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The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913. Why is it true?
I don't know, let's see...
In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem. The problem was to sum the reciprocals of perfect squares:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... = ???
Euler showed that the answer was π²/6:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... = π²/6
He also showed you could rewrite this sum as a product over primes:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... =
(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...
That's actually the easy part: it's a cute trick called the Euler product formula.
So we know
(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ... = π²/6
If you think about it, Ramanujan's formula is saying that
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ...
is 2/5 as big. So, proving it is the same as showing
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... = π²/15
Maybe the next step is to use the same idea as the Euler product formula. I think this gives
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... =
1/1² - 1/2² - 1/3² + 1/4² + 1/5² - 1/6² + 1/7² + ...
where the signs at right follow a fancy pattern: we get 1/n² whenever n is the product of an even number of primes, and -1/n² when n is the product of an odd number of primes. For example, 4 = 2 x 2 is the product of an even number of primes, so we get 1/4².
So I'm left wanting to know why this strange sum
1/1² - 1/2² - 1/3² + 1/4² + 1/5² - 1/6² + 1/7² + ...
equals π²/15. Ramanujan, dead since 1920, is still messing with my mind!
The formula is supposed to be in here:
• Srinivasa Ramanujan, Modular equations and approximations to π, Quart. J. Pure. Appl. Math. 45 (1913-1914), 350-372. Also available at ://ramanujan.sirinudi.org/Volumes/published/ram06.pdf.
But I don't see it!
Here you can see how Euler solved the Basel problem:
It's a great example of his brilliant tactics, many of which were far from rigorous by today's standards... but can be made rigorous.
Fry travels across Europe to find out how Gutenberg kept his development work secret, about the role of avaricious investors and unscrupulous competitors and why Gutenberg's approach started a cultural revolution. He then sets about building a copy of Gutenberg's press.
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