also, being a devout Brahman, this mathematical super-hero had a tough time adjusting with the culture and cuisine. 

5. 22nd December is called National Mathematics Day in India because of Ramanujan's birth-anniversary!

6. After a funny incident, 1729 is called Hardy-Ramanujam number in his honor, and such numbers are called Taxicab numbers.

After moving to England, Ramanujan had a lot of health disorders. A visit to hospital in a taxi resulted in one of the most celebrated anecdotes-

7. As a young man, he failed to get a degree, as he did not clear his fine arts courses, although he always performed exceptionally well in mathematics. 

Who says failure should be the dead-end in a person's life! This man is an inspiration for many that temporary failure can't ever decide your future. Don't let minor disappointments come your way and keep following your passion. 

                                                                                                                                                                                                                                                                                                 

THEORMS   BY   SRINIVAS  RAMANUJAN  

SIR 

1.Hardy–Ramanujan theorem

Jump to navigationJump to search

In mathematics, the Hardy–Ramanujan theorem, proved by G. H. Hardy and Srinivasa Ramanujan (1917), states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)).

Precise statement

A more precise version states that for any real-valued function ψ(n) that tends to infinity as n tends to infinity

${\displaystyle |\omega (n)-\log(\log(n))|<\psi (n){\sqrt {\log(\log(n))}}}$

or more traditionally

${\displaystyle |\omega (n)-\log(\log(n))|<{(\log(\log(n)))}^{{\frac {1}{2}}+\varepsilon }}$

for almost all (all but an infinitesimal proportion of) integers. That is, let g(x) be the number of positive integers n less than x for which the above inequality fails: then g(x)/x converges to zero as x goes to infinity.

History

A simple proof to the result Turán (1934) was given by Pál Turán, who used the Turán sieve to prove that

${\displaystyle \sum _{n\leq x}|\omega (n)-\log \log n|^{2}\ll x\log \log x\ .}$

Generalizations

The same results are true of Ω(n), the number of prime factors of n counted with multiplicity. This theorem is generalized by the Erdős–Kac theorem, which shows that ω(n) is essentially normally distributed.

2.Ramanujan's master theorem

Jump to navigationJump to search

In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan) is a technique that provides an analytic expression for the Mellin transform of an analytic function.

Page from Ramanujan's notebook stating his Master theorem.

The result is stated as follows:

If a complex-valued function ${\displaystyle f(x)}$ has an expansion of the form

${\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!}}(-x)^{k}}$

then the Mellin transform of ${\displaystyle f(x)}$ is given by

${\displaystyle \int _{0}^{\infty }x^{s-1}\,f(x)\,\operatorname {d} x=\Gamma (s)\,\varphi (-s)}$

where ${\displaystyle \Gamma (s)}$ is the gamma function.

It was widely used by Ramanujan to calculate definite integrals and infinite series.

Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).

A similar result was also obtained by Glaisher.

Alternative formalism

An alternative formulation of Ramanujan's master theorem is as follows:

${\displaystyle \int _{0}^{\infty }x^{s-1}\,\left(\,\lambda (0)-x\,\lambda (1)+x^{2}\,\lambda (2)-\,\cdots \,\right)\,\operatorname {d} x={\frac {\pi }{\,\sin(\pi s)\,}}\,\lambda (-s)}$

which gets converted to the above form after substituting ${\displaystyle \lambda (n)\equiv {\frac {\varphi (n)}{\,\Gamma (1+n)\,}}}$ and using the functional equation for the gamma function.

The integral above is convergent for ${\displaystyle 0<{\mathcal {Re}}(s)<1}$ subject to growth conditions on ${\displaystyle \varphi }$.

Proof

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the residue theorem and the well-known Mellin inversion theorem.

Application to Bernoulli polynomials

The generating function of the Bernoulli polynomials ${\displaystyle B_{k}(x)}$ is given by:

${\displaystyle {\frac {z\,e^{x\,z}}{\,e^{z}-1\,}}=\sum _{k=0}^{\infty }B_{k}(x)\,{\frac {z^{k}}{k!}}}$

These polynomials are given in terms of the Hurwitz zeta function:

${\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{\,(n+a)^{s}\,}}}$

by ${\displaystyle ~\zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}~}$ for ${\displaystyle ~n\geq 1~}$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:

${\displaystyle \int _{0}^{\infty }x^{s-1}\,\left({\frac {e^{-ax}}{\,1-e^{-x}\,}}-{\frac {1}{x}}\right)\,\operatorname {d} x=\Gamma (s)\,\zeta (s,a)\!}$

which is valid for ${\displaystyle ~0<{\mathcal {Re}}(s)<1~}$.

