# Why the Indian scientist Ramanujan Mathematitian is famous

## Srinivasa Ramanujan (1887-1920)

One of the classic questions of number theory is the problem of how many ways a natural number \ (n \) can be broken down into summands (the decomposition into a summand, i.e. the number itself, is also counted).

For example, the following applies to the number \ (p (n) \) of these so-called partitions: \ (p (3) = 3 \), since \ (3 = 2 + 1 = 1 + 1 + 1; \) \ (p (4 ) = 5 \), since \ (4 = 3 + 1 \) \ (= 2 + 2 = 2 + 1 + 1 \) \ (= 1 + 1 + 1 + 1 \) and \ (p (5) = 7 \), since \ (5 = 4 + 1 \) \ (= 3 + 2 = 3 + 1 + 1 \) \ (= 2 + 2 + 1 = 2 + 1 + 1 + 1 \) \ (= 1 + 1 + 1 + 1 + 1. \) The sequence \ (p (n) \) increases strongly; For example: \ (p (50) = 204226 \) and \ (p (100) = 190569292 \). Leonhard Euler succeeded in setting up a recursion formula for the calculation of \ (p (n) \). Ramanujan developed an approximation formula together with Hardy: \ (p (n) = \ frac {1} {4n \ sqrt {3}} e ^ {\ pi \ sqrt {2n / 3}} \)

Ramanujan is also interested in prime numbers \ ((p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 7, ...) \) and discovers "simple" relationships for certain infinite products:

\ [\ prod_ {k = 1} ^ \ infty \ frac {p_k ^ 2 + 1} {p_k ^ 2 - 1} = \ frac {5} {3} \ cdot \ frac {10} {8} \ cdot \ frac {26} {24} \ cdot ... = \ frac {5} {2} \]

and:

\ [\ prod_ {k = 1} ^ \ infty \ left (1+ \ frac {1} {p_k ^ 2} \ right) = \ left (1+ \ frac {1} {4} \ right) \ cdot \ left (1+ \ frac {1} {9} \ right) \ cdot \ left (1+ \ frac {1} {25} \ right) \ cdot ... = \ frac {15} {\ pi ^ 2} \]

Ramanujan's discoveries in number theory show that he has an "eye" for properties of numbers that are hidden from others. Two episodes may illustrate this: Hardy visits Ramanujan in the hospital and casually reports that the taxi he was using had the number 1729 - a number without any special properties, as he (Hardy) suspects. Ramanujan replies: 1729 is a very interesting number: 1729 is the smallest natural number that can be represented in two ways as the sum of cubic numbers: \ (1729 = 12 ^ 3 + 1 ^ 3 = 10 ^ 3 + 9 ^ 3 \) . Another time, Ramanujan is said to have solved the following puzzle without hesitation: The numbered houses of a street village are all on one side. Someone lives in a house with a house number for which the sum of the house numbers in front of and behind this house is the same. How many houses does the village have? Which house number is this? (Ramanujan solves the problem using a continued fraction ...)

Ramanujan, who grew up in the tropics, can hardly bear the climate in cold, rainy England; in the winter months he falls ill regularly and is hardly able to work scientifically; He also has great difficulty in eating an adequate vegetarian diet in accordance with the strict religious rules. It is not possible to return home during the war; in desperation, he tries to throw himself in front of an underground train in London. In 1917 Ramanujan fell seriously ill; fatalistically, he sees this as the destiny destined for him. Only when he received special honors did his will to live grow again and he began to work scientifically again: in 1918 he became a member of the *Cambridge Philosophical Society* as well as the *Trinity College* appointed, a few weeks later a member of the *Royal Society of London*.

When the war is over, Ramanujan travels back to his homeland, but dies six months later, presumably of tuberculosis. A dysentery epidemic, which was rampant in Madras at this time, may also play a role.

Ramanujan's records were long thought to be lost; they were not rediscovered until 1976; they contained about 600 formulas with no evidence; for some of them no evidence has yet been found.

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