1+2+3+4+… = -1/12 – the most natural explanation

Let’s treat the numbers in a specific way – as a set of unities arranged in the most natural way on the Numbers Plane.

A detailed explanation of the idea of the Numbers Plane (the most natural representation of the numbers) can be found here, and in the video linked there.

The red unities are relatively prime and the blue ones are the not relatively prime.

We know already that the resultant of all the coordinates of the relatively prime unities within given number n always equals to n*µ(n) (µ(n) is the value of the Möbius function – more about it here)

So for n=6 we are getting the value 6 (6*1), for 5 the value -5 (5*-1), and so on.

Looking at the well-known formula binding Möbius and Riemann Zeta functions

For s = -1 we’ve got at the left side the sum of mentioned above resultants of all the numbers n up to infinity and at the right side the reciprocal of the Riemann Zeta function (which for s=-1 giving a sum 1+2+3+... is equal to -1/12)

This means that we could expect, that the total resultant of coordinates of all the relatively primes (up to infinity) on the Numbers Plane is equal to -12.

And this is why we can sometimes replace the infinite sum of natural numbers (1+2+3….) with the value of -1/12, but only when we are looking at the numbers in such a specific, but also the most natural way - treating them as a kind of the field.