Application to the Gamma function

Weierstrass's definition of the Gamma function

${\displaystyle \Gamma (x)={\frac {\,e^{-\gamma \,x\,}}{x}}\,\prod _{n=1}^{\infty }\left(\,1+{\frac {x}{n}}\,\right)^{-1}e^{x/n}\!}$

is equivalent to expression

${\displaystyle \log \Gamma (1+x)=-\gamma \,x+\sum _{k=2}^{\infty }{\frac {\,\zeta (k)\,}{k}}\,(-x)^{k}}$

where ${\displaystyle \zeta (k)}$ is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

${\displaystyle \int _{0}^{\infty }x^{s-1}{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{2}}}\operatorname {d} x={\frac {\pi }{\sin(\pi s)}}{\frac {\zeta (2-s)}{2-s}}\!}$

valid for ${\displaystyle 0<{\mathcal {Re}}(s)<1}$.

Special cases of ${\displaystyle s={\frac {1}{2}}}$ and ${\displaystyle s={\frac {3}{4}}}$ are

${\displaystyle \int _{0}^{\infty }{\frac {\,\gamma x+\log \Gamma (1+x)\,}{x^{5/2}}}\,\operatorname {d} x={\frac {2\pi }{3}}\,\zeta \left({\frac {3}{2}}\right)}$

${\displaystyle \int _{0}^{\infty }{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{9/4}}}\,\operatorname {d} x={\sqrt {2}}{\frac {4\pi }{5}}\zeta \left({\frac {5}{4}}\right)}$

• 1+2+3+…=-1/12.
• Before saying anything, it should be informed that this identity is not true in the general sense. But this has a deep meaning which is why this is so famous. The constant on the right is extremely famous for being the analytical continuation of the Riemann zeta function at -1. Ramanujan devised an unique method to evaluate the Zeta function for negative integers. This technique is called the Ramanujan summation.
• Partition formula ( Hardy-Ramanujan-Rademacher asymptotic formula).
• This formula helped in establishing the fame of Ramanujan in the world of mathematics. He along with Hardy discovered and proved an asymptotic formula for the well known partition function which helps to approximate the function for large enough numbers. This function also has a lot of applications both within number theory and outside.
• Ramanujan congruences.
• Ramanujan independently discovered three congruences related to the partition function. These are perhaps the most beautiful results given by him. These congruences were later proved by Hardy and generalised by later mathematicians. One such congruence is p(5n+4)=0(mod 5).
• Ramanujan master theorem.
• This theorem is one of the most applied results of Ramanujan. This is used to evaluate the Mellin transforms of several complex valued functions. This has been generalised and now this is used in Quantum mechanics through Feynmann diagrams.
• Theta functions and their results.
• Ramanujan greatly studied a group of functions which he named as the theta functions. These functions have been found to have a large number of properties which are applicable in advanced physics and mathematics. For example, these functions are used in the working of nuclear reactors.
• Mock theta functions.
• Another class of functions discovered and studied by him. These are even more advanced and applicable than theta functions. Ramanujan described them in his last letter to Hardy. These functions are extremely useful in Quantum field theory and string theory. These functions are used for predicting the entropy of black holes.
• Rogers Ramanujan identities.
• Ramanujan discovered these identities while studying continued fractions. These identities were earlier discovered by Rogers. These identities also have beautiful properties and applications.
• Ramanujan's approximation for π.
• The most efficient approximation for π till date. Only the first term of this series can approximate π upto eight places after the decimal point which is a fair approximation for most practical calculations. Modern computers use this technique to approximate π upto trillions of decimal places.
• Ramanujan's results on highly composite numbers.
• Ramanujan got his BSc degree after writing his paper entitled “ Highly composite numbers.” He discovered these numbers and studied them and as a result discovered highly original results in the topic.
• Generalization of Bertrand's postulate.
• One of the three mathematicians who proved the Bertrand's postulate was Ramanujan. His short proof involved advanced properties of the gamma function and in this proof, he generalized the result which is the best generalization till date. Later on his methods of generalization were used by several mathematicians like Sondow who first defined and studied the Ramanujan primes.