I was watching a Numberphile video 'Why $-\frac{1}{12}$ is a gold nugget', in which Professor Edward Frenkel was discussing assigning values to divergent series'. He mentioned that Euler, being the mathematical gangster he was, used to manipulate infinite series in attempt to assign a value to them. For example:
$$1+2+3+4+...=-\frac{1}{12},$$ $$1^2+2^2+3^2+4^2+...=0,$$ $$1^3+2^3+3^3+4^3+...=-\frac{1}{120}.$$
Question 1
He mentions that around 100 years later Riemann proved Euler's suggestions with the Zeta function. I was just wondering what Euler's original method was in assigning these values?
Question 2
He also later mentions there is a now, mathematically rigorous framework in which these little 'gold nuggets' can be accepted as the value of the divergent series, on the basis you disregard the 'rest of the infinity', I was wondering what this framework is? So i can have a little read into it.
Notice: in the video, they show the first equation in a book. But that book does not say "$=$" in there. It has some other symbol. Line $(3)$ [written with "$=$"] is, clearly speaking, false.
FALSE.
Here is the Riemann zeta function explanation. Consider three equations.
$$ \zeta(-1) = -\frac{1}{12} \tag{1} $$
$$ 1+2+3+4+\dots = \zeta(-1) \tag{2} $$
$$ 1+2+3+4+\dots = -\frac{1}{12} \tag{3} $$
Here $(1)$ is true, but $(2)$ and $(3)$ are false. Also, $(3)$ is a consequence of $(1)$ and $(2)$. Note that
$$ \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots = \zeta(s),\qquad \text{if }s > 1 \tag{2a} $$ and if we plug in $s=-1$ we get $(2)$, provided $-1 > 1$.
The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \ne 1$, so $(1)$ makes perfect sense. It is just that $(2)$ is false, so we may not deduce $(3)$ with an "$=$" sign in there.