Straight away we can see a similarity to Ramanujan’s method in Euler’s except he has reached his equivalent of C 2 via stronger mathematical logic. Reaching this point I feel a quick pause is in order while we quickly go over the Riemann-Zeta function. This is a function that was first introduced by Euler and then generalised by Bernhard Riemann. It behaves as follows:
∞
3 ⁄ … = 1 − + 2 − + 3 − …
⁄
() = ∑ 1
= 1
1 ⁄ + 1
2 ⁄ + 1
=1
In practical terms, the Riemann-Zeta function is the infinite sum of 1, divided by integers beginning at 1, which have been placed to the power of the argument, s . Returning to the problem at hand, Euler’s proof continues:
(2 − )() = 2 − + 4 − + 6 − …
(1 − 2(2 − ))() = (1 + 2 − + 3 − … ) − 2(2 − + 4 − + 6 − … ) = 1 − 2 − + 3 − …
Set = −1
1 − 2(2 −−1 ) = −3
(−1) = 1 −−1 + 2 −−1 + 3 −−1 … = 1 + 2 + 3 …
1 − 2 −−1 + 3 −−1 − 4 −−1 … = 1 − 2 + 3 − 4 …
∴ −3(1 + 2 + 3 + 4 … ) = 1 − 2 + 3 − 4 … = 1 4⁄
1 + 2 + 3 + 4 + 5 … = −1
12⁄
It is notable that along the path of Euler’s proof there were close parallel at points to Ramanujan’s proof despite the fact that they were separated by around 250 years and Ramanujan worked, for much of his career, in isolation from the mathematical world, having taught himself and developed his own system of maths. Other than for the pleasure in maths, however, the sceptics are asking: “But what use is it?” and leaning back with a smug look smeared across their faces confident that a series reliant on infinity has no application in the real world. And these neigh-sayers would be correct if it weren’t for the great mathematical receptacle of quantum field theory that applies the -1/12 result in computing the Casimir effect. This result also has applications in bosonic string theory as it allows calculation of the number of physical dimensions and
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