Srinivasa Ramanujan

Srinivasa RamanujanThe mathematical genius of the 20th century

-The man who new infinity –

Refs: http://en.wikipedia.org/wiki/Srinivasa_Ramanujan

(22 December 1887 – 26 April 1920)

His home• Ramanujan's home

at SarangapaniStreet, Kumbakonam,Chennai

Deputy collector V. Ramaswamy Aiyer, seing Sri Ramanujan’s notebooks said

“I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.” (Ramanujan applied for a small job in revenue department, at that time)

Postal Stamp

• The Indian postage stamp issued in 1962 to commemorate the 75th anniversary of Ramanujan's birth.

The man ….

• Srīnivāsa Aiyangār Rāmānujam FRS, better known as Srinivasa Iyengar Ramanujan (Tamil:

or December 1887 – 26 April

1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions. Ramanujan's talent was said, by the prominent English mathematician G.H. Hardy, to be in the same league as legendary mathematicians such as Euler, Gauss, Newton and Archimedes

Age 10 -12

• Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney.[2] He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards.

Age 17

By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.

His work recognized …

• In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge

His works ….

• During his short lifetime, Ramanujanindependently compiled nearly 3900 results (mostly identities and equations).

• He stated results that were both original and highly unconventional, such as the Ramanujanprime and the Ramanujan theta function, and these have inspired a vast amount of further research

• The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.

Hardy–Ramanujan number 1729

• A common anecdote about Ramanujan relates to the number 1729. Hardy arrived at Ramanujan'sresidence in a cab numbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was actually a very interesting number mathematically, being the smallest natural number representable in two different ways as a sum of two cubes:

Ramanujan’s place in mathematics

• As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"

The problem posed by Ramanujam in a journal ..

• The solution given by him as no one answered (infinitely nested radicals)

On Bernoulli numbers …

• It will be observed that if n is even but not equal to zero,(i) Bn is a fraction and the numerator of in its lowest terms is a prime number,(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,(iii) is an integer and consequently is an odd integer.

Personality and spiritual life

• He often said, “An equation for me has no meaning, unless it represents a thought of God.”

• Hardy cites Ramanujan as remarking that all religions seemed equally true to him.

• Hardy remarked on Ramanujan's strict observance of vegetarianism.

Achievements

intriguing infinite series for π

In mathematics, the Ramanujan conjecture, named after Srinivasa

Ramanujan, states that the Fourier coefficients τ(n) of the cusp form Δ(z) of

weight 12, defined in modular form theory, satisfy

when p is a prime number.

In number theory, a branch of mathematics, Ramanujan's sum, usually

denoted cq(n), is a function of two positive integer variables q and n defined by

the formula

where (a, q) = 1 means that a only takes on values coprime to q.

and a lot more ....

Recognition

• Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day‘

• A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP)

• The Shanmugha Arts, Science, Technology, Research Academy (SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.

• An international feature film on Ramanujan's life was announced in 2006

• A Disappearing Number is a recent British production that explores the relationship between Hardy and Ramanujan.

r2s(n) (sums of squares)

r2s(n) is the number of way of representing n as the sum of 2s squares, counting

different orders and signs as different (e.g., r2(13) = 8, as 13 = (±2)2 + (±3)2 = (±3)2

+ (±2)2.)

Ramanujan defines a function δ2s(n) where he proved that r2s(n) = δ2s(n) for s = 1,

2, 3, and 4. For s > 4 he shows that δ2s(n) is a good approximation to r2s(n).

s = 1 has a special formula:

In the following formulas the signs repeat with a period of 4.If s ≡ 0 (mod 4),

r′2s(n) (sums of triangles)

r′2s(n) is the number of ways n can be represented as the sum of 2s triangular

numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...;

the nth triangular number is given by the formula n(n + 1)/2.)

The analysis here is similar to that for squares. Ramanujan refers to the same

paper as he did for the squares, where he showed that there is a function δ′2s(n)

such that r′2s(n) = δ′2s(n) for s = 1, 2, 3, and 4, and that for s > 4, δ′2s(n) is a good

approximation to r′2s(n).

Again, s = 1 requires a special formula:

If s is a multiple of 4,

http://en.wikipedia.org/wiki/Ramanujan’s_sum