When I first watched numberphile's 1+2+3+... = −112 I thought the sum actually equalled −112 without really understanding it.
Recently I read some wolframalpha pages and watched some videos and now I understand (I think), that −112 is just an associative value to the sum of all natural numbers when you analytically continue the riemann-zeta function. 3Blue1Brown's video really helped. What I don't really understand is why it gives the value −112 specifically. The value −112 seems arbitrary to me and I don't see any connection to the sum of all natural numbers. Is there any intuition behind why you get −112 when analytically continue the zeta function at ζ(−1)?
EDIT(just to make my question a little clearer):
I'll use an example here. Suppose you somehow didn't know about radians and never associated trig functions like sine to π but you knew about maclaurin expansion. By plugging in x=π to the series expansion of sine, you would get sine(π) = 0. You might have understood the process in which you get the value 0, the maclaurin expansion, but you wouldn't really know the intuition behind this connection between π and trig functions, namely the unit circle, which is essential in almost every branch of number theory.
Back to this question, I understand the analytic continuation of the zeta function and its continued form for s<0 ζ(s)=2sπs−1sinπs2Γ(1−s)ζ(1−s) and how when you plug in s = -1, things simplify down to −112 but I don't see any connection between the fraction and the infinite sum. I'm sure there is a beautiful connection between them, like the one between trig functions and π, but couldn't find any useful resources on the internet. Hope this clarified things.
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