Ramanujan found a proof of Chebyshev's theorem that there is always a
prime between a number and its double.
For example, gien any number, say 2, we double it (4). We can always
find a prime number between those two numbers (3 in this case).
Others were taken with the beauty of Ramanujan's approach.
Since Ramanujan did not have standard mathematical training, he did not
know that this had already been proven by Chebyshev.
Ramanujan's approach to the proof was very different than Chebyshev's.
The Prime Number Theorem gives an estimate of the
number of primes up to an integer n. As Gauss discovered, the estimate
gets better as n gets larger.
Integers below n | Actual number of primes | Gauss' estimated number of primes | Percentage of error |
1,000 | 168 | 145 | 16.0% |
1,000,000 | 78,498 | 72,382 | 8.4% |
1,000,000,000 | 50,847,478 | 48,254,942 | 5.4% |
Ramanujan thought he had improved on Gauss' estimates, but he had made a mistake. But he did come up with formulas for all sorts of other things related to primes that were correct and that Gauss didn't even consider.