Really, Really Large is not Infinity

I just happened to watch a refresher video on the famous Riemann Conjecture and his Zeta-function and ran into something which admittedly blew my mind. Now I have found the Riemann conjecture quite fascinating ever since I first heard about it, not because I'm a mathematician (I'm clearly not), but because

• it has been put forward by Bernhard Riemann already in 1859 and no-one has been able to prove it so far even though some of the greatest minds have tried very hard, and because
• a large number of other mathematical theorems depend on this hypothesis being true and thus would fall apart if only one counter-example were found, and also because
• it allows for statements regarding the distribution of prime numbers and thus connects a nice and orderly function like the Zeta-function with a seemingly unstructured and almost random area like the distribution of prime numbers - so it connects (as one might think) completely unrelated areas of mathematics, and lastly
• because it is probably the hardest approach to make $1 million by either proving or disproving the conjecture. Just send a counter-example to the Clay Institute and claim the money offered for solving one of the Millenium Problems. Good luck with that :-)

Now one of the examples touched on in this video mentioned more on the side (but which still caught my attention) was one special case of the Riemann Zeta-function for the argument -1 which yields this divergent infinite series

S = 1 + 2 + 3 + 4 + ...

with the very surprising result of the actual sum being -1/12. That's right: S = -1/12.

What in the world is going on?

Now to add up these numbers, one might think of the approach which is typically attributed to the young Carl Friedrich Gauss and write the sum for any N numbers in this series as

S = N * (N + 1) /2

It's not hard to see that this number gets larger and larger as N grows, so with N approaching infinity, the sum would also approach infinity, right? Yes, but for actual infinity: not so. It can be proven without doubt that this sum is really -1/12 and this is - at least I think - an impressive point to make on the difference between "really, really large" and "infinity". If you stop at any N, whatever it is, the result is different from -1/12. But if you continue until infinity (which obviously you can't), you would get -1/12.

I don't want to repeat the actual proof here as there is a great video which explains this really well. As a teaser, let me just say this: the proof is an interesting use of the sums of different infinite series put together, in this case the proof uses the infinite sums

S1 = 1 - 1 + 1 - 1 + 1 - ...

S2 = 1 - 2 + 3 - 4 + 5 - ...

the results of which can be combined in an intelligent way to come up with the surprising result for the infinite sum of the natural numbers.

Interestingly enough, this is not only something limited to pure mathematics, apparently there's actual applications of this in physics, notably in string theory, where one depends on being able to work with infinite sums.

So apologies for leaving a bit the perspectives of business and modern computer technology for a moment and digressing to mathematics, but at least for me the tangible difference between something "really, really large" and actual "infinity" was what caught my attention here. Maybe you like it too.

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